Date post: | 12-Apr-2015 |
Category: |
Documents |
Upload: | goodfellas121 |
View: | 463 times |
Download: | 38 times |
1
Chapter 1 Systems of Measurement Conceptual Problems
*1 • Determine the Concept The fundamental physical quantities in the SI system include mass, length, and time. Force, being the product of mass and acceleration, is not a fundamental quantity. correct. is )(c
2 • Picture the Problem We can express and simplify the ratio of m/s to m/s2 to determine the final units.
Express and simplify the ratio of m/s to m/s2: s
smsm
smsm
2
2
=⋅⋅
= and correct. is )(d
3 • Determine the Concept Consulting Table 1-1 we note that the prefix giga means 109. correct. is )(c
4 • Determine the Concept Consulting Table 1-1 we note that the prefix mega means 106. correct. is )(d
*5 • Determine the Concept Consulting Table 1-1 we note that the prefix pico means 10−12. correct. is )(a
6 • Determine the Concept Counting from left to right and ignoring zeros to the left of the first nonzero digit, the last significant figure is the first digit that is in doubt. Applying this criterion, the three zeros after the decimal point are not significant figures, but the last zero is significant. Hence, there are four significant figures in this number.
correct. is )(c
Chapter 1
2
7 • Determine the Concept Counting from left to right, the last significant figure is the first digit that is in doubt. Applying this criterion, there are six significant figures in this number. correct. is )(e
8 • Determine the Concept The advantage is that the length measure is always with you. The disadvantage is that arm lengths are not uniform; if you wish to purchase a board of ″two arm lengths″ it may be longer or shorter than you wish, or else you may have to physically go to the lumberyard to use your own arm as a measure of length.
9 • (a) True. You cannot add ″apples to oranges″ or a length (distance traveled) to a volume (liters of milk). (b) False. The distance traveled is the product of speed (length/time) multiplied by the time of travel (time). (c) True. Multiplying by any conversion factor is equivalent to multiplying by 1. Doing so does not change the value of a quantity; it changes its units.
Estimation and Approximation *10 •• Picture the Problem Because θ is small, we can approximate it by θ ≈ D/rm provided that it is in radian measure. We can solve this relationship for the diameter of the moon.
Express the moon’s diameter D in terms of the angle it subtends at the earth θ and the earth-moon distance rm:
mrD θ=
Find θ in radians: rad00915.0360
rad2524.0 =°
×°=πθ
Substitute and evaluate D: ( )( )
m1051.3
Mm384rad0.009156×=
=D
Systems of Measurement
3
*11 •• Picture the Problem We’ll assume that the sun is made up entirely of hydrogen. Then we can relate the mass of the sun to the number of hydrogen atoms and the mass of each. Express the mass of the sun MS as the product of the number of hydrogen atoms NH and the mass of each atom MH:
HHS MNM =
Solve for NH:
H
SH M
MN =
Substitute numerical values and evaluate NH:
5727
30
H 1019.1kg101.67kg101.99
×=××
= −N
12 •• Picture the Problem Let P represent the population of the United States, r the rate of consumption and N the number of aluminum cans used annually. The population of the United States is roughly 3×108 people. Let’s assume that, on average, each person drinks one can of soft drink every day. The mass of a soft-drink can is approximately 1.8 ×10−2 kg. (a) Express the number of cans N used annually in terms of the daily rate of consumption of soft drinks r and the population P:
trPN ∆=
Substitute numerical values and approximate N:
( )
( )
cans10
yd24.365y1
people103dperson
can1
11
8
≈
⎟⎟⎠
⎞⎜⎜⎝
⎛×
×⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
=N
(b) Express the total mass of aluminum used per year for soft drink cans M as a function of the number of cans consumed and the mass m per can:
NmM =
Chapter 1
4
Substitute numerical values and evaluate M:
( )( )kg/y102
kg/can108.1cans/y109
211
×≈
×= −M
(c) Express the value of the aluminum as the product of M and the value at recycling centers:
( )( )( )
dollars/ybillion 2
y/102$kg/y102kg/1$
kg/1$ Value
9
9
=
×=
×=
= M
13 •• Picture the Problem We can estimate the number of words in Encyclopedia Britannica by counting the number of volumes, estimating the average number of pages per volume, estimating the number of words per page, and finding the product of these measurements and estimates. Doing so in Encyclopedia Britannica leads to an estimate of approximately 200 million for the number of words. If we assume an average word length of five letters, then our estimate of the number of letters in Encyclopedia Britannica becomes 109. (a) Relate the area available for one letter s2 and the number of letters N to be written on the pinhead to the area of the pinhead:
22
4dNs π
= where d is the diameter of the
pinhead.
Solve for s to obtain:
Nds
4
2π=
Substitute numerical values and evaluate s:
( )
( ) m10104
incm54.2in
89
2
161
−≈⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
=π
s
(b) Express the number of atoms per letter n in terms of s and the atomic spacing in a metal datomic:
atomicdsn =
Substitute numerical values and evaluate n:
atoms20atoms/m105m10
10
8
≈×
= −
−
n
*14 •• Picture the Problem The population of the United States is roughly 3 × 108 people. Assuming that the average family has four people, with an average of two cars per
Systems of Measurement
5
family, there are about 1.5 × 108 cars in the United States. If we double that number to include trucks, cabs, etc., we have 3 × 108 vehicles. Let’s assume that each vehicle uses, on average, about 12 gallons of gasoline per week. (a) Find the daily consumption of gasoline G:
( )( )gal/d106
gal/d2vehicles1038
8
×=
×=G
Assuming a price per gallon P = $1.50, find the daily cost C of gasoline:
( )( )dollars/dbillion $1d/109$
gal/50.1$gal/d1068
8
≈×=
×== GPC
(b) Relate the number of barrels N of crude oil required annually to the yearly consumption of gasoline Y and the number of gallons of gasoline n that can be made from one barrel of crude oil:
ntG
nYN ∆==
Substitute numerical values and estimate N:
( ) ( )
barrels/y10
gal/barrel 19.4d/y24.365gal/d106
10
8
≈
×=N
15 •• Picture the Problem We’ll assume a population of 300 million (fairly accurate as of September, 2002) and a life expectancy of 76 y. We’ll also assume that a diaper has a volume of about half a liter. In (c) we’ll assume the disposal site is a rectangular hole in the ground and use the formula for the volume of such an opening to estimate the surface area required. (a) Express the total number N of disposable diapers used in the United States per year in terms of the number of children n in diapers and the number of diapers D used by each child in 2.5 y:
nDN =
Use the daily consumption, the number of days in a year, and the estimated length of time a child is in diapers to estimate the number of diapers D required per child:
ilddiapers/ch103
y5.2y
d24.365d
diapers3
3×≈
××=D
Chapter 1
6
Use the assumed life expectancy to estimate the number of children n in diapers:
( )
children10
children10300y76y5.2
7
6
≈
×⎟⎟⎠
⎞⎜⎜⎝
⎛=n
Substitute to obtain:
( )( )
diapers103
ilddiapers/ch103children10
10
3
7
×≈
××
=N
(b) Express the required landfill volume V in terms of the volume of diapers to be buried:
diaper oneNVV =
Substitute numerical values and evaluate V:
( )( )37
10
m105.1
L/diaper5.0diapers103
×≈
×=V
(c) Express the required volume in terms of the volume of a rectangular parallelepiped:
AhV =
Solve and evaluate h:
2637
m105.1m10
m105.1×=
×==
hVA
Use a conversion factor to express this area in square miles:
2
2
226
mi6.0
km2.590mi1m105.1
≈
××=A
16 ••• Picture the Problem The number of bits that can be stored on the disk can be found from the product of the capacity of the disk and the number of bits per byte. In part (b) we’ll need to estimate (i) the number of bits required for the alphabet, (ii) the average number of letters per word, (iii) an average number of words per line, (iv) an average number of lines per page, and (v) a book length in pages.
(a) Express the number of bits Nbits as a function of the number of bits per byte and the capacity of the hard disk Nbytes:
( )( )( )
bits1060.1
bits/byte8bytes102
bits/byte8
10
9
bytesbits
×=
×=
= NN
Systems of Measurement
7
(b) Assume an average of 8 letters/word and 8 bits/character to estimate the number of bytes required per word:
wordbytes8
wordbits64
wordcharacters8
characterbits8
=
=×
Assume 10 words/line and 60 lines/page:
pagebytes4800
wordbytes8
pagewords600 =×
Assume a book length of 300 pages and approximate the number bytes required:
bytes1044.1pagebytes4800pages300 6×=×
Divide the number of bytes per disk by our estimated number of bytes required per book to obtain an estimate of the number of books the 2-gigabyte hard disk can hold:
books1400
bytes/book1044.1bytes102
6
9
books
≈
××
=N
*17 •• Picture the Problem Assume that, on average, four cars go through each toll station per minute. Let R represent the yearly revenue from the tolls. We can estimate the yearly revenue from the number of lanes N, the number of cars per minute n, and the $6 toll per car C.
M177$car
6$yd24.365
dh24
hmin60
mincars 4lanes14 =×××××== NnCR
Units 18 • Picture the Problem We can use the metric prefixes listed in Table 1-1 and the abbreviations on page EP-1 to express each of these quantities.
(a)
MW1
watts10watts000,000,1 6
=
=
(c) m3meter103 6 µ=× −
(b) mg2g102gram002.0 3 =×= −
(d) ks30s1003seconds000,30 3 =×=
Chapter 1
8
19 • Picture the Problem We can use the definitions of the metric prefixes listed in Table 1-1 to express each of these quantities without prefixes.
(a)
W0.000040W1040W40 6 =×= −µ (c)
W000,000,3W103MW3 6 =×=
(b) s40.00000000s104ns4 9 =×= −
(d) m000,52m1025km25 3 =×=
*20 • Picture the Problem We can use the definitions of the metric prefixes listed in Table 1-1 to express each of these quantities without abbreviations.
(a) picoboo1boo10 12 =− (e) megaphone1phone106 =
(b) gigalow1low109 = (f) nanogoat1goat10 9 =−
(c) microphone1phone10 6 =− (g) terabull1bull1012 =
(d) attoboy1boy10 18 =−
21 •• Picture the Problem We can determine the SI units of each term on the right-hand side of the equations from the units of the physical quantity on the left-hand side.
(a) Because x is in meters, C1 and C2t must be in meters:
m/sin is m;in is 21 CC
(b) Because x is in meters, ½C1t2 must be in meters:
21 m/sin is C
(c) Because v2 is in m2/s2, 2C1x must be in m2/s2:
21 m/sin is C
(d) The argument of trigonometric function must be dimensionless; i.e. without units. Therefore, because x
121 sin is m;in is −CC
Systems of Measurement
9
is in meters: (e) The argument of an exponential function must be dimensionless; i.e. without units. Therefore, because v is in m/s:
121 sin is m/s;in is −CC
22 •• Picture the Problem We can determine the US customary units of each term on the right-hand side of the equations from the units of the physical quantity on the left-hand side.
(a) Because x is in feet, C1 and C2t must be in feet:
ft/sin is ft;in is 21 CC
(b) Because x is in feet, ½C1t2 must be in feet:
21 ft/sin is C
(c) Because v2 is in ft2/s2, 2C1x must be in ft2/s2:
21 ft/sin is C
(d) The argument of trigonometric function must be dimensionless; i.e. without units. Therefore, because x is in feet:
121 sin is ft;in is −CC
(e) The argument of an exponential function must be dimensionless; i.e. without units. Therefore, because v is in ft/s:
121 sin is ft/s;in is −CC
Conversion of Units 23 • Picture the Problem We can use the formula for the circumference of a circle to find the radius of the earth and the conversion factor 1 mi = 1.61 km to convert distances in meters into distances in miles.
(a) The Pole-Equator distance is one-fourth of the circumference:
m104 7×=c
Chapter 1
10
(b) Use the formula for the circumference of a circle to obtain:
m1037.62
m1042
67
×=×
==−
ππcR
(c) Use the conversion factors 1 km = 1000 m and 1 mi = 1.61 km:
mi102.48
km1.61mi1
m10km 1m104
4
37
×=
×××=c
and
mi1096.3
km1.61mi1
m10km1m1037.6
3
36
×=
×××=R
24 • Picture the Problem We can use the conversion factor 1 mi = 1.61 km to convert speeds in km/h into mi/h.
Find the speed of the plane in km/s: ( )
km/h2450
hs3600
m10km1
sm680
m/s680m/s3402
3
=
⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
==v
Convert v into mi/h:
mi/h1520
km1.61mi1
hkm2450
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=v
*25 • Picture the Problem We’ll first express his height in inches and then use the conversion factor 1 in = 2.54 cm.
Express the player’s height into inches:
in82.5in10.5ftin12ft6 =+×=h
Convert h into cm: cm210
incm2.54in2.58 =×=h
26 • Picture the Problem We can use the conversion factors 1 mi = 1.61 km, 1 in = 2.54 cm, and 1 m = 1.094 yd to complete these conversions.
Systems of Measurement
11
(a) h
mi62.1km1.61
mi1h
km100h
km100 =×=
(b) in23.6
cm2.54in1cm60cm60 =×=
(c) m91.4
yd1.094m1yd100yd100 =×=
27 • Picture the Problem We can use the conversion factor 1.609 km = 5280 ft to convert the length of the main span of the Golden Gate Bridge into kilometers. Convert 4200 ft into km: km1.28
ft5280km1.609ft4200ft4200 =×=
*28 • Picture the Problem Let v be the speed of an object in mi/h. We can use the conversion factor 1 mi = 1.61 km to convert this speed to km/h.
Multiply v mi/h by 1.61 km/mi to convert v to km/h:
km/h61.1mi
km1.61h
mih
mi vvv =×=
29 • Picture the Problem Use the conversion factors 1 h = 3600 s, 1.609 km = 1 mi, and 1 mi = 5280 ft to make these conversions.
(a) sh
km36.0s3600
h1hkm10296.1
hkm10296.1 2
52
5
⋅=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ ×=×
(b) 2
32
25
25
sm10.0
kmm10
s3600h1
hkm10296.1
hkm10296.1 =⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ ×=×
(c) sft0.88
s3600h1
mi1ft5280
hmi60
hmi60 =⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
(d) sm8.26
s3600h1
kmm10
mi1km1.609
hmi60
hmi60
3
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
Chapter 1
12
30 • Picture the Problem We can use the conversion factor 1 L = 1.057 qt to convert gallons into liters and then use this gallons-to-liters conversion factor to convert barrels into cubic meters.
(a) ( ) L784.3qt057.1
L1gal
qt4gal1gal1 =⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
(b) ( ) 333
m1589.0L
m10gal
L3.784barrel
gal42barrel1barrel1 =⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
−
31 • Picture the Problem We can use the conversion factor given in the problem statement and the fact that 1 mi = 1.609 km to express the number of square meters in one acre.
Multiply by 1 twice, properly chosen, to convert one acre into square miles, and then into square meters:
( )
2
22
m4050
mim1609
acres640mi1acre1acre1
=
⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
32 •• Picture the Problem The volume of a right circular cylinder is the area of its base multiplied by its height. Let d represent the diameter and h the height of the right circular cylinder; use conversion factors to express the volume V in the given units.
(a) Express the volume of the cylinder: hdV 2
41 π=
Substitute numerical values and evaluate V:
( ) ( )
( ) ( )
3
22
41
241
ft504.0
in12ft1ft2in6.8
ft2in6.8
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=
π
πV
(b) Use the fact that 1 m = 3.281 ft to convert the volume in cubic feet into cubic meters:
( )3
33
m0.0143
ft3.281m1ft0.504
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=V
(c) Because 1 L = 10−3 m3: ( ) L14.3
m10L1
0.0143m 333 =⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−V
Systems of Measurement
13
*33 •• Picture the Problem We can treat the SI units as though they are algebraic quantities to simplify each of these combinations of physical quantities and constants.
(a) Express and simplify the units of v2/x:
( )22
22
sm
smm
msm
=⋅
=
(b) Express and simplify the units of ax :
ssm/s
m 22 ==
(c) Noting that the constant factor 21 has no units, express and simplify
the units of 221 at :
( ) ( ) mssms
sm 2
22
2 =⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
Dimensions of Physical Quantities 34 • Picture the Problem We can use the facts that each term in an equation must have the same dimensions and that the arguments of a trigonometric or exponential function must be dimensionless to determine the dimensions of the constants.
(a) x = C1 + C2 t
TTLLL
(d) x = C1 cos C2 t
TT
LL 1
(b) 2
121 tCx =
22 T
TLL
(e) v = C1 exp( −C2 t)
TTT
LTL 1
(c) xCv 1
2 2=
LTL
TL
22
2
35 •• Picture the Problem Because the exponent of the exponential function must be dimensionlthe dimension of λ must be .1−T
Chapter 1
14
*36 •• Picture the Problem We can solve Newton’s law of gravitation for G and substitute the dimensions of the variables. Treating them as algebraic quantities will allow us to express the dimensions in their simplest form. Finally, we can substitute the SI units for the dimensions to find the units of G.
Solve Newton’s law of gravitation for G to obtain:
21
2
mmFrG =
Substitute the dimensions of the variables:
2
3
2
22
MTL
M
LTML
G =×
=
Use the SI units for L, M, and T:
2
3
skgm are of Units⋅
G
37 •• Picture the Problem Let m represent the mass of the object, v its speed, and r the radius of the circle in which it moves. We can express the force as the product of m, v, and r (each raised to a power) and then use the dimensions of force F, mass m, speed v, and radius r to obtain three equations in the assumed powers. Solving these equations simultaneously will give us the dependence of F on m, v, and r. Express the force in terms of powers of the variables:
cba rvmF =
Substitute the dimensions of the physical quantities:
cb
a LTLMMLT ⎟⎠⎞
⎜⎝⎛=−2
Simplify to obtain:
bcba TLMMLT −+− =2
Equate the exponents to obtain: a = 1, b + c = 1, and −b = −2
Solve this system of equations to obtain:
a = 1, b = 2, and c = −1
Substitute in equation (1): r
vmrmvF2
12 == −
Systems of Measurement
15
38 •• Picture the Problem We note from Table 1-2 that the dimensions of power are ML2/T3. The dimensions of mass, acceleration, and speed are M, L/T2, and L/T respectively.
Express the dimensions of mav: [ ] 3
2
2 TML
TL
TLMmav =××=
From Table 1-2: [ ] 3
2
TMLP =
power. of dimensions thehas speed andon,accelerati mass, ofproduct that thesee weresults, theseComparing
39 •• Picture the Problem The dimensions of mass and velocity are M and L/T, respectively. We note from Table 1-2 that the dimensions of force are ML/T2.
Express the dimensions of momentum: [ ]
TML
TLMmv =×=
From Table 1-2: [ ] 2T
MLF =
Express the dimensions of force multiplied by time:
[ ]T
MLTTMLFt =×= 2
by time. multiplied force of dimensions thehas momentum that see weresults, theseComparing
40 •• Picture the Problem Let X represent the physical quantity of interest. Then we can express the dimensional relationship between F, X, and P and solve this relationship for the dimensions of X.
Express the relationship of X to force and power dimensionally:
[ ][ ] [ ]PXF =
Solve for [ ]X : [ ] [ ][ ]FPX =
Chapter 1
16
Substitute the dimensions of force and power and simplify to obtain: [ ]
TL
TMLT
ML
X ==
2
3
2
Because the dimensions of velocity are L/T, we can conclude that:
[ ] [ ][ ]vFP =
Remarks: While it is true that P = Fv, dimensional analysis does not reveal the presence of dimensionless constants. For example, if πFvP = , the analysis shown above would fail to establish the factor of π. *41 •• Picture the Problem We can find the dimensions of C by solving the drag force equation for C and substituting the dimensions of force, area, and velocity. Solve the drag force equation for the constant C: 2
air
AvFC =
Express this equation dimensionally:
[ ] [ ][ ][ ]2
air
vAFC =
Substitute the dimensions of force, area, and velocity and simplify to obtain:
[ ] 322
2
LM
TLL
TML
C =
⎟⎠⎞
⎜⎝⎛
=
42 •• Picture the Problem We can express the period of a planet as the product of these factors (each raised to a power) and then perform dimensional analysis to determine the values of the exponents.
Express the period T of a planet as the product of cba MGr Sand,, :
cba MGCrT S= (1)
where C is a dimensionless constant.
Solve the law of gravitation for the constant G:
21
2
mmFrG =
Express this equation dimensionally: [ ] [ ][ ]
[ ][ ]21
2
mmrFG =
Systems of Measurement
17
Substitute the dimensions of F, r, and m: [ ]
( )2
32
2
MTL
MM
LTML
G =×
×=
Noting that the dimension of time is represented by the same letter as is the period of a planet, substitute the dimensions in equation (1) to obtain:
( ) ( )cb
a MMT
LLT ⎟⎟⎠
⎞⎜⎜⎝
⎛= 2
3
Introduce the product of M 0 and L0 in the left hand side of the equation and simplify to obtain:
bbabc TLMTLM 23100 −+−=
Equate the exponents on the two sides of the equation to obtain:
0 = c – b, 0 = a + 3b, and 1 = –2b
Solve these equations simultaneously to obtain:
21
21
23 and,, −=−== cba
Substitute in equation (1): 23
S
21S
2123 rGMCMGCrT == −−
Scientific Notation and Significant Figures *43 • Picture the Problem We can use the rules governing scientific notation to express each of these numbers as a decimal number.
(a) 000,30103 4 =× (c) 000004.0104 6 =× −
(b) 0062.0102.6 3 =× − (d) 000,2171017.2 5 =×
44 • Picture the Problem We can use the rules governing scientific notation to express each of these measurements in scientific notation.
(a) W103.1GW3.1 9×= (c) s102.3fs3.2 15−×=
Chapter 1
18
(b) m10m1010pm10 1112 −− =×= (d) s104s4 6−×=µ
45 • Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions.
(a) The number of significant figures in each factor is three; therefore the result has three significant figures:
( )( ) 54 1014.11099.914.1 ×=×
(b) Express both terms with the same power of 10. Because the first measurement has only two digits after the decimal point, the result can have only two digits after the decimal point:
( ) ( )( )
8
8
98
1025.2
10531.078.21031.51078.2
−
−
−−
×=
×−=
×−×
(c) We’ll assume that 12 is exact. Hence, the answer will have three significant figures:
33 1027.8
1056.412
×=× −
π
(d) Proceed as in (b): ( )
2
2
1027.6
6275996.271099.56.27
×=
=+=×+
46 • Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions.
(a) Note that both factors have four significant figures.
( )( ) 510144.13.5699.200 ×=
(b) Express the first factor in scientific notation and note that both factors have three significant figures.
( )( )( )( )
2
77
7
1020.3
103.621013.5103.62000000513.0
×=
××=
×−
Systems of Measurement
19
(c) Express both terms in scientific notation and note that the second has only three significant figures. Hence the result will have only three significant figures.
( )( ) ( )( )
4
4
44
4
1062.8
1078.5841.21078.510841.2
1078.528401
×=
×+=
×+×=
×+
(d) Because the divisor has three significant figures, the result will have three significant figures.
43 1052.1
1017.425.63
×=× −
*47 • Picture the Problem Let N represent the required number of membranes and express N in terms of the thickness of each cell membrane.
Express N in terms of the thickness of a single membrane:
nm7in1
=N
Convert the units into SI units and simplify to obtain:
6
9
104
m10nm1
cm100m1
incm2.54
nm7in1
×=
×××= −N
48 • Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions.
(a) Both factors and the result have three significant figures:
( )( ) 324 1022.11010.61000.2 ×=×× −
(b) Because the second factor has three significant figures, the result will have three significant figures:
( )( ) 65 1026.11000.4141592.3 ×=×
(c) Both factors and the result have three significant figures:
58
3
1000.21016.11032.2 −×=
××
(d) Write both terms using the same power of 10. Note that the result will have only three significant figures:
( ) ( )( ) ( )( )
3
3
33
23
1042.5
10278.014.510278.01014.5
1078.21014.5
×=
×+=
×+×=
×+×
Chapter 1
20
(e) Follow the same procedure used in (d):
( ) ( )( ) ( )
2
22
52
1099.1
10000000999.01099.11099.91099.1
×=
×+×=
×+× −
*49 • Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions. (a) The second factor and the result have three significant figures:
( ) 32 1069.12.23141592654.3 ×=×
(b) We’ll assume that 2 is exact. Therefore, the result will have two significant figures:
8.476.0141592654.32 =××
(c) We’ll assume that 4/3 is exact. Therefore the result will have two significant figures:
( ) 6.51.134 3 =×π
(d) Because 2.0 has two significant figures, the result has two significant figures:
( ) 10141592654.3
0.2 5
=
General Problems 50 • Picture the Problem We can use the conversion factor 1 mi = 1.61 km to convert 100 km/h into mi/h.
Multiply 100 km/h by 1 mi/1.61 km to obtain:
mi/h1.62
km1.61mi1
hkm100
hkm100
=
×=
*51 • Picture the Problem We can use a series of conversion factors to convert 1 billion seconds into years.
Multiply 1 billion seconds by the appropriate conversion factors to convert into years:
Systems of Measurement
21
y31.7days365.24
y1h24
day1s3600
h1s10s10 99 =×××=
52 • Picture the Problem In both the examples cited we can equate expressions for the physical quantities, expressed in different units, and then divide both sides of the equation by one of the expressions to obtain the desired conversion factor.
(a) Divide both sides of the equation expressing the speed of light in the two systems of measurement by 186,000 mi/s to obtain: km/mi61.1
m10km1
mim1061.1
m/mi1061.1mi/h101.86
m/s1031
33
35
8
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ ×=
×=××
=
(b) Find the volume of 1.00 kg of water:
Volume of 1.00 kg = 103 g is 103 cm3
Express 103 cm3 in ft3: ( )
3
333
ft0.0353
in12ft1
cm2.54in1cm10
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
Relate the weight of 1 ft3 of water to the volume occupied by 1 kg of water:
33 ftlb62.4
ft0.0353kg1.00
=
Divide both sides of the equation by the left-hand side to obtain: lb/kg20.2
ft0.0353kg1.00ftlb62.4
13
3==
53 •• Picture the Problem We can use the given information to equate the ratios of the number of uranium atoms in 8 g of pure uranium and of 1 atom to its mass.
Express the proportion relating the number of uranium atoms NU in 8 g of pure uranium to the mass of 1 atom:
kg104.0atom1
g8 26U
−×=
N
Chapter 1
22
Solve for and evaluate NU: ( )
23
26U
100.2
kg104.0atom1g8
×=
⎟⎟⎠
⎞⎜⎜⎝
⎛×
= −N
54 •• Picture the Problem We can relate the weight of the water to its weight per unit volume and the volume it occupies.
Express the weight w of water falling on the acre in terms of the weight of one cubic foot of water, the depth d of the water, and the area A over which the rain falls:
Adw ⎟⎠⎞
⎜⎝⎛= 3ft
lb4.62
Find the area A in ft2: ( )
24
22
ft104.356
mift5280
acre640mi1acre1
×=
⎟⎠⎞
⎜⎝⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=A
Substitute numerical values and evaluate w:
( )( ) lb1017.3in12ft1in1.4ft104.356
ftlb4.62 524
3 ×=⎟⎟⎠
⎞⎜⎜⎝
⎛×⎟
⎠⎞
⎜⎝⎛=w
55 •• Picture the Problem We can use the definition of density and the formula for the volume of a sphere to find the density of iron. Once we know the density of iron, we can use these same relationships to find what the radius of the earth would be if it had the same mass per unit volume as iron.
(a) Using its definition, express the density of iron:
Vm
=ρ
Assuming it to be spherical, express the volume of an iron nucleus as a function of its radius:
334 rV π=
Substitute to obtain: 34
3r
mπ
ρ = (1)
Systems of Measurement
23
Substitute numerical values and evaluate ρ:
( )( )
317
315
26
kg/m1041.1
m104.54kg103.93
×=
×
×=
−
−
πρ
(b) Because equation (1) relates the density of any spherical object to its mass and radius, we can solve for r to obtain:
343πρmr =
Substitute numerical values and evaluate r:
( )( ) m216
kg/m1041.14kg1098.53
3317
24
=××
=π
r
56 •• Picture the Problem Apply the general rules concerning the multiplication, division, addition, and subtraction of measurements to evaluate each of the given expressions.
(a) Because all of the factors have two significant figures, the result will have two significant figures:
( ) ( )
( ) ( )
2
12
65
12
5
108.1
104.2105.7106.5
104.20000075.0106.5
×=
×××
=
××
−
−−
−
−
(b) Because the factor with the fewest significant figures in the first term has two significant figures, the result will have two significant figures. Because its last significant figure is in the tenth’s position, the difference between the first and second term will have its last significant figure in the tenth’s position:
( )( )( )4.306.48.7
06.4102.8104.62.14 97
=−=
−×× −
(c) Because all of the factors have two significant figures, the result will have two significant figures:
( ) ( )( )
82111
3426
109.2106.3
106.3101.6×=
×
××−
−
Chapter 1
24
(d) Because the factor with the fewest significant figures has two significant figures, the result will have two significant figures.
( )( )( )
( )( ) ( )
45.0
10490108.12104.6
10490108.12000064.0
2113
315
2113
31
=
××
×=
××
−−
−
−−
*57 •• Picture the Problem We can use the relationship between an angle θ, measured in radians, subtended at the center of a circle, the radius R of the circle, and the length L of the arc to answer these questions concerning the astronomical units of measure.
(a) Relate the angle θ subtended by an arc of length S to the distance R:
RS
=θ (1)
Solve for and evaluate S: ( )( )
parsec1085.4
360rad2
min601
s60min1s1parsec1
6−×=
⎟⎠⎞
⎜⎝⎛
°⎟⎟⎠
⎞⎜⎜⎝
⎛ °×
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=
π
RS θ
(b) Solve equation (1) for and evaluate R:
( )
m1009.3
360rad2
min601
s60min1s1
m10496.1
16
11
×=
⎟⎠⎞
⎜⎝⎛
°⎟⎟⎠
⎞⎜⎜⎝
⎛ °⎟⎟⎠
⎞⎜⎜⎝
⎛×
=
=
π
SRθ
(c) Relate the distance D light travels in a given interval of time ∆t to its speed c and evaluate D for ∆t = 1 y:
( )
m1047.9
ys103.156y1
sm103
15
78
×=
⎟⎟⎠
⎞⎜⎜⎝
⎛×⎟
⎠⎞
⎜⎝⎛ ×=
∆= tcD
Systems of Measurement
25
(d) Use the definition of 1 AU and the result from part (c) to obtain: ( )
AU106.33
m101.496AU1m109.47y1
4
1115
×=
⎟⎟⎠
⎞⎜⎜⎝
⎛×
×=⋅c
(e) Combine the results of parts (b) and (c) to obtain:
( )
y25.3
m109.47y1
m1008.3parsec1
15
16
⋅=
⎟⎟⎠
⎞⎜⎜⎝
⎛×⋅
×
×=
c
c
58 •• Picture the Problem Let Ne and Np represent the number of electrons and the number of protons, respectively and ρ the critical average density of the universe. We can relate these quantities to the masses of the electron and proton using the definition of density.
(a) Using its definition, relate the required density ρ to the electron density Ne/V:
VmN
Vm ee==ρ
Solve for Ne/V:
e
e
mVN ρ
= (1)
Substitute numerical values and evaluate Ne/V: 33
31
327e
melectrons/1059.6
nkg/electro109.11kg/m106
×=
××
= −
−
VN
(b) Express and evaluate the ratio of the masses of an electron and a proton:
427
31
p
e 1046.5kg101.67kg109.11 −
−
−
×=××
=mm
Rewrite equation (1) in terms of protons:
p
p
mVN ρ
= (2)
Divide equation (2) by equation (1) to obtain:
p
e
e
p
mm
VNVN
= or ⎟⎠⎞
⎜⎝⎛=
VN
mm
VN e
p
ep
Chapter 1
26
Substitute numerical values and use the result from part (a) to evaluate Np/V:
( )( )
3
33
4p
protons/m59.3
protons/m1059.6
1046.5
=
××
×= −
VN
*59 •• Picture the Problem We can use the definition of density to relate the mass of the water in the cylinder to its volume and the formula for the volume of a cylinder to express the volume of water used in the detector’s cylinder. To convert our answer in kg to lb, we can use the fact that 1 kg weighs about 2.205 lb. Relate the mass of water contained in the cylinder to its density and volume:
Vm ρ=
Express the volume of a cylinder in terms of its diameter d and height h:
hdhAV 2base 4
π==
Substitute to obtain: hdm 2
4πρ=
Substitute numerical values and evaluate m:
( ) ( ) ( )
kg1002.5
m4.41m3.394
kg/m10
7
233
×=
⎟⎠⎞
⎜⎝⎛=πm
Convert 5.02 × 107 kg to tons:
ton104.55lb2000
ton1kg
lb2.205kg1002.5
3
7
×=
×××=m
tons.55,000 tocloser is weight actual The ve.conservati is claimton 50,000 The −
60 ••• Picture the Problem We’ll solve this problem two ways. First, we’ll substitute two of the ordered pairs in the given equation to obtain two equations in C and n that we can solve simultaneously. Then we’ll use a spreadsheet program to create a graph of log T as a function of log m and use its curve-fitting capability to find n and C. Finally, we can identify the data points that deviate the most from a straight-line plot by examination of the graph.
Systems of Measurement
27
1st Solution for (a)
(a) To estimate C and n, we can apply the relation T = Cm n to two arbitrarily selected data points. We’ll use the 1st and 6th ordered pairs. This will produce simultaneous equations that can be solved for C and n.
nCmT 11 =
and nCmT 66 =
Divide the second equation by the first to obtain:
nn
mm
CmCm
TT
⎟⎟⎠
⎞⎜⎜⎝
⎛==
1
621
6
1
6
Substitute numerical values and solve for n to obtain:
n
⎟⎟⎠
⎞⎜⎜⎝
⎛=
kg0.1kg1
s0.56s75.1
or n10125.3 = ⇒ 4948.0=n
and so a ″judicial″ guess is that n = 0.5.
Substituting this value into the second equation gives:
5.055 CmT =
so ( ) 5.0kg1Cs75.1 =
Solving for C gives:
0.5s/kg75.1=C
2nd Solution for (a)
Take the logarithm (we’ll arbitrarily use base 10) of both sides of T = Cmn and simplify to obtain:
( ) ( )Cmn
mCCmT nn
loglogloglogloglog
+=+==
which, we note, is of the form bmxy += .
Hence a graph of log T vs. log m should be linear with a slope of n and a log T-intercept log C.
The graph of log T vs. log m shown below was created using a spreadsheet program. The equation shown on the graph was obtained using Excel’s ″Add Trendline″ function. (Excel’s ″Add Trendline″ function uses regression analysis to generate the trendline.)
Chapter 1
28
log T = 0.4987log m + 0.2479
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2
log m
log
T
Comparing the equation on the graph generated by the Add Trendline function to log T( ) = n log m + log C , we observe:
499.0=n
and 212479.0 s/kg77.110 ==C
or ( ) 499.021s/kg77.1 mT =
(b) From the graph we see that the data points that deviate the most from a straight-line plot are: s 2.22 kg, 1.50
and s, 0.471 kg, 0.02
==
==
Tm
Tm
(b) graph. on the plotted points thefit tobest thengrepresenti
line thefrommost thedeviate s) 1.05 kg, (0.4 and s) 0.471 kg, (0.02 pairs data theusing generated points that thesee graph we theFrom
Remarks: Still another way to find n and C is to use your graphing calculator to perform regression analysis on the given set of data for log T versus log m. The slope yields n and the y-intercept yields log C. 61 ••• Picture the Problem We can plot log T versus log r and find the slope of the best-fit line to determine the exponent n. We can then use any of the ordered pairs to evaluate C. Once we know n and C, we can solve T = Crn for r as a function of T.
Systems of Measurement
29
(a) Take the logarithm (we’ll arbitrarily use base 10) of both sides of T = Crn and simplify to obtain:
( ) ( )Crn
rCCrT nn
loglogloglogloglog
+=+==
Note that this equation is of the form bmxy += . Hence a graph of log T vs.
log r should be linear with a slope of n and a log T -intercept log C.
The graph of log T versus log r shown below was created using a spreadsheet program. The equation shown on the graph was obtained using Excel’s ″Add Trendline″ function. (Excel’s ″Add Trendline″ function uses regression analysis to generate the trendline.)
y = 1.5036x + 1.2311
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
-1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2
log r
log
T
From the regression analysis we observe that:
50.1=n
and
( ) 232311.1 Gmy/0.1710 ==C
or ( )( ) 50.123Gmy/0.17 rT = (1)
(b) Solve equation (1) for the radius of the planet’s orbit:
( )
32
23Gm/y0.17 ⎟⎟⎠
⎞⎜⎜⎝
⎛=
Tr
Substitute numerical values and evaluate r: ( )
Gm510.0Gmy/0.17
y20.632
23 =⎟⎟⎠
⎞⎜⎜⎝
⎛=r
Chapter 1
30
*62 ••• Picture the Problem We can express the relationship between the period T of the pendulum, its length L, and the acceleration of gravity g as ba gCLT =
and perform dimensional analysis to find the values of a and b and, hence, the function relating these variables. Once we’ve performed the experiment called for in part (b), we can determine an experimental value for C.
(a) Express T as the product of L and g raised to powers a and b:
ba gCLT = (1)
where C is a dimensionless constant.
Write this equation in dimensional form:
[ ] [ ] [ ]ba gLT =
Noting that the symbols for the dimension of the period and length of the pendulum are the same as those representing the physical quantities, substitute the dimensions to obtain:
ba
TLLT ⎟⎠⎞
⎜⎝⎛= 2
Because L does not appear on the left-hand side of the equation, we can write this equation as:
bba TLTL 210 −+=
Equate the exponents to obtain:
12and0 =−=+ bba
Solve these equations simultaneously to find a and b:
21
21 and −== ba
Substitute in equation (1) to obtain: g
LCgCLT == − 2121 (2)
(b) If you use pendulums of lengths 1 m and 0.5 m; the periods should be about:
( )
( ) s4.1m5.0
and
s2m1
=
=
T
T
(c) Solve equation (2) for C:
LgTC =
Systems of Measurement
31
Evaluate C with L = 1 m and T = 2 s: ( ) π226.6
m1m/s9.81s2
2
≈==C
Substitute in equation (2) to obtain:
gLT π2=
63 ••• Picture the Problem The weight of the earth’s atmosphere per unit area is known as the atmospheric pressure. We can use this definition to express the weight w of the earth’s atmosphere as the product of the atmospheric pressure and the surface area of the earth.
Using its definition, relate atmospheric pressure to the weight of the earth’s atmosphere:
AwP =
Solve for w: PAw =
Relate the surface area of the earth to its radius R:
24 RA π=
Substitute to obtain:
PRw 24π=
Substitute numerical values and evaluate w:
( ) lb1016.1inlb14.7
min39.37
kmm10km63704 19
2
2232 ×=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛= πw
Chapter 1
32
33
Chapter 2 Motion in One Dimension Conceptual Problems 1 • Determine the Concept The "average velocity" is being requested as opposed to "average speed".
The average velocity is defined as the change in position or displacement divided by the change in time.
tyv
∆∆
=av
The change in position for any "round trip" is zero by definition. So the average velocity for any round trip must also be zero.
00av =
∆=
∆∆
=tt
yv
*2 • Determine the Concept The important concept here is that "average speed" is being requested as opposed to "average velocity". Under all circumstances, including constant acceleration, the definition of the average speed is the ratio of the total distance traveled (H + H) to the total time elapsed, in this case 2H/T. correct. is )(d
Remarks: Because this motion involves a round trip, if the question asked for "average velocity," the answer would be zero. 3 • Determine the Concept Flying with the wind, the speed of the plane relative to the ground (vPG) is the sum of the speed of the wind relative to the ground (vWG) and the speed of the plane relative to the air (vPG = vWG + vPA). Flying into or against the wind the speed relative to the ground is the difference between the wind speed and the true air speed of the plane (vg = vw – vt). Because the ground speed landing against the wind is smaller than the ground speed landing with the wind, it is safer to land against the wind. 4 • Determine the Concept The important concept here is that a = dv/dt, where a is the acceleration and v is the velocity. Thus, the acceleration is positive if dv is positive; the acceleration is negative if dv is negative. (a) Let’s take the direction a car is moving to be the positive direction:
Because the car is moving in the direction we’ve chosen to be positive, its velocity is positive (dx > 0). If the car is braking, then its velocity is decreasing (dv < 0) and its acceleration (dv/dt) is negative.
(b) Consider a car that is moving to Because the car is moving in the direction
Chapter 2
34
the right but choose the positive direction to be to the left:
opposite to that we’ve chosen to be positive, its velocity is negative (dx < 0). If the car is braking, then its velocity is increasing (dv > 0) and its acceleration (dv/dt) is positive.
*5 • Determine the Concept The important concept is that when both the acceleration and the velocity are in the same direction, the speed increases. On the other hand, when the acceleration and the velocity are in opposite directions, the speed decreases.
(a) . be
must nt displacemeyour negative, remainsity your veloc Becausenegative
(b) reached. is wall theuntil walking,of speed theslowgradually steps five
last theDuring direction. negative theas your trip ofdirection theDefine
(c) A graph of v as a function of t that is consistent with the conditions stated in the problem is shown below:
-5
-4
-3
-2
-1
0
0 0.5 1 1.5 2 2.5
t (s)
v (m
/s)
6 • Determine the Concept True. We can use the definition of average velocity to express the displacement ∆x as ∆x = vav∆t. Note that, if the acceleration is constant, the average velocity is also given by vav = (vi + vf)/2.
7 • Determine the Concept Acceleration is the slope of the velocity versus time curve, a = dv/dt; while velocity is the slope of the position versus time curve, v = dx/dt. The speed of an object is the magnitude of its velocity.
Motion in One Dimension
35
(a) True. Zero acceleration implies that the velocity is constant. If the velocity is constant (including zero), the speed must also be constant. (b) True in one dimension. Remarks: The answer to (b) would be False in more than one dimension. In one dimension, if the speed remains constant, then the object cannot speed up, slow down, or reverse direction. Thus, if the speed remains constant, the velocity remains constant, which implies that the acceleration remains zero. (In more than one-dimensional motion, an object can change direction while maintaining constant speed. This constitutes a change in the direction of the velocity.) Consider a ball moving in a circle at a constant rotation rate. The speed (magnitude of the velocity) is constant while the velocity is tangent to the circle and always changing. The acceleration is always pointing inward and is certainly NOT zero. *8 •• Determine the Concept Velocity is the slope of the position versus time curve and acceleration is the slope of the velocity versus time curve. See the graphs below.
0
1
2
3
4
5
6
7
0 5 10 15 20 25
time (s)
posi
tion
(m)
-4
-3
-2
-1
0
1
2
3
0 5 10 15 20 25
time (s)
acce
lera
tion
(m/s2 )
Chapter 2
36
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0 5 10 15 20 25
time (s)
velo
city
(m/s
)
9 • Determine the Concept False. The average velocity is defined (for any acceleration) as the change in position (the displacement) divided by the change in time txv ∆∆=av . It is always valid. If the acceleration remains constant the average velocity is also given by
2fi
avvvv +
=
Consider an engine piston moving up and down as an example of non-constant velocity. For one complete cycle, vf = vi and xi = xf so vav = ∆x/∆t is zero. The formula involving the mean of vf and vi cannot be applied because the acceleration is not constant, and yields an incorrect nonzero value of vi. 10 • Determine the Concept This can occur if the rocks have different initial speeds. Ignoring air resistance, the acceleration is constant. Choose a coordinate system in which the origin is at the point of release and upward is the positive direction. From the constant-acceleration equation
221
00 attvyy ++=
we see that the only way two objects can have the same acceleration (–g in this case) and cover the same distance, ∆y = y – y0, in different times would be if the initial velocities of the two rocks were different. Actually, the answer would be the same whether or not the acceleration is constant. It is just easier to see for the special case of constant acceleration. *11 •• Determine the Concept Neglecting air resistance, the balls are in free fall, each with the same free-fall acceleration, which is a constant. At the time the second ball is released, the first ball is already moving. Thus, during any time interval their velocities will increase by exactly the same amount. What can be said about the speeds of the two balls? The first ball will always be moving faster than the second ball. This being the case, what happens to the separation of the two balls while they are both
Motion in One Dimension
37
falling? Their separation increases. correct. is )(a
12 •• Determine the Concept The slope of an x(t) curve at any point in time represents the speed at that instant. The way the slope changes as time increases gives the sign of the acceleration. If the slope becomes less negative or more positive as time increases (as you move to the right on the time axis), then the acceleration is positive. If the slope becomes less positive or more negative, then the acceleration is negative. The slope of the slope of an x(t) curve at any point in time represents the acceleration at that instant. The slope of curve (a) is negative and becomes more negative as time increases.
Therefore, the velocity is negative and the acceleration is negative.
The slope of curve (b) is positive and constant and so the velocity is positive and constant.
Therefore, the acceleration is zero.
The slope of curve (c) is positive and decreasing.
Therefore, the velocity is positive and the acceleration is negative.
The slope of curve (d) is positive and increasing.
Therefore, the velocity and acceleration are positive. We need more information to conclude that a is constant.
The slope of curve (e) is zero. Therefore, the velocity and acceleration are zero.
on.accelerati positiveconstanth motion wit showsbest )(d
*13 • Determine the Concept The slope of a v(t) curve at any point in time represents the acceleration at that instant. Only one curve has a constant and positive slope.
( ) correct. is b
14 • Determine the Concept No. The word average implies an interval of time rather than an instant in time; therefore, the statement makes no sense. *15 • Determine the Concept Note that the ″average velocity″ is being requested as opposed to the ″average speed.″
Chapter 2
38
Yes. In any roundtrip, A to B, and back to A, the average velocity is zero.
( )
( )
0
0BAAB
BAABABAav
=∆
=∆
∆−+∆=
∆∆+∆
=∆∆
=→→
ttxxt
xxtxv
On the other hand, the average velocity between A and B is not generally zero.
( ) 0ABBAav ≠
∆∆
=→ txv
Remarks: Consider an object launched up in the air. Its average velocity on the way up is NOT zero. Neither is it zero on the way down. However, over the round trip, it is zero. 16 • Determine the Concept An object is farthest from the origin when it is farthest from the time axis. In one-dimensional motion starting from the origin, the point located farthest from the time axis in a distance-versus-time plot is the farthest from its starting point. Because the object’s initial position is at x = 0, point B represents the instant that the object is farthest from x = 0. correct. is )(b
17 • Determine the Concept No. If the velocity is constant, a graph of position as a function of time is linear with a constant slope equal to the velocity. 18 • Determine the Concept Yes. The average velocity in a time interval is defined as the displacement divided by the elapsed time txv ∆∆=av . The fact that vav = 0 for some time interval, ∆t, implies that the displacement ∆x over this interval is also zero. Because the instantaneous velocity is defined as ( )txv t ∆∆= →∆ /lim 0 , it follows that v must also be zero. As an example, in the following graph of x versus t, over the interval between t = 0 and t ≈ 21 s, ∆x = 0. Consequently, vav = 0 for this interval. Note that the instantaneous velocity is zero only at t ≈ 10 s.
Motion in One Dimension
39
0
100
200
300
400
500
600
0 5 10 15 20
t (s)
x (m
)
19 •• Determine the Concept In the one-dimensional motion shown in the figure, the velocity is a minimum when the slope of a position-versus-time plot goes to zero (i.e., the curve becomes horizontal). At these points, the slope of the position-versus-time curve is zero; therefore, the speed is zero. correct. is )(b
*20 •• Determine the Concept In one-dimensional motion, the velocity is the slope of a position-versus-time plot and can be either positive or negative. On the other hand, the speed is the magnitude of the velocity and can only be positive. We’ll use v to denote velocity and the word “speed” for how fast the object is moving.
(a) curve a: ( ) ( )12 tvtv < curve b: ( ) ( )12 tvtv = curve c: ( ) ( )12 tvtv > curve d: ( ) ( )12 tvtv <
(b) curve a: ( ) ( )12 speedspeed tt < curve b: ( ) ( )12 speedspeed tt = curve c: ( ) ( )12 speedspeed tt < curve d: ( ) ( )12 speedspeed tt >
21 • Determine the Concept Acceleration is the slope of the velocity-versus-time curve, a = dv/dt, while velocity is the slope of the position-versus-time curve, v = dx/dt.
(a) False. Zero acceleration implies that the velocity is not changing. The velocity could be any constant (including zero). But, if the velocity is constant and nonzero, the particle must be moving.
(b) True. Again, zero acceleration implies that the velocity remains constant. This means that the x-versus-t curve has a constant slope (i.e., a straight line). Note: This does not necessarily mean a zero-slope line.
Chapter 2
40
22 • Determine the Concept Yes. If the velocity is changing the acceleration is not zero. The velocity is zero and the acceleration is nonzero any time an object is momentarily at rest. If the acceleration were also zero, the velocity would never change; therefore, the object would have to remain at rest. Remarks: It is important conceptually to note that when both the acceleration and the velocity have the same sign, the speed increases. On the other hand, when the acceleration and the velocity have opposite signs, the speed decreases.
23 • Determine the Concept In the absence of air resistance, the ball will experience a constant acceleration. Choose a coordinate system in which the origin is at the point of release and the upward direction is positive. The graph shows the velocity of a ball that has been thrown straight upward with an initial speed of 30 m/s as a function of time. Note that the slope of this graph, the acceleration, is the same at every point, including the point at which v = 0 (at the top of its flight). Thus, 0flight of top =v and ga −=flight of top .
-30
-20
-10
0
10
20
30
0 1 2 3 4 5 6
t (s)
v (m
/s)
The acceleration is the slope (–g).
24 • Determine the Concept The "average speed" is being requested as opposed to "average velocity." We can use the definition of average speed as distance traveled divided by the elapsed time and expression for the average speed of an object when it is experiencing constant acceleration to express vav in terms of v0. The average speed is defined as the total distance traveled divided by the change in time:
TH
THH
v
2timetotal
traveleddistancetotalav
=+
=
=
Motion in One Dimension
41
Find the average speed for the upward flight of the object: T
Hvv21
0upav, 2
0=
+=
Solve for H to obtain:
TvH 041=
Find the average speed for the downward flight of the object: T
Hvv21
0downav, 2
0=
+=
Solve for H to obtain:
TvH 041=
Substitute in our expression for vav to obtain:
( )2
2 0041
avv
TTvv ==
Because 00 ≠v , the average speed is not zero.
Remarks: 1) Because this motion involves a roundtrip, if the question asked for ″average velocity″, the answer would be zero. 2) Another easy way to obtain this result is take the absolute value of the velocity of the object to obtain a graph of its speed as a function of time. A simple geometric argument leads to the result we obtained above. 25 • Determine the Concept In the absence of air resistance, the bowling ball will experience constant acceleration. Choose a coordinate system with the origin at the point of release and upward as the positive direction. Whether the ball is moving upward and slowing down, is momentarily at the top of its trajectory, or is moving downward with ever increasing velocity, its acceleration is constant and equal to the acceleration due to gravity. correct. is )(b
26 • Determine the Concept Both objects experience the same constant acceleration. Choose a coordinate system in which downward is the positive direction and use a constant-acceleration equation to express the position of each object as a function of time. Using constant-acceleration equations, express the positions of both objects as functions of time:
221
0A0,A gttvxx ++= and
221
0B,0B gttvxx ++= where v0 = 0.
Express the separation of the two objects by evaluating xB − xA:
m10A.0B,0AB =−=− xxxx
and correct. is )(d
*27 •• Determine the Concept Because the Porsche accelerates uniformly, we need to look for a graph that represents constant acceleration. We are told that the Porsche has a constant acceleration that is positive (the velocity is increasing); therefore we must look for a velocity-versus-time curve with a positive constant slope and a nonzero intercept.
Chapter 2
42
( ) correct. is c
*28 •• Determine the Concept In the absence of air resistance, the object experiences constant acceleration. Choose a coordinate system in which the downward direction is positive. Express the distance D that an object, released from rest, falls in time t:
221 gtD =
Because the distance fallen varies with the square of the time, during the first two seconds it falls four times the distance it falls during the first second.
( ) correct. is a
29 •• Determine the Concept In the absence of air resistance, the acceleration of the ball is constant. Choose a coordinate system in which the point of release is the origin and upward is the positive y direction. The displacement of the ball halfway to its highest point is:
2maxyy ∆
=∆
Using a constant-acceleration equation, relate the ball’s initial and final velocities to its displacement and solve for the displacement:
ygvyavv ∆−=∆+= 22 20
20
2
Substitute v0 = 0 to determine the maximum displacement of the ball:
( ) gv
gvy
22
20
20
max =−
−=∆
Express the velocity of the ball at half its maximum height:
22
222
20
202
0max20
max20
20
2
vg
vgvygv
ygvygvv
=−=∆−=
∆−=∆−=
Solve for v:
00 707.022 vvv ≈=
and ( ) correct. is c
30 • Determine the Concept As long as the acceleration remains constant the following constant-acceleration equations hold. If the acceleration is not constant, they do not, in general, give correct results except by coincidence.
221
00 attvxx ++= atvv += 0 xavv ∆+= 220
2 2
fiav
vvv +=
Motion in One Dimension
43
(a) False. From the first equation, we see that (a) is true if and only if the acceleration is constant. (b) False. Consider a rock thrown straight up into the air. At the "top" of its flight, the velocity is zero but it is changing (otherwise the velocity would remain zero and the rock would hover); therefore the acceleration is not zero. (c) True. The definition of average velocity, txv ∆∆=av , requires that this always be true. *31 • Determine the Concept Because the acceleration of the object is constant, the constant-acceleration equations can be used to describe its motion. The special expression for
average velocity for constant acceleration is2
fiav
vvv += . ( ) correct. is c
32 • Determine the Concept The constant slope of the x-versus-t graph tells us that the velocity is constant and the acceleration is zero. A linear position versus time curve implies a constant velocity. The negative slope indicates a constant negative velocity. The fact that the velocity is constant implies that the acceleration is also constant and zero. ( ) correct. is e
33 •• Determine the Concept The velocity is the slope of the tangent to the curve, and the acceleration is the rate of change of this slope. Velocity is the slope of the position-versus-time curve. A parabolic x(t) curve opening upward implies an increasing velocity. The acceleration is positive. ( ) correct. is a
34 •• Determine the Concept The acceleration is the slope of the tangent to the velocity as a function of time curve. For constant acceleration, a velocity-versus- time curve must be a straight line whose slope is the acceleration. Zero acceleration means that slope of v(t) must also be zero. ( ) correct. is c
35 •• Determine the Concept The acceleration is the slope of the tangent to the velocity as a function of time curve. For constant acceleration, a velocity-versus- time curve must be a straight line whose slope is the acceleration. The acceleration and therefore the slope can be positive, negative, or zero. ( ) correct. is d
36 •• Determine the Concept The velocity is positive if the curve is above the v = 0 line (the t axis), and the acceleration is negative if the tangent to the curve has a negative slope. Only graphs (a), (c), and (e) have positive v. Of these, only graph (e) has a negative slope. ( ) correct. is e
Chapter 2
44
37 •• Determine the Concept The velocity is positive if the curve is above the v = 0 line (the t axis), and the acceleration is negative if the tangent to the curve has a negative slope. Only graphs (b) and (d) have negative v. Of these, only graph (d) has a negative slope.
( ) correct. is d
38 •• Determine the Concept A linear velocity-versus-time curve implies constant acceleration. The displacement from time t = 0 can be determined by integrating v-versus-t — that is, by finding the area under the curve. The initial velocity at t = 0 can be read directly from the graph of v-versus-t as the v-intercept; i.e., v(0). The acceleration of the object is the slope of v(t) . The average velocity of the object is given by drawing a horizontal line that has the same area under it as the area under the curve. Because all of these quantities can be determined ( ) correct. is e
*39 •• Determine the Concept The velocity is the slope of a position versus time curve and the acceleration is the rate at which the velocity, and thus the slope, changes.
Velocity
(a) Negative at t0 and t1. (b) Positive at t3, t4, t6, and t7. (c) Zero at t2 and t5.
Acceleration The acceleration is positive at points where the slope increases as you move toward the right.
(a) Negative at t4. (b) Positive at t2 and t6. (c) Zero at t0, t1, t3, t5, and t7.
40 •• Determine the Concept Acceleration is the slope of a velocity-versus-time curve. (a) Acceleration is zero and constant while velocity is not zero.
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3
t
v
Motion in One Dimension
45
(b) Acceleration is constant but not zero.
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3
t
v
(c) Velocity and acceleration are both positive.
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3
t
v
(d) Velocity and acceleration are both negative.
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3
t
v
Chapter 2
46
(e) Velocity is positive and acceleration is negative.
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3
t
v
(f) Velocity is negative and acceleration is positive.
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3
t
v
Motion in One Dimension
47
(g) Velocity is momentarily zero at the intercept with the t axis but the acceleration is not zero.
-3
-2
-1
0
1
2
3
0 0.5 1 1.5 2 2.5 3
t
v
41 •• Determine the Concept Velocity is the slope and acceleration is the slope of the slope of a position-versus-time curve. Acceleration is the slope of a velocity- versus-time curve.
(a) For constant velocity, x-versus-t must be a straight line; v-versus-t must be a horizontal straight line; and a-versus-t must be a straight horizontal line at a = 0.
(a), (f), and (i) are the correct answers.
(b) For velocity to reverse its direction x-versus-t must have a slope that changes sign and v-versus-t must cross the time axis. The acceleration cannot remain zero at all times.
(c) and (d) are the correct answers.
(c) For constant acceleration, x-versus-t must be a straight line or a parabola, v-versus-t must be a straight line, and a-versus-t must be a horizontal straight line.
(a), (d), (e), (f), (h), and (i) are the correct answers.
(d) For non-constant acceleration, x-versus-t must not be a straight line or a parabola; v-versus-t must not be a straight line, or a-versus-t must not be a horizontal straight line.
(b), (c), and (g) are the correct answers.
Chapter 2
48
For two graphs to be mutually consistent, the curves must be consistent with the definitions of velocity and acceleration.
Graphs (a) and (i) are mutually consistent. Graphs (d) and (h) are mutually consistent. Graphs (f) and (i) are also mutually consistent.
Estimation and Approximation 42 • Picture the Problem Assume that your heart beats at a constant rate. It does not, but the average is pretty stable. (a) We will use an average pulse rate of 70 bpm for a seated (resting) adult. One’s pulse rate is defined as the number of heartbeats per unit time:
Timeheartbeats of #rate Pulse =
and Timerate Pulse heartbeats of # ×=
The time required to drive 1 mi at 60 mph is (1/60) h or 1 min:
( )( )beats 70
min1beats/min 70 heartbeats of #
=
=
(b) Express the number of heartbeats during a lifetime in terms of the pulse rate and the life span of an individual:
TimeratePulseheartbeatsof# ×=
Assuming a 95-y life span, calculate the time in minutes:
( )( )( )( ) min1000.5hmin/60h/d24d/y25.365y95Time 7×==
Substitute numerical values and evaluate the number of heartbeats:
( )( ) beats1050.3min1000.5min/beats70heartbeatsof# 97 ×=×=
*43 •• Picture the Problem In the absence of air resistance, Carlos’ acceleration is constant. Because all the motion is downward, let’s use a coordinate system in which downward is positive and the origin is at the point at which the fall began. (a) Using a constant-acceleration equation, relate Carlos’ final velocity to his initial velocity, acceleration, and distance fallen and solve for his final velocity:
yavv ∆+= 220
2 and, because v0 = 0 and a = g,
ygv ∆= 2
Substitute numerical values and evaluate v:
( )( ) m/s2.54m150m/s81.92 2 ==v
Motion in One Dimension
49
(b) While his acceleration by the snow is not constant, solve the same constant- acceleration equation to get an estimate of his average acceleration:
yvva
∆−
=2
20
2
Substitute numerical values and evaluate a: ( )
( )g
a
123
m/s1020.1m22.12
m/s54 2322
−=
×−=−
=
Remarks: The final velocity we obtained in part (a), approximately 121 mph, is about the same as the terminal velocity for an "average" man. This solution is probably only good to about 20% accuracy. 44 •• Picture the Problem Because we’re assuming that the accelerations of the skydiver and the mouse are constant to one-half their terminal velocities, we can use constant-acceleration equations to find the times required for them to reach their ″upper-bound″ velocities and their distances of fall. Let’s use a coordinate system in which downward is the positive y direction. (a) Using a constant-acceleration equation, relate the upper-bound velocity to the free-fall acceleration and the time required to reach this velocity:
tgvv ∆+= 0boundupper or, because v0 = 0,
tgv ∆=boundupper
Solve for ∆t:
gv
t boundupper=∆
Substitute numerical values and evaluate ∆t:
s55.2m/s9.81m/s52
2 ==∆t
Using a constant-acceleration equation, relate the skydiver’s distance of fall to the elapsed time ∆t:
( )221
0 tatvy ∆+∆=∆ or, because v0 = 0 and a = g,
( )221 tgy ∆=∆
Substitute numerical values and evaluate ∆y:
( ) ( ) m31.9s2.55m/s9.81 2221 ==∆y
(b) Proceed as in (a) with vupper bound = 0.5 m/s to obtain: s0510.0
m/s9.81m/s5.0
2 ==∆t
and ( )( ) cm27.1s0510.0m/s9.81 22
21 ==∆y
Chapter 2
50
45 •• Picture the Problem This is a constant-acceleration problem. Choose a coordinate system in which the direction Greene is running is the positive x direction. During the first 3 s of the race his acceleration is positive and during the rest of the race it is zero. The pictorial representation summarizes what we know about Greene’s race.
Express the total distance covered by Greene in terms of the distances covered in the two phases of his race:
1201m100 xx ∆+∆=
Express the distance he runs getting to his maximum velocity:
( ) ( )2212
010121
01001 s3atatvx =∆+∆=∆
Express the distance covered during the rest of the race at the constant maximum velocity:
( )( )
( )( )s79.6s31201
212122
112max12
atta
tatvx
=∆∆=
∆+∆=∆
Substitute for these displacements and solve for a:
( ) ( )( )s79.6s3s3m100 221 aa +=
and 2m/s02.4=a
*46 •• Determine the Concept This is a constant-acceleration problem with a = −g if we take upward to be the positive direction. At the maximum height the ball will reach, its speed will be near zero and when the ball has just been tossed in the air its speed is near its maximum value. What conclusion can you draw from the image of the ball near its maximum height?
Because the ball is moving slowly its blur is relatively short (i.e., there is less blurring).
To estimate the initial speed of the ball:
Motion in One Dimension
51
a) Estimate how far the ball being tossed moves in 1/30 s:
The ball moves about 3 ball diameters in 1/30 s.
b) Estimate the diameter of a tennis ball:
The diameter of a tennis ball is approximately 5 cm.
c) Now one can calculate the approximate distance the ball moved in 1/30 s:
( )( )cm 15
rcm/diamete 5diameters 3 traveledDistance
=×
=
d) Calculate the average speed of the tennis ball over this distance:
m/s4.50
cm/s 450s
301cm 15speed Average
=
==
e) Because the time interval is very short, the average speed of the ball is a good approximation to its initial speed:
∴ v0 = 4.5 m/s
f) Finally, use the constant-acceleration equation
yavv ∆+= 220
2 to solve for and evaluate ∆y:
( )( ) m03.1
m/s81.92m/s5.4
2 2
220 =
−−
=−
=∆avy
Remarks: This maximum height is in good agreement with the height of the higher ball in the photograph.
*47 •• Picture the Problem The average speed of a nerve impulse is approximately 120 m/s. Assume an average height of 1.7 m and use the definition of average speed to estimate the travel time for the nerve impulse.
Using the definition of average speed, express the travel time for the nerve impulse:
avvxt ∆
=∆
Substitute numerical values and evaluate ∆t: ms14.2
m/s120m1.7
==∆t
Speed, Displacement, and Velocity
48 • Picture the Problem Think of the electron as traveling in a straight line at constant speed and use the definition of average speed.
Chapter 2
52
(a) Using its definition, express the average speed of the electron:
ts
∆∆
=
=flight of time
traveleddistancespeed Average
Solve for and evaluate the time of flight:
ns00.4s104
sm104m16.0
speed Average9
7
=×=
×=
∆=∆
−
st
(b) Calculate the time of flight for an electron in a 16-cm long current carrying wire similarly.
min7.66s104
sm104m16.0
speed Average3
5
=×=
×=
∆=∆ −
st
*49 • Picture the Problem In this problem the runner is traveling in a straight line but not at constant speed - first she runs, then she walks. Let’s choose a coordinate system in which her initial direction of motion is taken as the positive x direction.
(a) Using the definition of average velocity, calculate the average velocity for the first 9 min:
min/km278.0min9
km5.2av ==
∆∆
=txv
(b) Using the definition of average velocity, calculate her average speed for the 30 min spent walking:
min/km0833.0
min30km5.2
av
−=
−=
∆∆
=txv
(c) Express her average velocity for the whole trip: 00tripround
av =∆
=∆
∆=
ttx
v
(d) Finally, express her average speed for the whole trip:
min/km128.0
min9min30)km5.2(2
timeelapsedtraveleddistancespeed Average
=
+=
=
50 • Picture the Problem The car is traveling in a straight line but not at constant speed. Let the direction of motion be the positive x direction. (a) Express the total displacement of the car for the entire trip: 21total xxx ∆+∆=∆
Motion in One Dimension
53
Find the displacement for each leg of the trip:
( )( )km200
h5.2km/h8011,1
=
=∆=∆ tvx av
and ( )( )
km0.60h5.1km/h4022,2
=
=∆=∆ tvx av
Add the individual displacements to get the total displacement:
km260
km0.60km20021total
=
+=∆+∆=∆ xxx
(b) As long as the car continues to move in the same direction, the average velocity for the total trip is given by: hkm0.65
h5.1h5.2km260
total
total
=
+=
∆∆
≡txvav
51 • Picture the Problem However unlikely it may seem, imagine that both jets are flying in a straight line at constant speed. (a) The time of flight is the ratio of the distance traveled to the speed of the supersonic jet.
( )( )h25.2
s/h3600km/s340.02km5500
speedsupersonic
Atlanticsupersonic
=
=
=st
(b) The time of flight is the ratio of the distance traveled to the speed of the subsonic jet.
( )( )h99.4
s/h3600km/s340.09.0km5500
speedsubsonic
Atlanticsubsonic
=
=
=st
(c) Adding 2 h on both the front and the back of the supersonic trip, we obtain the average speed of the supersonic flight. hkm880
h00.4h25.2km5500speed supersonic av,
=
+=
(d) Adding 2 h on both the front and the back of the subsonic trip, we obtain the average speed of the subsonic flight. hkm611
h00.4h00.5km5500speed subsonic av,
=
+=
Chapter 2
54
*52 • Picture the Problem In free space, light travels in a straight line at constant speed, c. (a) Using the definition of average speed, solve for and evaluate the time required for light to travel from the sun to the earth:
ts
=speedaverage
and
min33.8s500
m/s103m101.5
speedaverage 8
11
==
××
==st
(b) Proceed as in (a) this time using the moon-earth distance:
s28.1m/s103
m1084.38
8
=×
×=t
(c) One light-year is the distance light travels in a vacuum in one year: ( )( )
mi1089.5
kmmi/1.611km109.48
km1048.9m1048.9year-light1
12
12
1215
×=
×=
×=×=
53 • Picture the Problem In free space, light travels in a straight line at constant speed, c. (a) Using the definition of average speed (equal here to the assumed constant speed of light), solve for the time required to travel the distance to Proxima Centauri:
y4.33 s1037.1
sm103m101.4
light of speed traveleddistance
8
8
16
=×=
××
==t
(b) Traveling at 10-4c, the delivery time (ttotal) will be the sum of the time for the order to reach Hoboken and the time for the pizza to be delivered to Proxima Centauri:
( )( )
y1033.4y1033.4y33.4
sm10310km101.433.4
6
6
84
13delivered be order toHoboken sent to be order tototal
×≈
×+=
××
+=
+=
−y
ttt
pay. tohavenot doesGregor y,1000y10 4.33 Since 6 >>×
54 • Picture the Problem The time for the second 50 km is equal to the time for the entire journey less the time for the first 50 km. We can use this time to determine the average speed for the second 50 km interval from the definition of average speed. Using the definition of average speed, find the time required for the total journey:
h2hkm50
km100speed average
distance totaltotal ===t
Motion in One Dimension
55
Find the time required for the first 50 km:
h25.1hkm40
km50km501st ==t
Find the time remaining to travel the last 50 km: h0.75
h1.25h2km 50st 1totalkm 50 2nd
=−=−= ttt
Finally, use the time remaining to travel the last 50 km to determine the average speed over this distance:
hkm7.66h75.0
km50time
traveleddistancespeed Average
km 50 2nd
km 50 2nd
km 50 2nd
==
=
*55 •• Picture the Problem Note that both the arrow and the sound travel a distance d. We can use the relationship between distance traveled, the speed of sound, the speed of the arrow, and the elapsed time to find the distance separating the archer and the target. Express the elapsed time between the archer firing the arrow and hearing it strike the target:
soundarrows1 ttt ∆+∆==∆
Express the transit times for the arrow and the sound in terms of the distance, d, and their speeds:
m/s40arrowarrow
dv
dt ==∆
and
m/s340soundsound
dv
dt ==∆
Substitute these two relationships in the expression obtained in step 1 and solve for d:
s1m/s340m/s40
=+dd
and m8.35=d
56 •• Picture the Problem Assume both runners travel parallel paths in a straight line along the track. (a) Using the definition of average speed, find the time for Marcia:
( )
( ) s5.14sm61.15
m 100speed sJohn'15.1run distance
speed sMarcia'run distance
Marcia
==
=
=t
Chapter 2
56
Find the distance covered by John in 14.5 s and the difference between that distance and 100 m:
( )( ) m 0.87s 14.5sm6John ==x and Marcia wins by
m 13.0 m 87 m 100 =−
(b) Using the definition of average speed, find the time required by John to complete the 100-m run:
s 7.16sm6
m 100speedsJohn'
rundistanceJohn ===t
Marsha wins by 16.7 s – 14.5 s = 2.2 s Alternatively, the time required by John to travel the last 13.0 m is (13 m)/(6 m/s) = s 17.2
57 • Picture the Problem The average velocity in a time interval is defined as the displacement divided by the time elapsed; that is txv ∆∆= /av .
(a) ∆xa = 0 0av =v
(b) ∆xb = 1 m and ∆tb = 3 s m/s333.0av =v
(c) ∆xc = –6 m and ∆tc = 3 s m/s00.2av −=v
(d) ∆xd = 3 m and ∆td = 3 s m/s00.1av =v
58 •• Picture the Problem In free space, light travels in a straight line at constant speed c. We can use Hubble’s law to find the speed of the two planets. (a) Using Hubble’s law, calculate the speed of the first galaxy:
( )( )m/s1090.7
s1058.1m1054
11822a
×=
××= −−v
(b) Using Hubble’s law, calculate the speed of the second galaxy:
( )( )m/s1016.3
s1058.1m1027
11825b
×=
××= −−v
(c) Using the relationship between distance, speed, and time for both galaxies, determine how long ago they were both located at the same place as the earth: yearsbillion 1.20
y1020.1s1033.6
1
917
=
×=×=
===HrH
rvrt
Motion in One Dimension
57
*59 •• Picture the Problem Ignoring the time intervals during which members of this relay time get up to their running speeds, their accelerations are zero and their average speed can be found from its definition. Using its definition, relate the average speed to the total distance traveled and the elapsed time: timeelapsed
traveleddistanceav =v
Express the time required for each animal to travel a distance L:
sailfishsailfish
falconfalcon
cheetahcheetah
and
,
,
vLt
vLt
vLt
=
=
=
Express the total time, ∆t: ⎟⎟
⎠
⎞⎜⎜⎝
⎛++=∆
sailfishfalconcheetah
111vvv
Lt
Use the total distance traveled by the relay team and the elapsed time to calculate the average speed:
km/h122
km/h1051
km/h1611
km/h1131
3av =
⎟⎟⎠
⎞⎜⎜⎝
⎛++
=L
Lv
Calculate the average of the three speeds:
avspeedsthree 03.1km/h1263
km/h105km/h161km/h113Average v==++
=
60 •• Picture the Problem Perhaps the easiest way to solve this problem is to think in terms of the relative velocity of one car relative to the other. Solve this problem from the reference frame of car A. In this frame, car A remains at rest. Find the velocity of car B relative to car A:
vrel = vB – vA = (110 – 80) km/h = 30 km/h
Find the time before car B reaches car A:
h1.5km/h30km45
rel
==∆
=∆v
xt
Find the distance traveled, relative to the road, by car A in 1.5 h:
( )( ) km120km/h80h1.5 ==d
Chapter 2
58
*61 •• Picture the Problem One way to solve this problem is by using a graphing calculator to plot the positions of each car as a function of time. Plotting these positions as functions of time allows us to visualize the motion of the two cars relative to the (fixed) ground. More importantly, it allows us to see the motion of the two cars relative to each other. We can, for example, tell how far apart the cars are at any given time by determining the length of a vertical line segment from one curve to the other. (a) Letting the origin of our coordinate system be at the intersection, the position of the slower car, x1(t), is given by:
x1(t) = 20t where x1 is in meters if t is in seconds.
Because the faster car is also moving at a constant speed, we know that the position of this car is given by a function of the form:
x2(t) = 30t + b
We know that when t = 5 s, this second car is at the intersection (i.e., x2(5 s) = 0). Using this information, you can convince yourself that:
b = −150 m
Thus, the position of the faster car is given by:
( ) 150302 −= ttx
One can use a graphing calculator, graphing paper, or a spreadsheet to obtain the graphs of x1(t) (the solid line) and x2(t) (the dashed line) shown below:
0
50
100
150
200
250
300
350
0 2 4 6 8 10 12 14 16
t (s)
x (m
)
(b) Use the time coordinate of the intersection of the two lines to determine the time at which the second car overtakes the first:
From the intersection of the two lines, one can see that the second car will "overtake" (catch up to) the first car at s 15 =t .
Motion in One Dimension
59
(c) Use the position coordinate of the intersection of the two lines to determine the distance from the intersection at which the second car catches up to the first car:
From the intersection of the two lines, one can see that the distance from the intersection is m 300 .
(d) Draw a vertical line from t = 5 s to the red line and then read the position coordinate of the intersection of this line and the red line to determine the position of the first car when the second car went through the intersection:
From the graph, when the second car passes the intersection, the first car was
ahead m 100 .
62 • Picture the Problem Sally’s velocity relative to the ground (vSG) is the sum of her velocity relative to the moving belt (vSB) and the velocity of the belt relative to the ground (vBG). Joe’s velocity relative to the ground is the same as the velocity of the belt relative to the ground. Let D be the length of the moving sidewalk. Express D in terms of vBG (Joe’s speed relative to the ground):
( ) BGmin2 vD =
Solve for vBG:
min2BGDv =
Express D in terms of vBG + vSG (Sally’s speed relative to the ground):
( )( )
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=
SG
SGBG
min2min1
min1
vD
vvD
Solve for vSG: min2min2min1SG
DDDv =−=
Express D in terms of vBG + 2vSB (Sally’s speed for a fast walk relative to the ground):
( )
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=+=
min23
min22
min22
f
fSBBGf
Dt
DDtvvtD
Solve for tf as time for Sally's fast walk: s0.40
3min2
f ==t
Chapter 2
60
63 •• Picture the Problem The speed of Margaret’s boat relative to the riverbank ( BRv ) is the
sum or difference of the speed of her boat relative to the water ( BWv ) and the speed of
the water relative to the riverbank ( RvW ), depending on whether she is heading with or against the current. Let D be the distance to the marina. Express the total time for the trip:
21tot ttt +=
Express the times of travel with the motor running in terms of D, RvW
and BWv :
h4WRBW
1 =−
=vv
Dt
and
WRBW2 vv
Dt+
=
Express the time required to drift distance D and solve for WRv :
h8
and
h8
WR
WR3
Dv
vDt
=
==
From t1 = 4 h, find BWv : h8
3h8h4h4 WRBW
DDDvDv =+=+=
Solve for t2: h2
h8h83
WRBW2 =
+=
+= DD
Dvv
Dt
Add t1 and t2 to find the total time: h621tot =+= ttt
Acceleration 64 • Picture the Problem In part (a), we can apply the definition of average acceleration to find aav. In part (b), we can find the change in the car’s velocity in one second and add this change to its velocity at the beginning of the interval to find its speed one second later. (a) Apply the definition of average acceleration:
shkm8.70
s3.7km/h48.3km/h80.5
av
⋅=
−=
∆∆
=tva
Motion in One Dimension
61
Convert to m/s2:
2
3av
m/s42.2
s3600h1
shm108.70
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⋅×=a
(b) Express the speed of the car at the end of 4.7 s:
( ) ( )s1
s1
km/h5.80
s7.3s7.4
v
vvv
∆+=
∆+=
Find the change in the speed of the car in 1 s: ( )
km/h8.70
s1sh
km8.70av
=
⎟⎠⎞
⎜⎝⎛
⋅=∆=∆ tav
Substitute and evaluate v(4.7 s): ( )
km/h2.89
km/h7.8km/h5.80s7.4
=
+=v
65 • Picture the Problem Average acceleration is defined as aav = ∆v/∆t. The average acceleration is defined as the change in velocity divided by the change in time:
( ) ( )( ) ( )
2
av
m/s00.2
s5s8m/s5m/s1
−=
−−−
=∆∆
=tva
66 •• Picture the Problem The important concept here is the difference between average acceleration and instantaneous acceleration. (a) The average acceleration is defined as the change in velocity divided by the change in time:
aav = ∆v/∆t
Determine v at t = 3 s, t = 4 s, and t = 5 s:
v(3 s) = 17 m/s v(4 s) = 25 m/s v(5 s) = 33 m/s
Find aav for the two 1-s intervals:
aav(3 s to 4 s) = (25 m/s – 17 m/s)/(1 s) = 8 m/s2 and aav(4 s to 5 s) = (33 m/s – 25 m/s)/(1 s) = 8 m/s2
The instantaneous acceleration is defined as the time derivative of the velocity or the slope of the velocity- versus-time curve:
2m/s00.8==dtdva
Chapter 2
62
(b) The given function was used to plot the following spreadsheet-graph of v-versus-t:
-10
-5
0
5
10
15
20
25
30
35
0 1 2 3 4 5
t (s)
v (m
/s)
67 •• Picture the Problem We can closely approximate the instantaneous velocity by the average velocity in the limit as the time interval of the average becomes small. This is important because all we can ever obtain from any measurement is the average velocity, vav, which we use to approximate the instantaneous velocity v. (a) Find x(4 s) and x(3 s):
x(4 s) = (4)2 – 5(4) + 1 = –3 m and x(3 s) = (3)2 – 5(3) + 1 = −5 m
Find ∆x:
∆x = x(4 s) – x(3 s) = (–3 m) – (–5 m)
= m2
Use the definition of average velocity: vav = ∆x/∆t = (2 m)/(1 s) = m/s 2
(b) Find x(t + ∆t):
x(t + ∆t) = (t + ∆t)2 − 5(t + ∆t) + 1 = (t2 + 2t∆t + (∆t)2) –
5(t + ∆t) + 1
Express x(t + ∆t) – x(t) = ∆x: ( ) ( )252 tttx ∆+∆−=∆
where ∆x is in meters if t is in seconds.
(c) From (b) find ∆x/∆t as ∆t → 0: ( ) ( )
ttt
ttttx
∆+−=∆
∆+∆−=
∆∆
52
52 2
and
Motion in One Dimension
63
( ) 52/lim 0 −=∆∆= →∆ ttxv t
where v is in m/s if t is in seconds.
Alternatively, we can take the derivative of x(t) with respect to time to obtain the instantaneous velocity.
( ) ( ) ( )522
12
−=+=
++==
tbat
btatdtddttdxtv
*68 •• Picture the Problem The instantaneous velocity is dtdx and the acceleration is dtdv . Using the definitions of instantaneous velocity and acceleration, determine v and a:
[ ] BAtCBtAtdtd
dtdxv −=+−== 22
and
[ ] ABAtdtd
drdva 22 =−==
Substitute numerical values for A and B and evaluate v and a:
( )( ) m/s6m/s 16
m/s 6m/s822
2
−=
−=
t
tv
and
( ) 22 m/s 0.16m/s 82 ==a
69 •• Picture the Problem We can use the definition of average acceleration (aav = ∆v/∆t) to find aav for the three intervals of constant acceleration shown on the graph.
(a) Using the definition of average acceleration, find aav for the interval AB:
2AB av, m/s33.3
s3m/s5m/s15
=−
=a
Find aav for the interval BC: 0
s3m/s15m/s15
BC av, =−
=a
Find aav for the interval CE:
2CE av, m/s50.7
s415m/sm/s15
−=−−
=a
(b) Use the formulas for the areas of trapezoids and triangles to find the area under the graph of v as a function of t.
Chapter 2
64
( ) ( ) ( ) ( )( )( )
m0.75
s) m/s)(2 15()sm/s)(215(s) m/s)(3 (15s3m/s15m/s5 21
21
21
EDDCCBBA
=
−++++=∆+∆+∆+∆=∆ →→→→ xxxxx
(c) The graph of displacement, x, as a function of time, t, is shown in the following figure. In the region from B to C the velocity is constant so the x- versus-t curve is a straight line.
0
20
40
60
80
100
0 2 4 6 8 10
t (s)
x (m
)
(d) Reading directly from the figure, we can find the time when the particle is moving the slowest. 0. , thereforeaxis; timethe
crossesgraph thes, 8 D,point At =
=v
t
Constant Acceleration and Free-Fall *70 • Picture the Problem Because the acceleration is constant (–g) we can use a constant-acceleration equation to find the height of the projectile. Using a constant-acceleration equation, express the height of the object as a function of its initial velocity, the acceleration due to gravity, and its displacement:
yavv ∆+= 220
2
Solve for ∆ymax = h:
Because v(h) = 0,
( ) 22
20
20
gv
gvh =
−−
=
From this expression for h we see that the maximum height attained is proportional to the square of the launch speed:
20vh ∝
Motion in One Dimension
65
Therefore, doubling the initial speed gives four times the height:
( )00
42
42
2 20
20
2v vhg
vg
vh =⎟⎟⎠
⎞⎜⎜⎝
⎛==
and ( ) correct. is a
71 • Picture the Problem Because the acceleration of the car is constant we can use constant-acceleration equations to describe its motion. (a) Uing a constant-acceleration equation, relate the velocity to the acceleration and the time:
( )( )sm0.80
s10sm80 20
=
+=+= atvv
(b) sing a constant-acceleration equation, relate the displacement to the acceleration and the time:
200 2
tatvxxx +=−=∆
Substitute numerical values and evaluate ∆x: ( ) ( ) m400s10sm8
21 22 ==∆x
(c) Use the definition of vav: m/s40.0
s10m400
av ==∆∆
=txv
Remarks: Because the area under a velocity-versus-time graph is the displacement of the object, we could solve this problem graphically. 72 • Picture the Problem Because the acceleration of the object is constant we can use constant-acceleration equations to describe its motion. Using a constant-acceleration equation, relate the velocity to the acceleration and the displacement:
xavv ∆+= 220
2
Solve for and evaluate the displacement:
( )( )
m0.50
sm22sm515
2 2
222220
2
=
−=
−=∆
avvx
*73 • Picture the Problem Because the acceleration of the object is constant we can use constant-acceleration equations to describe its motion. Using a constant-acceleration equation, relate the velocity to the acceleration and the displacement:
xavv ∆+= 220
2
Chapter 2
66
Solve for the acceleration: xvva
∆−
=2
20
2
Substitute numerical values and evaluate a:
( )( )
22222
sm6.15m42
sm1015=
−=a
74 • Picture the Problem Because the acceleration of the object is constant we can use constant-acceleration equations to describe its motion. Using a constant-acceleration equation, relate the velocity to the acceleration and the displacement:
xavv ∆+= 220
2
Solve for and evaluate v: ( ) ( )( )
m/s00.3
m1sm42sm1 22
=
+=v
Using the definition of average acceleration, solve for the time: s500.0
sm4sm1sm3
2av
=−
=∆
=a
vt
75 •• Picture the Problem In the absence of air resistance, the ball experiences constant acceleration. Choose a coordinate system with the origin at the point of release and the positive direction upward. (a) Using a constant-acceleration equation, relate the displacement of the ball to the acceleration and the time:
221
0 attvy +=∆
Setting ∆y = 0 (the displacement for a round trip), solve for and evaluate the time required for the ball to return to its starting position:
( ) s08.4m/s9.81m/s2022
20
tripround ===gvt
(b) Using a constant-acceleration equation, relate the final speed of the ball to its initial speed, the acceleration, and its displacement:
yavv ∆+= 220
2top
or, because vtop = 0 and a = −g, ( )Hg−+= 2v0 2
0
Solve for and evaluate H:
( )( ) m4.20
sm81.92sm20
2 2
220 ===g
vH
(c) Using the same constant-acceleration equation with which we began part (a), express the displacement as a function of time:
221
0 attvy +=∆
Motion in One Dimension
67
Substitute numerical values to obtain: ( ) 2
2
2m/s81.9m/s20m15 tt ⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
Solve the quadratic equation for the times at which the displacement of the ball is 15 m:
The solutions are s991.0=t (this
corresponds to passing 15 m on the way up) and s09.3=t (this corresponds to
passing 15 m on the way down). 76 •• Picture the Problem This is a multipart constant-acceleration problem using two different constant accelerations. We’ll choose a coordinate system in which downward is the positive direction and apply constant-acceleration equations to find the required times. (a) Using a constant-acceleration equation, relate the time for the slide to the distance of fall and the acceleration:
212
1100 0 attvhyyy +=−=−=∆
or, because v0 = 0, 212
1 ath =
Solve for t1:
ght 2
1 =
Substitute numerical values and evaluate t1:
( ) s68.9sm81.9
m460221 ==t
(b) Using a constant-acceleration equation, relate the velocity at the bottom of the mountain to the acceleration and time:
1101 tavv += or, because v0 = 0 and a1 = g,
11 gtv =
Substitute numerical values and evaluate v1:
( )( ) sm0.95s68.9sm81.9 21 ==v
(c) Using a constant-acceleration equation, relate the time required to stop the mass of rock and mud to its average speed and the distance it slides:
avvxt ∆
=∆
Because the acceleration is constant:
220
211f1
avvvvvv =
+=
+=
Substitute to obtain:
1
2v
xt ∆=∆
Chapter 2
68
Substitute numerical values and evaluate ∆t:
( ) s168sm0.95
m80002==∆t
*77 •• Picture the Problem In the absence of air resistance, the brick experiences constant acceleration and we can use constant-acceleration equations to describe its motion. Constant acceleration implies a parabolic position-versus-time curve. (a) Using a constant-acceleration equation, relate the position of the brick to its initial position, initial velocity, acceleration, and time into its fall:
( )( ) ( ) 22
221
00
sm91.4sm5m6 tt
tgtvyy
−+=
−++=
The following graph of ( ) ( ) 22sm91.4sm5m6 tty −+= was plotted using a spreadsheet program:
0
1
2
3
4
5
6
7
8
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
t (s)
y (m
)
(b) Relate the greatest height reached by the brick to its height when it falls off the load and the additional height it rises ∆ymax:
max0 ∆yyh +=
Using a constant-acceleration equation, relate the height reached by the brick to its acceleration and initial velocity:
( ) max20
2top 2 ygvv ∆−+=
or, because vtop = 0, ( ) max
20 20 ygv ∆−+=
Solve for ∆ymax:
gvy2
20
max =∆
Substitute numerical values and evaluate ∆ymax:
( )( ) m27.1
sm81.92sm5
2
2
max ==∆y
Motion in One Dimension
69
Substitute to obtain: m7.27m1.27m6∆ max0 =+=+= yyhNote: The graph shown above confirms this result.
(c) Using the quadratic formula, solve for t in the equation obtained in part (a):
( )
( )⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆−±⎟⎟
⎠
⎞⎜⎜⎝
⎛=
⎟⎠⎞
⎜⎝⎛ −
∆−⎟⎠⎞
⎜⎝⎛ −
−±−=
20
0
200
211
22
24
vyg
gv
g
ygvvt
With ybottom = 0 and yo = 6 m or ∆y = –6 m, we obtain:
s73.1=t and t = –0.708 s. Note: The second solution is nonphysical.
(d) Using a constant-acceleration equation, relate the speed of the brick on impact to its acceleration and displacement, and solve for its speed:
ghv 2=
Substitute numerical values and evaluate v:
( )( ) sm9.11m27.7sm81.92 2 ==v
78 •• Picture the Problem In the absence of air resistance, the acceleration of the bolt is constant. Choose a coordinate system in which upward is positive and the origin is at the bottom of the shaft (y = 0). (a) Using a constant-acceleration equation, relate the position of the bolt to its initial position, initial velocity, and fall time:
( ) 221
00
bottom 0
tgtvy
y
−++=
=
Solve for the position of the bolt when it came loose:
221
00 gttvy +−=
Substitute numerical values and evaluate y0:
( )( ) ( )( )m1.26
s3sm81.9s3sm6 2221
0
=
+−=y
(b) Using a constant-acceleration equation, relate the speed of the bolt to its initial speed, acceleration, and fall time:
atvv += 0
Chapter 2
70
Substitute numerical values and evaluate v :
( )( ) sm4.23s3sm81.9sm6 2 −=−=v and
sm23.4 =v
*79 •• Picture the Problem In the absence of air resistance, the object’s acceleration is constant. Choose a coordinate system in which downward is positive and the origin is at the point of release. In this coordinate system, a = g and y = 120 m at the bottom of the fall. Express the distance fallen in the last second in terms of the object’s position at impact and its position 1 s before impact:
impactbefores1secondlast m120 yy −=∆ (1)
Using a constant-acceleration equation, relate the object’s position upon impact to its initial position, initial velocity, and fall time:
221
00 gttvyy ++= or, because y0 = 0 and v0 = 0,
2fall2
1 gty =
Solve for the fall time:
gyt 2
fall =
Substitute numerical values and evaluate tfall:
( ) s95.4m/s81.9
m12022fall ==t
We know that, one second before impact, the object has fallen for 3.95 s. Using the same constant-acceleration equation, calculate the object’s position 3.95 s into its fall:
( )( )m4.76
s95.3m/s81.9s) (3.95 2221
=
=y
Substitute in equation (1) to obtain: m6.43m4.76m120secondlast =−=∆y 80 •• Picture the Problem In the absence of air resistance, the acceleration of the object is constant. Choose a coordinate system with the origin at the point of release and downward as the positive direction. Using a constant-acceleration equation, relate the height to the initial and final velocities and the acceleration; solve for the height:
yavv ∆+= 220
2f
or, because v0 = 0,
gvh2
2f= (1)
Motion in One Dimension
71
Using the definition of average velocity, find the average velocity of the object during its final second of fall:
sm38s1m38
2fs1-f
av ==∆∆
=+
=tyvv
v
Express the sum of the final velocity and the velocity 1 s before impact:
( ) sm76sm382fs1-f ==+ vv
From the definition of acceleration, we know that the change in velocity of the object, during 1 s of fall, is 9.81 m/s:
sm81.9s1-ff =−=∆ vvv
Add the equations that express the sum and difference of vf – 1 s and vf and solve for vf:
sm9.422
sm81.9sm76f =
+=v
Substitute in equation (1) and evaluate h:
( )( ) m8.93
sm81.92sm9.42
2
2
==h
*81 • Picture the Problem In the absence of air resistance, the acceleration of the stone is constant. Choose a coordinate system with the origin at the bottom of the trajectory and the upward direction positive. Let 21-fv be the speed one-half second before impact
and fv the speed at impact. Using a constant-acceleration equation, express the final speed of the stone in terms of its initial speed, acceleration, and displacement:
yavv ∆+= 220
2f
Solve for the initial speed of the stone:
ygvv ∆+= 22f0 (1)
Find the average speed in the last half second:
sm90s0.5
m452
second halflast f21-fav
=
=∆
∆=
+=
txvv
v
and ( ) sm180sm902f21-f ==+ vv
Using a constant-acceleration equation, express the change in speed of the stone in the last half second in terms of the acceleration and the elapsed time; solve for the change in its speed:
( )( )sm91.4
s5.0sm81.9 2
21-ff
==
∆=−=∆ tgvvv
Chapter 2
72
Add the equations that express the sum and difference of vf – ½ and vf and solve for vf:
sm5.922
sm91.4sm180f =
+=v
Substitute in equation (1) and evaluate v0:
( ) ( ) ( )sm1.68
m200sm81.92sm5.92 220
=
−+=v
Remarks: The stone may be thrown either up or down from the cliff and the results after it passes the cliff on the way down are the same. 82 •• Picture the Problem In the absence of air resistance, the acceleration of the object is constant. Choose a coordinate system in which downward is the positive direction and the object starts from rest. Apply constant-acceleration equations to find the average velocity of the object during its descent. Express the average velocity of the falling object in terms of its initial and final velocities:
2f0
avvvv +
=
Using a constant-acceleration equation, express the displacement of the object during the 1st second in terms of its acceleration and the elapsed time:
hgty 0.4 m 91.42
2
second1st ===∆
Solve for the displacement to obtain:
h = 12.3 m
Using a constant-acceleration equation, express the final velocity of the object in terms of its initial velocity, acceleration, and displacement:
220
2f ygvv ∆+=
or, because v0 = 0, 2f ygv ∆=
Substitute numerical values and evaluate the final velocity of the object:
( ) ( ) sm5.15m3.12sm81.92 2f ==v
Substitute in the equation for the average velocity to obtain: sm77.7
2sm5.150
av =+
=v
83 •• Picture the Problem This is a three-part constant-acceleration problem. The bus starts from rest and accelerates for a given period of time, and then it travels at a constant velocity for another period of time, and, finally, decelerates uniformly to a stop. The pictorial representation will help us organize the information in the problem and develop our solution strategy.
Motion in One Dimension
73
(a) Express the total displacement of the bus during the three intervals of time.
( ) ( )( )ends37
s37s12s120total
→∆+→∆+→∆=∆
xxxx
Using a constant-acceleration equation, express the displacement of the bus during its first 12 s of motion in terms of its initial velocity, acceleration, and the elapsed time; solve for its displacement:
( ) 221
0s120 attvx +=→∆ or, because v0 = 0,
( ) m108s120 221 ==→∆ atx
Using a constant-acceleration equation, express the velocity of the bus after 12 seconds in terms of its initial velocity, acceleration, and the elapsed time; solve for its velocity at the end of 12 s:
( )( )m/s18
s12m/s5.1 2s1200s12
=
=∆+= → tavv
During the next 25 s, the bus moves with a constant velocity. Using the definition of average velocity, express the displacement of the bus during this interval in terms of its average (constant) velocity and the elapsed time:
( ) ( )( )m450
s25m/s18s37s12 s12
=
=∆=→∆ tvx
Because the bus slows down at the same rate that its velocity increased during the first 12 s of motion, we can conclude that its displacement during this braking period is the same as during its acceleration period and the time to brake to a stop is equal to the time that was required for the bus to accelerate to its cruising speed of 18 m/s. Hence:
( ) m108s49s37 =→∆x
Add the displacements to find the distance the bus traveled:
m 666
m 108 m 450 m 108total
=
++=∆x
Chapter 2
74
(b) Use the definition of average velocity to calculate the average velocity of the bus during this trip:
sm6.13s49m666total
av ==∆
∆=
txv
Remarks: One can also solve this problem graphically. Recall that the area under a velocity as a function-of-time graph equals the displacement of the moving object. *84 •• Picture the Problem While we can solve this problem analytically, there are many physical situations in which it is not easy to do so and one has to rely on numerical methods; for example, see the spreadsheet solution shown below. Because we’re neglecting the height of the release point, the position of the ball as a function of time is given by 2
21
0 gttvy −= . The formulas used to calculate the quantities in the columns are as follows:
Cell Content/Formula Algebraic FormB1 20 v0 B2 9.81 g B5 0 t B6 B5 + 0.1 tt ∆+ C6 $B$1*B6 − 0.5*$B$2*B6^2 2
21
0 gttv − (a)
A B C 1 v0 = 20 m/s 2 g = 9.81 m/s^2 3 t height 4 (s) (m) 5 0.0 0.00 6 0.1 1.95 7 0.2 3.80
44 3.9 3.39 45 4.0 1.52 46 4.1 −0.45
The graph shown below was generated from the data in the previous table. Note that the maximum height reached is a little more than 20 m and the time of flight is about 4 s.
Motion in One Dimension
75
0
5
10
15
20
25
0 1 2 3 4
t (s )
heig
ht (m
)
(b) In the spreadsheet, change the value in cell B1 from 20 to 10. The graph should automatically update. With an initial velocity of 10 m/s, the maximum height achieved is approximately 5 m and the time-of-flight is approximately 2 s.
0
1
2
3
4
5
6
0.0 0.5 1.0 1.5 2.0
t (s)
heig
ht (m
)
*85 •• Picture the Problem Because the accelerations of both Al and Bert are constant, constant-acceleration equations can be used to describe their motions. Choose the origin of the coordinate system to be where Al decides to begin his sprint. (a) Using a constant-acceleration equation, relate Al's initial velocity, his acceleration, and the time to reach the end of the trail to his
221
0 attvx +=∆
Chapter 2
76
displacement in reaching the end of the trail:
Substitute numerical values to obtain:
2221 )m/s5.0( m/s) (0.75m35 tt +=
Solve for the time required for Al to reach the end of the trail:
s4.10=t
(b) Using constant-acceleration equations, express the positions of Bert and Al as functions of time. At the instant Al turns around at the end of the trail, t = 0. Also, x = 0 at a point 35 m from the end of the trail:
( )and
m/s75.0Bert,0Bert txx +=
( )( )t
txxm/s0.85m35m/s85.00,AlAl
−=
−=
Calculate Bert’s position at t = 0. At that time he has been running for 10.4 s:
( )( ) m80.7s4.10m/s75.0Bert,0 ==x
Because Bert and Al will be at the same location when they meet, equate their position functions and solve for t:
( ) ( )tt m/s85.0m35m/s75.0m80.7 −=+and
s0.17=t
To determine the elapsed time from when Al began his accelerated run, we need to add 10.4 s to this time:
s4.27s4.10s0.17start =+=t
(c) Express Bert’s distance from the end of the trail when he and Al meet:
Al meets he until runsBert
Bert,0 trailof end m35d
xd−
−=
Substitute numerical values and evaluate dend of trail: ( )
m5.14
m/s0.75s)17(m80.7m35 trailof end
=
−−=d
86 •• Picture the Problem Generate two curves on one graph with the first curve representing Al's position as a function of time and the second curve representing Bert’s position as a function of time. Al’s position, as he runs toward the end of the trail, is given by
2Al2
10Al tatvx += and Bert’s position by tvxx BertBert0,Bert += . Al’s position, once he’s
reached the end of the trail and is running back toward Bert, is given by ( )s5.10AlAl,0Al −+= tvxx . The coordinates of the intersection of the two curves give the time and place where they meet. A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows:
Cell Content/Formula Algebraic Form
Motion in One Dimension
77
B1 0.75 v0 B2 0.50 aAl B3 −0.85 t
B10 B9 + 0.25 t + ∆t C10 $B$1*B10 + 0.5*$B$2*B10^2 2
Al21
0 tatv + C52 $C$51 + $B$3*(B52 − $B$51) ( )s5.10AlAl,0 −+ tvx F10 $F$9 + $B$1*B10 tvx BertBert0, +
(b) and (c)
A B C D E F
1 v0 = 0.75 m/s 2 a(Al) = 0.5 m/s^2 3 v(Al) = −0.85 m/s 4 5 t (s) x (m) x (m) 6 7 8 Al Bert 9 0.00 0.00 0.00
10 0.25 0.20 0.19 11 0.50 0.44 0.38
49 10.00 32.50 7.50 50 10.25 33.95 7.69 51 10.50 35.44 *Al reaches 7.88 52 10.75 35.23 end of trail 8.06 53 11.00 35.01 and starts 8.25 54 11.25 34.80 back toward 8.44 55 11.50 34.59 Bert 8.63
56 11.75 34.38 8.81 119 27.50 20.99 20.63 120 27.75 20.78 20.81 121 28.00 20.56 21.00 122 28.25 20.35 21.19 127 29.50 19.29 22.13 128 29.75 19.08 22.31 129 30.00 18.86 22.50
The graph shown below was generated from the spreadsheet; the positions of both Al and Bert were calculated as functions of time. The dashed curve shows Al’s position as a function of time for the two parts of his motion. The solid line that is linear from the origin shows Bert’s position as a function of time.
Chapter 2
78
0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30
t (s)
Posi
tion
on tr
ail (
m)
AlBert
Note that the spreadsheet and the graph (constructed from the spreadsheet data) confirm the results in Problem 85 by showing Al and Bert meeting at about 14.5 m from the end of the trail after an elapsed time of approximately 28 s. 87 •• Picture the Problem This is a two-part constant-acceleration problem. Choose a coordinate system in which the upward direction is positive. The pictorial representation will help us organize the information in the problem and develop our solution strategy.
(a) Express the highest point the rocket reaches, h, as the sum of its displacements during the first two stages of its flight:
stage2ndstage1st xxh ∆+∆=
Using a constant-acceleration equation, express the altitude reached in the first stage in terms of the rocket’s initial velocity, acceleration, and burn time; solve for the first stage altitude:
m 6250s) 25)(m/s (20 22
21
2stagst12
100stage1st
=
=
++= tatvxx e
Using a constant-acceleration equation, express the velocity of the rocket at the end of its first stage in terms of its initial velocity, acceleration, and displacement; calculate its end-of-first-stage velocity:
m/s 500 s) 25)(m/s (20 2
stagest10stagest1
==
+= tavv
Motion in One Dimension
79
Using a constant-acceleration equation, express the final velocity of the rocket during the remainder of its climb in terms of its shut-off velocity, free-fall acceleration, and displacement; solve for its displacement:
stage2ndstagend22shutoff
2pointhighest 2 yavv ∆+=
and, because vhighest point = 0, ( )( )m102742.1
sm81.92sm500
2
4
2
22shutoff
stagend2
×=
=−
−=∆
gvy
Substitute in the expression for the total height to obtain:
km0.19m1027.1m6250 4 =×+=h
(b) Express the total time the rocket is in the air in terms of the three segments of its flight: descentsegment2nd
descentsegment2ndclimbpoweredtotal
s25 tt
tttt
∆+∆+=
∆+∆+∆=∆
Express ∆t2nd segment in terms of the rocket’s displacement and average velocity:
velocityAveragentDisplaceme
segment2nd =∆t
Substitute numerical values and evaluate ∆t2nd segment:
s 50.97
2m/s5000
m102742.1 4
segment2nd =⎟⎠⎞
⎜⎝⎛ +
×=∆t
Using a constant-acceleration equation, relate the fall distance to the descent time:
( )2descent2
10 tgtvy ∆+=∆
or, because v0 = 0, ( )2
descent21 tgy ∆=∆
Solve for ∆tdescent: g
yt ∆=∆
2descent
Substitute numerical values and evaluate ∆tdescent:
( )s2.62
m/s9.81m1090.12
2
4
descent =×
=∆t
Substitute and calculate the total time the rocket is in the air: s18min2
s138s62.2s50.97s25
=
=++=∆t
(c) Using a constant-acceleration equation, express the impact velocity of the rocket in terms of its initial downward velocity, acceleration under free-fall, and time of descent; solve for its impact velocity:
descent0impact tgvv ∆+= and, because v0 = 0,
( )( )m/s610
s2.62m/s81.9 2impact
=
=∆= tgv
Chapter 2
80
88 •• Picture the Problem In the absence of air resistance, the acceleration of the flowerpot is constant. Choose a coordinate system in which downward is positive and the origin is at the point from which the flowerpot fell. Let t = time when the pot is at the top of the window, and t + ∆t the time when the pot is at the bottom of the window. To find the distance from the ledge to the top of the window, first find the time ttop that it takes the pot to fall to the top of the window. Using a constant-acceleration equation, express the distance y below the ledge from which the pot fell as a function of time:
221
00
221
00
,0=and = Since
gty
yvgaattvyy
=
=++=
Express the position of the pot as it reaches the top of the window:
2top2
1top gty =
Express the position of the pot as it reaches the bottom of the window:
( )2windowtop2
1bottom ttgy ∆+=
where ∆twindow = t top − tbottom
Subtract ybottom from ytop to obtain an expression for the displacement ∆ywindow of the pot as it passes the window:
( )[ ]( )[ ]2
windowwindowtop21
2top
2windowtop2
1window
2 tttg
tttgy
∆+∆=
−∆+=∆
Solve for ttop: ( )
window
2window
window
top 2g
2
t
ty
t∆
∆−∆
=
Substitute numerical values and evaluate ttop:
( ) ( )
( ) s839.1s2.02
s2.0m/s9.81m42 2
2
top =−
=t
Substitute this value for ttop to obtain the distance from the ledge to the top of the window:
m 18.4s) )(1.939m/s (9.81 2221
top ==y
*89 •• Picture the Problem The acceleration of the glider on the air track is constant. Its average acceleration is equal to the instantaneous (constant) acceleration. Choose a coordinate system in which the initial direction of the glider’s motion is the positive direction. Using the definition of acceleration, express the average acceleration of the glider in terms of the glider’s velocity change and the elapsed time:
tvaa
∆∆
== av
Motion in One Dimension
81
Using a constant-acceleration equation, express the average velocity of the glider in terms of the displacement of the glider and the elapsed time:
20
avvv
txv +
=∆∆
=
Solve for and evaluate the initial velocity:
( )
cm/s 0.40
cm/s) 15(s8cm10022
0
=
−−=−∆∆
= vtxv
Substitute this value of v0 and evaluate the average acceleration of the glider:
2cm/s 6.88s 8
cm/s) (40 cm/s 15
−=
−−=a
90 •• Picture the Problem In the absence of air resistance, the acceleration of the rock is constant and its motion can be described using the constant-acceleration equations. Choose a coordinate system in which the downward direction is positive and let the height of the cliff, which equals the displacement of the rock, be represented by h. Using a constant-acceleration equation, express the height h of the cliff in terms of the initial velocity of the rock, acceleration, and time of fall:
221
0 attvy +=∆ or, because v0 = 0, a = g, and ∆y = h,
221 gth =
Using this equation, express the displacement of the rock during the
a) first two-thirds of its fall, and
221
32 gth = (1)
b) its complete fall in terms of the time required for it to fall this distance.
( )221 s1+= tgh (2)
Substitute equation (2) in equation (1) to obtain a quadratic equation in t:
t2 – (4 s)t – 2 s2 = 0
Solve for the positive root:
s45.4=t
Evaluate ∆t = t + 1 s:
∆t = 4.45 s + 1 s = 5.45 s
Substitute numerical values in equation (2) and evaluate h:
( )( ) m146s45.5m/s81.9 2221 ==h
Chapter 2
82
91 ••• Picture the Problem Assume that the acceleration of the car is constant. The total distance the car travels while stopping is the sum of the distances it travels during the driver’s reaction time and the time it travels while braking. Choose a coordinate system in which the positive direction is the direction of motion of the automobile and apply a constant-acceleration equation to obtain a quadratic equation in the car’s initial speed v0. (a) Using a constant-acceleration equation, relate the velocity of the car to its initial velocity, acceleration, and displacement during braking:
brk20
2 2 xavv ∆+= or, because the final velocity is zero,
brk20 20 xav ∆+=
Solve for the distance traveled during braking: a
vx2
20
brk −=∆
Express the total distance traveled by the car as the sum of the distance traveled during the reaction time and the distance traveled while slowing down:
avtv
xxx
2
20
react0
brkreacttot
−∆=
∆+∆=∆
Rearrange this quadratic equation to obtain:
022 tot0react20 =∆+∆− xavtav
Substitute numerical values and simplify to obtain:
( )( )( ) 0m)4(m/s72
s5.0m/s7220
220
=−+
−− vv
or ( ) 0s/m56m/s7 22
020 =−+ vv
Solve the quadratic equation for the positive root to obtain:
m/s7613558.40 =v
Convert this speed to mi/h:: ( )
mi/h7.10
m/s0.477mi/h1m/s7613558.40
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=v
(b) Find the reaction-time distance:
m 2.38s) m/s)(0.5 (4.76react0react
==∆=∆ tvx
Express and evaluate the ratio of the reaction distance to the total distance: 595.0
m 4m 2.38
tot
react ==∆∆
xx
Motion in One Dimension
83
92 •• Picture the Problem Assume that the accelerations of the trains are constant. Choose a coordinate system in which the direction of the motion of the train on the left is the positive direction. Take xo = 0 as the position of the train on the left at t = 0. Using a constant-acceleration equation, relate the distance the train on the left will travel before the trains pass to its acceleration and the time-to-passing:
( )( ) 22
22212
L21
L
sm7.0
sm4.1
t
ttax
=
==
Using a constant-acceleration equation, relate the position of the train on the right to its initial velocity, position, and acceleration:
( ) 2221
2R2
1
sm2.2m40
m40
t
taxR
−=
−=
Equate xL and xR and solve for t:
s71.4and
1.1407.0 22
=
−=
t
tt
Find the position of the train initially on the left, xL, as they pass:
m 15.6s) 71.4)(m/s (1.4 2221
L ==x
Remarks: One can also solve this problem by graphing the functions for xL and xR. The coordinates of the intersection of the two curves give one the time-to-passing and the distance traveled by the train on the left.
93 •• Picture the Problem In the absence of air resistance, the acceleration of the stones is constant. Choose a coordinate system in which the downward direction is positive and the origin is at the point of release of the stones. Using constant-acceleration equations, relate the positions of the two stones to their initial positions, accelerations, and time-of-fall:
221
2
221
1
s)1.6(and
−=
=
tgx
gtx
Express the difference between x1 and x2:
x1 − x2 = 36 m
Substitute for x1 and x2 to obtain: ( )2212
21 s6.1m36 −−= tggt
Chapter 2
84
Solve this equation for the time t at which the stones will be separated by 36 m:
t = 3.09 s
Substitute this result in the expression for x2 and solve for x2:
( )( )m9.10
s1.6s3.09sm81.9 2221
2
=
−=x
*94 •• Picture the Problem The acceleration of the police officer’s car is positive and constant and the acceleration of the speeder’s car is zero. Choose a coordinate system such that the direction of motion of the two vehicles is the positive direction and the origin is at the stop sign. Express the velocity of the car in terms of the distance it will travel until the police officer catches up to it and the time that will elapse during this chase:
car
caughtcar t
dv =
Letting t1 be the time during which she accelerates and t2 the time of travel at v1 = 110 km/h, express the time of travel of the police officer:
21office ttt r +=
Convert 110 km/h into m/s:
( )m/s30.6
s) h/3600 (1m/km) 10(km/h 110 31
==v
Express and evaluate t1: s 94.4
m/s 6.2m/s 6.30
2motorcycle
11 ===
avt
Express and evaluate d1: m 75.6 s) m/s)(4.94 (30.62
1112
11 === tvd
Determine d2:
m1324.4
m 75.6 m 1400 1caught2
=
−=−= ddd
Express and evaluate t2: s 43.3
m/s 30.6m 1324.4
1
22 ===
vdt
Express the time of travel of the car:
tcar = 2.0 s + 4.93 s + 43.3 s = 50.2 s
Motion in One Dimension
85
Finally, find the speed of the car:
( )
mi/h62.4
m/s0.447
mi/h1m/s 27.9
m/s 27.9 s 50.2m 1400
car
caughtcar
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
===t
dv
95 •• Picture the Problem In the absence of air resistance, the acceleration of the stone is constant. Choose a coordinate system in which downward is positive and the origin is at the point of release of the stone and apply constant-acceleration equations. Using a constant-acceleration equation, express the height of the cliff in terms of the initial position of the stones, acceleration due to gravity, and time for the first stone to hit the water:
212
1 gth =
Express the displacement of the second stone when it hits the water in terms of its initial velocity, acceleration, and time required for it to hit the water.
222
12022 gttvd +=
where t2 = t1 – 1.6 s.
Because the stones will travel the same distances before hitting the water, equate h and d2 and solve for t.
222
1202
212
1 gttvgt += or
( ) ( )( )( ) ( )2
12
21
121
221
s6.1m/s81.9
s6.1m/s32m/s81.9
−+
−=
t
tt
Solve for t1 to obtain:
s37.21 =t
Substitute for t1 and evaluate h:
m 27.6s) 37.2)(m/s (9.81 2221 ==h
96 ••• Picture the Problem Assume that the acceleration of the passenger train is constant. Let xp = 0 be the location of the passenger train engine at the moment of sighting the freight train’s end; let t = 0 be the instant the passenger train begins to slow (0.4 s after the passenger train engineer sees the freight train ahead). Choose a coordinate system in which the direction of motion of the trains is the positive direction and use constant-acceleration equations to express the positions of the trains in terms of their initial positions, speeds, accelerations, and elapsed time. (a) Using constant-acceleration equations, write expressions for the positions of the front of the passenger train and the rear of the
( )( ) s4.0m/s29 221
p attx −+=
( ) ( ) s)0.4 (t m/s6m360f ++=x where xp and xf are in meters if t is in
Chapter 2
86
freight train, xp and xf, respectively: seconds.
Equate xf = xp to obtain an equation for t:
( ) 0m8.350m/s23221 =+− tat
Find the discriminant ( )AC4B2 −=D of this equation:
( ) ( )m8.3502
4m/s23 2⎟⎠⎞
⎜⎝⎛−=
aD
The equation must have real roots if it is to describe a collision. The necessary condition for real roots is that the discriminant be greater than or equal to zero:
If (23 m/s)2 – a (701.6 m) ≥ 0, then 2m/s754.0≤a
(b) Express the relative speed of the trains:
fppfrel vvvv −== (1)
Repeat the previous steps with a = 0.754 m/s2 and a 0.8 s reaction time. The quadratic equation that guarantees real roots with the longer reaction time is:
( ) ( )0m6.341
m/s23m/s754.0 2221
=+− tt
Solve for t to obtain the collision times:
t = 25.6 s and t = 35.4 s
Note that at t = 35.4 s, the trains have already collided; therefore this root is not a meaningful solution to our problem.
Note: In the graph shown below, you will see why we keep only the smaller of the two solutions.
Now we can substitute our value for t in the constant-acceleration equation for the passenger train and solve for the distance the train has moved prior to the collision:
xp = (29 m/s)(25.6 s + 0.8 s) – ( 0.377 m/s2)(25.6 s)2
= m518
Find the speeds of the two trains: vp = vop + at = (29 m/s) + (–0.754 m/s2)(25.5 s) = 9.77 m/s and vf = vof = 6 m/s
Substitute in equation (1) and evaluate the relative speed of the trains:
m/s77.3m/s6.00-m/s77.9rel ==v
The graph shows the location of both trains as functions of time. The solid straight line is for the constant velocity freight train; the dashed curves are for the passenger train, with reaction times of 0.4 s for the lower curve and 0.8 s for the upper curve.
Motion in One Dimension
87
0
100
200
300
400
500
600
700
0 10 20 30 40
t (s)
x (m
)0.4 s reaction time
Freight train
0.8 s reaction time
Remarks: A collision occurs the first time the curve for the passenger train crosses the curve for the freight train. The smaller of two solutions will always give the time of the collision. 97 • Picture the Problem In the absence of air resistance, the acceleration of an object near the surface of the earth is constant. Choose a coordinate system in which the upward direction is positive and the origin is at the surface of the earth and apply constant-acceleration equations. Using a constant-acceleration equation, relate the velocity to the acceleration and displacement:
yavv ∆+= 220
2 or, because v = 0 and a = −g,
ygv ∆−= 20 20
Solve for the height to which the projectile will rise: g
vyh2
20=∆=
Substitute numerical values and evaluate h:
( )( ) km59.4
m/s81.92m/s300
2
2
==h
*98 • Picture the Problem This is a composite of two constant accelerations with the acceleration equal to one constant prior to the elevator hitting the roof, and equal to a different constant after crashing through it. Choose a coordinate system in which the upward direction is positive and apply constant-acceleration equations. (a) Using a constant-acceleration equation, relate the velocity to the acceleration and displacement:
yavv ∆+= 220
2 or, because v = 0 and a = −g,
ygv ∆−= 20 20
Chapter 2
88
Solve for v0:
ygv ∆= 20
Substitute numerical values and evaluate v0:
( )( ) m/s443m10m/s81.92 420 ==v
(b) Find the velocity of the elevator just before it crashed through the roof:
vf = 2 × 443 m/s = 886 m/s
Using the same constant-acceleration equation, this time with v0 = 0, solve for the acceleration:
yav ∆= 22
Substitute numerical values and evaluate a:
( )( )
g
a
267
m/s1062.2m1502
m/s886 232
=
×==
99 •• Picture the Problem Choose a coordinate system in which the upward direction is positive. We can use a constant-acceleration equation to find the beetle’s velocity as its feet lose contact with the ground and then use this velocity to calculate the height of its jump. Using a constant-acceleration equation, relate the beetle’s maximum height to its launch velocity, velocity at the top of its trajectory, and acceleration once it is airborne; solve for its maximum height:
( )hgv
yavv
−+=
∆+=
2
22launch
fallfree2launch
2pointhighest
Because vhighest point = 0: g
vh2
2launch=
Now, in order to determine the beetle’s launch velocity, relate its time of contact with the ground to its acceleration and push-off distance:
launch20
2launch 2 yavv ∆+=
or, because v0 = 0,
launch2launch 2 yav ∆=
Substitute numerical values and evaluate 2
launchv : ( )( )( )
22
222launch
/sm1.47
m106.0m/s81.94002
=
×= −v
Substitute to find the height to which the beetle can jump: ( ) m40.2
m/s81.92/sm1.47
2 2
222launch ===
gvh
Motion in One Dimension
89
Using a constant-acceleration equation, relate the velocity of the beetle at its maximum height to its launch velocity, free-fall acceleration while in the air, and time-to-maximum height:
heightmax.launchheightmax.
0
orgtvv
atvv
−=
+=
and, because vmax height = 0, heightmax.launch0 gtv −=
Solve for tmax height:
gvt launch
heightmax =
For zero displacement and constant acceleration, the time-of-flight is twice the time-to-maximum height:
( ) s40.1m/s81.9
m/s86.62
22
2
launchheightmax.flight
==
==g
vtt
100 • Picture the Problem Because its acceleration is constant we can use the constant- acceleration equations to describe the motion of the automobile. Using a constant-acceleration equation, relate the velocity to the acceleration and displacement:
xavv ∆+= 220
2 or, because v = 0,
xav ∆+= 20 20
Solve for the acceleration a: x
va∆
−=
2
20
Substitute numerical values and evaluate a: ( )( )( )[ ]
( )2
23
sm41.7
m502s 3600h1kmm10hkm98
−=
−=a
Express the ratio of a to g and then solve for a:
755.0sm81.9sm41.72
2
−=−
=ga
and ga 755.0−=
Using the definition of average acceleration, solve for the stopping time:
tva
∆∆
=av ⇒ avavt ∆
=∆
Substitute numerical values and evaluate ∆t:
( )( )( )
s67.3
sm7.41s3600h1kmm10hkm98
2
3
=
−−
=∆t
Chapter 2
90
*101 •• Picture the Problem In the absence of air resistance, the puck experiences constant acceleration and we can use constant-acceleration equations to describe its position as a function of time. Choose a coordinate system in which downward is positive, the particle starts from rest (vo = 0), and the starting height is zero (y0 = 0).
Using a constant-acceleration equation, relate the position of the falling puck to the acceleration and the time. Evaluate the y-position at successive equal time intervals ∆t, 2∆t, 3∆t, etc:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
etc.
)16(2
42
)9(2
32
)4(2
22
22
224
223
222
221
tgtgy
tgtgy
tgtgy
tgtgy
∆−
=∆−
=
∆−
=∆−
=
∆−
=∆−
=
∆−
=∆−
=
Evaluate the changes in those positions in each time interval:
( )
( )
( )
( )
etc.
72
7
52
5
32
3
20
102
3443
102
2332
102
1221
2110
ytgyyy
ytgyyy
ytgyyy
tgyy
∆=∆⎟⎠⎞
⎜⎝⎛ −
=−=∆
∆=∆⎟⎠⎞
⎜⎝⎛ −
=−=∆
∆=∆⎟⎠⎞
⎜⎝⎛ −
=−=∆
∆⎟⎠⎞
⎜⎝⎛ −
=−=∆
102 •• Picture the Problem Because the particle moves with a constant acceleration we can use the constant-acceleration equations to describe its motion. A pictorial representation will help us organize the information in the problem and develop our solution strategy.
Using a constant-acceleration equation, find the position x at t = 6 s. To find x at t = 6 s, we first need to find v0 and x0:
221
00 attvxx ++=
Motion in One Dimension
91
Using the information that when t = 4 s, x = 100 m, obtain an equation in x0 and v0:
( )( ) ( )( )22
21
00 s4m/s3s4
m100s4
++=
=
vx
x
or ( ) m76s4 00 =+ vx
Using the information that when t = 6 s, v = 15 m/s, obtain a second equation in x0 and v0:
( ) ( )( )s6m/s3s6 20 += vv
Solve for v0 to obtain:
v0 = −3 m/s
Substitute this value for v0 in the previous equation and solve for x0:
x0 = 88 m
Substitute for x0 and v0 and evaluate x at t = 6 s:
( ) ( ) ( ) ( ) ( ) m124s6m/s3s6m/s3m88s6 2221 =+−+=x
*103 •• Picture the Problem We can use constant-acceleration equations with the final velocity v = 0 to find the acceleration and stopping time of the plane. (a) Using a constant-acceleration equation, relate the known velocities to the acceleration and displacement:
xavv ∆+= 220
2
Solve for a:
xv
xvva
∆−
=∆−
=22
20
20
2
Substitute numerical values and evaluate a:
( )( )
22
sm7.25m702
sm60−=
−=a
(b) Using a constant-acceleration equation, relate the final and initial speeds of the plane to its acceleration and stopping time:
tavv ∆+= 0
Solve for and evaluate the stopping time: s33.2
m/s7.25sm6002
0 =−
−=
−=∆
avvt
104 •• Picture the Problem This is a multipart constant-acceleration problem using three different constant accelerations (+2 m/s2 for 20 s, then zero for 20 s, and then –3 m/s2 until the automobile stops). The final velocity is zero. The pictorial representation will help us organize the information in the problem and develop our solution strategy.
Chapter 2
92
Add up all the displacements to get the total:
∆x03 = ∆x01 + ∆x12 + ∆x23
Using constant-acceleration formulas, find the first displacement:
m 400=s) 20)(m/s 2(0 2221
21012
11001
+=
+=∆ tatvx
The speed is constant for the second displacement. Find the displacement:
( )and0 vwhere 10110101
12112
tatavttvx
+=+=−=∆
( )m 800=s) s)(20 20)(m/s 2( 2
1210112
=
−=∆ tttax
Find the displacement during the braking interval: and0 and = where
2
310112
232322
23
==∆+=
vtavvxavv
( ) [ ]( )
m267sm32
s) 20(m/s) 2(2
02
2
23
2101
2
23
=−
−=
−=∆
atax
Add the displacements to get the total: km 1.47=
m 146723120103 =∆+∆+∆=∆ xxxx
Remarks: Because the area under the curve of a velocity-versus-time graph equals the displacement of the object experiencing the acceleration, we could solve this problem by plotting the velocity as a function of time and finding the area bounded by it and the time axis. *105 •• Picture the Problem Note: No material body can travel at speeds faster than light. When one is dealing with problems of this sort, the kinematic formulae for displacement, velocity and acceleration are no longer valid, and one must invoke the special theory of relativity to answer questions such as these. For now, ignore such subtleties. Although the formulas you are using (i.e., the constant- acceleration equations) are not quite correct, your answer to part (b) will be wrong by about 1%. (a) This part of the problem is an exercise in the conversion of units. Make use of the fact that 1 c⋅y = 9.47×1015 m and 1 y = 3.16×107 s:
( ) ( )( )
22
27
152 y/y03.1
y1s1016.3
m1047.9y1m/s81.9 ⋅=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ×⎟⎟⎠
⎞⎜⎜⎝
⎛×
⋅= ccg
Motion in One Dimension
93
(b) Let t1/2 represent the time it takes to reach the halfway point. Then the total trip time is:
t = 2 t1/2 (1)
Use a constant- acceleration equation to relate the half-distance to Mars ∆x to the initial speed, acceleration, and half-trip time t1/2 :
2212
10 attvx +=∆
Because v0 = 0 and a = g:
axt ∆
=2
2/1
The distance from Earth to Mars at closest approach is 7.8 × 1010 m. Substitute numerical values and evaluate t1/2 :
( ) s1092.8m/s81.9
m109.32 42
10
2/1 ×=×
=t
Substitute for t1/2 in equation (1) to obtain:
( ) d2s1078.1s1092.82 54 ≈×=×=t
Remarks: Our result in part (b) seems remarkably short, considering how far Mars is and how low the acceleration is. 106 • Picture the Problem Because the elevator accelerates uniformly for half the distance and uniformly decelerates for the second half, we can use constant-acceleration equations to describe its motion Let t1/2 = 40 s be the time it takes to reach the halfway mark. Use the constant-acceleration equation that relates the acceleration to the known variables to obtain:
221
0 attvy +=∆ or, because v0 = 0,
221 aty =∆
Solve for a: 2
2/1
2t
ya ∆=
Substitute numerical values and evaluate a:
( )( )( )( )
g
a
0228.0
m/s223.0s40
ftm/3.2811ft11732 22
21
=
==
107 •• Picture the Problem Because the acceleration is constant, we can describe the motions of the train using constant-acceleration equations. Find expressions for the distances traveled, separately, by the train and the passenger. When are they equal? Note that the train is accelerating and the passenger runs at a constant minimum velocity (zero acceleration) such that she can just catch the train.
Chapter 2
94
1. Using the subscripts ″train″ and ″p″ to refer to the train and the passenger and the subscript ″c″ to identify ″critical″ conditions, express the position of the train and the passenger:
( )
( ) ( )ttvtx
tatx
∆−=
=
ccp,ccp,
2c
traincctrain,
and
2
Express the critical conditions that must be satisfied if the passenger is to catch the train:
cp,ctrain, vv = and
cp,ctrain, xx =
2. Express the train’s average velocity. 22
0) to(0 ctrain,ctrain,
cavvv
tv =+
=
3. Using the definition of average velocity, express vav in terms of xp,c and tc. c
cp,
c
cp,av 0
0t
xt
xtxv =
++
=∆∆
≡
4. Combine steps 2 and 3 and solve for xp,c. 2
cctrain,cp,
tvx =
5. Combine steps 1 and 4 and solve for tc. ( )
2
or
2
cc
cctrain,ccp,
ttt
tvttv
=∆−
=∆−
and tc = 2 ∆t = 2 (6 s) = 12 s
6. Finally, combine steps 1 and 5 and solve for vtrain, c.
( )( )m/s 80.4
s12m/s4.0 2ctrainctrain,cp,
=
=== tavv
The graph shows the location of both the passenger and the train as a function of time. The parabolic solid curve is the graph of xtrain(t) for the accelerating train. The straight dashed line is passenger's position xp(t) if she arrives at ∆t = 6.0 s after the train departs. When the passenger catches the train, our graph shows that her speed and that of the train must be equal ( cp,ctrain, vv = ). Do you see why?
Motion in One Dimension
95
05
101520253035404550
0 4 8 12 16
t (s)
x (m
)
Train
Passenger
108 ••• Picture the Problem Both balls experience constant acceleration once they are in flight. Choose a coordinate system with the origin at the ground and the upward direction positive. When the balls collide they are at the same height above the ground. Using constant-acceleration equations, express the positions of both balls as functions of time. At the ground y = 0.
221
0B
221
A
and
gttvy
gthy
−=
−=
The conditions at collision are that the heights are equal and the velocities are related: BA
BA
2 and
vv
yy
−=
=
Express the velocities of both balls as functions of time:
gtvv
gtv
−=
−=
0B
A
and
Substituting the position and velocity functions into the conditions at collision gives: ( )c0c
2c2
1c0
2c2
1
2 and
gtvgt
gttvgth
−−=−
−=−
where tc is the time of collision.
We now have two equations and two unknowns, tc and v0. Solving the equations for the unknowns gives:
23 and
32
0cghv
ght ==
Substitute the expression for tc into the equation for yA to obtain the height at collision: 3
232
21
Ah
ghghy =⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
Chapter 2
96
Remarks: We can also solve this problem graphically by plotting velocity- versus-time for both balls. Because ball A starts from rest its velocity is given by gtvA −= . Ball B initially moves with an unknown velocity vB0 and its velocity is given by gtvv B0B −= . The graphs of these equations are shown below with T representing the time at which they collide.
The height of the building is the sum of the sum of the distances traveled by the balls. Each of these distances is equal to the magnitude of the area ″under″ the corresponding v-versus-t curve. Thus, the height of the building equals the area of the parallelogram, which is vB0T. The distance that A falls is the area of the lower triangle, which is (1/3) vB0T. Therefore, the ratio of the distance fallen by A to the height of the building is 1/3, so the collision takes place at 2/3 the height of the building. 109 ••• Picture the Problem Both balls are moving with constant acceleration. Take the origin of the coordinate system to be at the ground and the upward direction to be positive. When the balls collide they are at the same height above the ground. The velocities at collision are related by vA = 4vB. Using constant-acceleration equations, express the positions of both balls as functions of time:
221
0B
221
A
y and
gttv
gthy
−=
−=
The conditions at collision are that the heights are equal and the velocities are related: BA
BA
4 and
vv
yy
=
=
Express the velocities of both balls as functions of time:
gtvvgtv −=−= 0BA and
Motion in One Dimension
97
Substitute the position and velocity functions into the conditions at collision to obtain: ( )c0c
2c2
1c0
2c2
1
4and
gtvgt
gttvgth
−=−
−=−
where tc is the time of collision.
We now have two equations and two unknowns, tc and v0. Solving the equations for the unknowns gives:
43vand
34
0cgh
ght ==
Substitute the expression for tc into the equation for yA to obtain the height at collision: 33
421
Ah
ghghy =⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
*110 •• Determine the Concept The problem describes two intervals of constant acceleration; one when the train’s velocity is increasing, and a second when it is decreasing. (a) Using a constant-acceleration equation, relate the half-distance ∆x between stations to the initial speed v0, the acceleration a of the train, and the time-to-midpoint ∆t:
( )221
0 tatvx ∆+∆=∆ or, because v0 = 0,
( )221 tax ∆=∆
Solve for ∆t: a
xt ∆=∆
2
Substitute numerical values and evaluate the time-to-midpoint ∆t:
( ) s0.30m/s1
m45022 ==∆t
Because the train accelerates uniformly and from rest, the first part of its velocity graph will be linear, pass through the origin, and last for 30 s. Because it slows down uniformly and at the same rate for the second half of its journey, this part of its graph will also be linear but with a negative slope. The graph of v as a function of t is shown below.
Chapter 2
98
0
5
10
15
20
25
30
0 10 20 30 40 50 60
t (s)
v (m
/s)
(b) The graph of x as a function of t is obtained from the graph of v as a function of t by finding the area under the velocity curve. Looking at the velocity graph, note that when the train has been in motion for 10 s, it will have traveled a distance of
( )( ) m50m/s10s1021 =
and that this distance is plotted above 10 s on the following graph.
0
100
200
300
400
500
600
700
800
900
0 10 20 30 40 50 60
t (s)
x (m
)
Selecting additional points from the velocity graph and calculating the areas under the curve will confirm the graph of x as a function of t that is shown. 111 •• Picture the Problem This is a two-stage constant-acceleration problem. Choose a coordinate system in which the direction of the motion of the cars is the positive direction. The pictorial representation summarizes what we know about the motion of the speeder’s car and the patrol car.
Motion in One Dimension
99
Convert the speeds of the vehicles and the acceleration of the police car into SI units:
2m/s22.2s3600
h1sh
km8sh
km8 =×⋅
=⋅
,
m/s7.34s3600
h1h
km125h
km125 =×= ,
and
m/s8.52s3600
h1h
km190h
km190 =×=
(a) Express the condition that determines when the police car catches the speeder; i.e., that their displacements will be the same:
∆xP,02 = ∆xS,02
Using a constant-acceleration equation, relate the displacement of the patrol car to its displacement while accelerating and its displacement once it reaches its maximum velocity:
( )121,PP,01
P,12P,01P,02
ttvxxxx
−+∆=
∆+∆=∆
Using a constant-acceleration equation, relate the displacement of the speeder to its constant velocity and the time it takes the patrol car to catch it:
( ) 2
02S,02S,02
m/s7.34 ttvx
=
∆=∆
Calculate the time during which the police car is speeding up:
s8.23m/s2.22
0m/s52.82
P,01
0P,1,P
P,01
P,01P,01
=−
=
−=
∆=∆
avv
av
t
Chapter 2
100
Express the displacement of the patrol car: ( )( )
m629s8.23m/s22.20 22
21
201,P01,P2
1P,01P,0P,01
=+=
∆+∆=∆ tatvx
Equate the displacements of the two vehicles: ( )
( ) )s8.23(m/s8.52m629 2
121,PP,01
P,12P,01P,02
−+=
−+∆=
∆+∆=∆
tttvx
xxx
Solve for the time to catch up to obtain:
(34.7 m/s) t2 = 629 m + (52.8 m/s)(t2 – 23.8 s) s7.342 =∴ t
(b) The distance traveled is the displacement, ∆x02,S, of the speeder during the catch:
( )( )km1.20
s34.7m/s34.702S,02S,02
=
=∆=∆ tvx
(c) The graphs of xS and xP are shown below. The straight line (solid) represents xS(t) and the parabola (dashed) represents xP(t).
0
200
400
600
800
1000
1200
1400
0 10 20 30 40
t (s)
x (m
)
SpeederOfficer
112 •• Picture the Problem The accelerations of both cars are constant and we can use constant-acceleration equations to describe their motions. Choose a coordinate system in which the direction of motion of the cars is the positive direction, and the origin is at the initial position of the police car. (a) The collision will not occur if, during braking, the displacements of the two cars differ by less than 100 m.
∆xP − ∆xS < 100 m.
Motion in One Dimension
101
Using a constant-acceleration equation, relate the speeder’s initial and final speeds to its displacement and acceleration and solve for the displacement:
ss2
s,02s 2 xavv ∆+=
or, because vs = 0,
s
2s,0
s 2av
x−
=∆
Substitute numerical values and evaluate ∆xs:
( )( ) m100
m/s62m/s7.34
2
2
S =−
−=∆x
Using a constant-acceleration equation, relate the patrol car’s initial and final speeds to its displacement and acceleration and solve for the displacement:
pp2
p,02p 2 xavv ∆+=
or, assuming vp = 0,
p
2p,0
p 2av
x−
=∆
Substitute numerical values and evaluate ∆xp:
( )( ) m232
m/s62m/s8.52
2
2
P =−
−=∆x
Finally, substitute these displacements into the inequality that determines whether a collision occurs:
232 m − 100 m = 132 m Because this difference is greater than 100 m, collide cars the .
(b) Using constant-acceleration equations, relate the positions of both vehicles to their initial positions, initial velocities, accelerations, and time in motion:
( ) ( )
( ) ( ) 22P
22S
m/s3m/s8.52and
m/s3m/s7.34m100
ttx
ttx
−=
−+=
Equate these expressions and solve for t:
100 m + (34.7 m/s) t – (3 m/s2) t2 = (52.8 m/s) t – (3m/s2) t2 and
s52.5=t
(c) severe. more be andsooner
occur willcollision theaccount, into imereaction t theyou take If
113 •• Picture the Problem Lou’s acceleration is constant during both parts of his trip. Let t1 be the time when the brake is applied; L1 the distance traveled from t = 0 to t = t1. Let tfin be the time when Lou's car comes to rest at a distance L from the starting line. A pictorial representation will help organize the given information and plan the solution.
Chapter 2
102
(a) Express the total length, L, of the course in terms of the distance over which Lou will be accelerating, ∆x01, and the distance over which he will be braking, ∆x12:
L = ∆x01 + ∆x12
Express the final velocity over the first portion of the course in terms of the initial velocity, acceleration, and displacement; solve for the displacement:
010120
21 2 xavv ∆+=
or, because v0 = 0, ∆x01 = L1, and a01 = a,
av
avLx
22
2max
21
101 ===∆
Express the final velocity over the second portion of the course in terms of the initial velocity, acceleration, and displacement; solve for the displacement:
121221
22 2 xavv ∆+=
or, because v2 = 0 and a12 = −2a,
241
21
12L
avx ==∆
Substitute for ∆x01 and ∆x12 to obtain:
123
121
11201 LLLxxL =+=∆+∆= and
LL 32
1 =
(b) Using the fact that the acceleration was constant during both legs of the trip, express Lou’s average velocity over each leg:
2max
12,av01,avvvv ==
Express the time for Lou to reach his maximum velocity as a function of L1 and his maximum velocity:
max
1
av,01
0101
2v
Lv
xt =∆
=∆
and
LLt32
101 =∝∆
Having just shown that the time required for the first segment of the trip is proportional to the length of the segment, use this result to express ∆t01 (= t1) in terms tfin:
fin32 tt =
Motion in One Dimension
103
114 •• Picture the Problem There are three intervals of constant acceleration described in this problem. Choose a coordinate system in which the upward direction (shown to the left below) is positive. A pictorial representation will help organize the details of the problem and plan the solution.
(a) The graphs of a(t) (dashed lines) and v(t) (solid lines) are shown below.
-80
-60
-40
-20
0
20
0 2 4 6 8 10 12 14 16
t (s)
v (m
/s) a
nd a
(m/s
^2)
VelocityAcceleration
(b) Using a constant-acceleration equation, express her speed in terms of her acceleration and the elapsed time; solve for her speed after 8 s of fall:
( )( )m/s5.78
s8m/s81.90 210101
=
+=
+= tavv
(c) Using the same constant- acceleration equation that you used in part (b), determine the duration of her constant upward acceleration:
121212 tavv ∆+= ( )
s90.4
m/s15m/s78.5m/s5
212
1212
=
−−−=
−=∆
avvt
(d) Find her average speed as she slows from 78.5 m/s to 5 m/s:
m/s8.412
m/s5m/s78.52
21av
=
+=
+=
vvv
Chapter 2
104
Use this value to calculate how far she travels in 4.90 s:
( )m204
s) 90.4(m/s 8.4112av12
==∆=∆ tvy
down.slowingwhilem204travelsShe
(e) Express the total time in terms of the times for each segment of her descent:
231201total tttt ∆+∆+∆=
We know the times for the intervals from 0 to 1 and 1 to 2 so we only need to determine the time for the interval from 2 to 3. We can calculate ∆t23 from her displacement and constant velocity during that segment of her descent.
( )
m0.57
m204s82m/s5.78m575
1201total23
=
−⎟⎠⎞
⎜⎝⎛−=
∆−∆−∆=∆ yyyy
Add the times to get the total time:
s3.24
m/s5m0.75s9.4s8
231201total
=
++=
++= tttt
(f) Using its definition, calculate her average velocity: m/s18.7
s209m1500
av −=−
=∆∆
=txv
Integration of the Equations of Motion *115 • Picture the Problem The integral of a function is equal to the "area" between the curve for that function and the independent-variable axis. (a) The graph is shown below:
0
5
10
15
20
25
30
35
0 1 2 3 4 5
t (s)
v (m
/s)
Motion in One Dimension
105
The distance is found by determining the area under the curve. You can accomplish this easily because the shape of the area under the curve is a trapezoid.
A = (36 blocks)(2.5 m/block) = m90
or
( ) m90s0s52
m/s3m/s33=−⎟
⎠⎞
⎜⎝⎛ +
=A
Alternatively, we could just count the blocks and fractions thereof.
There are approximately 36 blocks each having an area of (5 m/s)(0.5 s) = 2.5 m.
(b) To find the position function x(t), we integrate the velocity function v(t) over the time interval in question:
( ) ( )
( ) ( )[ ] 'dm/s3'm/s6
'd'
0
2
0
tt
ttvtx
t
t
∫
∫
+=
=
and ( ) ( ) ( )tttx m/s3m/s3 22 +=
Now evaluate x(t) at 0 s and 5 s respectively and subtract to obtain ∆x:
( ) ( )m 90.0
m 0 m 90 s0s5
=
−=−=∆ xxx
116 • Picture the Problem The integral of v(t) over a time interval is the displacement (change in position) during that time interval. The integral of a function is equivalent to the "area" between the curve for that function and the independent-variable axis. Count the grid boxes.
(a) Find the area of the shaded gridbox:
( )( ) boxper m 1s 1m/s 1 ==Area
(b) Find the approximate area under curve for 1 s ≤ t ≤ 2 s:
∆x1 s to 2 s = m 1.2
Find the approximate area under curve for 2 s ≤ t ≤ 3 s:
∆x2 s to 3 s = m 2.3
(c) Sum the displacements to obtain the total in the interval 1 s ≤ t ≤ 3 s:
∆x1 s to 3 s = 1.2 m + 3.2 m = 4.4 m
Using its definition, express and evaluate vav:
m/s2.20s2m4.4
t s 3 tos 1
s 3 tos 1av ==
∆∆
=xv
(d) Because the velocity of the particle is dx/dt, separate the
( )dtdx 3m/s5.0= so
Chapter 2
106
variables and integrate over the interval 1 s ≤ t ≤ 3 s to determine the displacement in this time interval:
( )
( ) m33.43
m/s5.0
''m/s5.0'
s3
s1
33
s3
s1
23s3s1
0
=⎥⎦
⎤⎢⎣
⎡ ′=
==∆ ∫∫→
t
dttdxxx
x
This result is a little smaller than the sum of the displacements found in part (b).
Calculate the average velocity over the 2-s interval from 1 s to 3 s: m/s17.2
s2m33.4
s3s1
s3s1s)3s1( av ==
∆∆
=−
−− t
xv
Calculate the initial and final velocities of the particle over the same interval:
( ) ( )( )( ) ( )( ) m/s5.4s3m/s5.0s3
m/s5.0s1m/s5.0s123
23
==
==
v
v
Finally, calculate the average value
of the velocities at t = 1 s and t = 3 s: m/s50.2
2m/s 4.5 m/s 0.5
2s) (3 s) (1
=
+=
+ vv
This average is not equal to the average velocity calculated above.
Remarks: The fact that the average velocity was not equal to the average of the velocities at the beginning and the end of the time interval in part (d) is a consequence of the acceleration not being constant. *117 •• Picture the Problem Because the velocity of the particle varies with the square of the time, the acceleration is not constant. The displacement of the particle is found by integration. Express the velocity of a particle as the derivative of its position function:
( ) ( )dt
tdxtv =
Separate the variables to obtain: ( ) ( )dttvtdx =
Express the integral of x from xo = 0 to x and t from t0 = 0 to t: ( ) ( )∫∫
==
==t
t
tx
t
dttvdxtx0
)(
0 00
'''
Substitute for v(t′) to obtain: ( ) ( ) ( )[ ]
( ) ( )tt
dtttxt
t
m/s5m/s
'm/s5'm/s7
3337
0
23
0
−=
−= ∫=
Motion in One Dimension
107
118 •• Picture the Problem The graph is one of constant negative acceleration. Because
vx = v(t) is a linear function of t, we can make use of the slope-intercept form of the equation of a straight line to find the relationship between these variables. We can then differentiate v(t) to obtain a(t) and integrate v(t) to obtain x(t).
Find the acceleration (the slope of the graph) and the velocity at time 0 (the v-intercept) and use the slope-intercept form of the equation of a straight line to express vx(t):
2m/s10−=a
ttvx )m/s 10(m/s50)( 2−+=
Find x(t) by integrating v(t):
( ) ( )[ ]( ) ( ) Cm/s5m/s50
m/s50m/s1022
2
+−=
+−= ∫tt
dtttx
Using the fact that x = 0 when t = 0, evaluate C:
( )( ) ( )( ) C0m/s50m/s500 22 +−= and C = 0
Substitute to obtain: ( ) ( ) ( ) 22m/s5m/s50 tttx −=
Note that this expression is quadratic in t
and that the coefficient of t2 is negative and equal in magnitude to half the constant acceleration.
Remarks: We can check our result for x(t) by evaluating it over the 10-s interval shown and comparing this result with the area bounded by this curve and the time axis. 119 ••• Picture the Problem During any time interval, the integral of a(t) is the change in velocity and the integral of v(t) is the displacement. The integral of a function equals the "area" between the curve for that function and the independent-variable axis.
(a) Find the area of the shaded grid box in Figure 2-37:
Area = (0.5 m/s2)(0.5 s) = boxper m/s 0.250
(b) We start from rest (vo = 0) at t = 0. For the velocities at the other times, count boxes and multiply by the 0.25 m/s per box that we found in part (a):
Examples: v(1 s) = (3.7 boxes)[(0.25 m/s)/box] = m/s 925.0
v(2 s) = (12.9 boxes)[(0.25 m/s)/box] = m/s 3.22
and
Chapter 2
108
v(3 s) = (24.6 boxes)[(0.25 m/s)/box] = m/s 6.15
(c) The graph of v as a function of t is shown below:
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2 2.5 3
t (s)
v (m
/s)
Area = (1.0 m/s)(1.0 s) = 1.0 m per box
Count the boxes under the v(t) curve to find the distance traveled:
( ) ( )( ) ( )[ ]
m00.7
box/m0.1boxes7s30s3
=
=→∆= xx
120 •• Picture the Problem The integral of v(t) over a time interval is the displacement (change in position) during that time interval. The integral of a function equals the "area" between the curve for that function and the independent-variable axis. Because acceleration is the slope of a velocity versus time curve, this is a non-constant-acceleration problem. The derivative of a function is equal to the "slope" of the function at that value of the independent variable.
(a) To obtain the data for x(t), we must estimate the accumulated area under the v(t) curve at each time interval: Find the area of a shaded grid box in Figure 2-38:
A = (1 m/s)(0.5 s) = 0.5 m per box.
We start from rest (vo = 0) at to= 0. For the position at the other times, count boxes and multiply by the 0.5 m per box that we found above. Remember to add the offset from the origin, xo = 5 m, and that boxes below the v = 0 line are counted as negative:
Examples:
( ) ( )
m17.9
m5box
m0.5boxes25.8s3
=
+⎟⎠⎞
⎜⎝⎛=x
( ) ( )
m0.92
m5box
m0.5boxes0.84s5
=
+⎟⎠⎞
⎜⎝⎛=x
Motion in One Dimension
109
( ) ( )
( )
m5.12
m5box
m0.5boxes0.36
boxm0.5boxes0.51s10
=
+⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛=x
A graph of x as a function of t follows:
0
5
10
15
20
25
30
35
0 2 4 6 8 10
t (s)
x (m
)
(b) To obtain the data for a(t), we must estimate the slope (∆v/∆t) of the v(t) curve at each time. A good way to get reasonably reliable readings from the graph is to enlarge several fold:
Examples:
( ) ( ) ( )
2m/s3.8s0.5
m/s3.0m/s4.9s5.0
s75.0s25.1s1
=−
=
−=
vva
( ) ( ) ( )
2m/s2.4s0.5
m/s4.0m/s7.1s5.0
s75.5s25.6s6
−=−−
=
−=
vva
Chapter 2
110
A graph of a as a function of t follows:
-6
-4
-2
0
2
4
6
0 2 4 6 8 10
t (s)
a (m
/s^2
)
*121 •• Picture the Problem Because the position of the body is not described by a parabolic function, the acceleration is not constant. Select a series of points on the graph of x(t) (e.g., at the extreme values and where the graph crosses the t axis), draw tangent lines at those points, and measure their slopes. In doing this, you are evaluating v = dx/dt at these points. Plot these slopes above the times at which you measured the slopes. Your graph should closely resemble the following graph.
-8
-6
-4
-2
0
2
4
6
8
0 0.2 0.4 0.6 0.8 1 1.2 1.4
t
v
Select a series of points on the graph of v(t) (e.g., at the extreme values and where the graph crosses the t axis), draw tangent lines at those points, and measure their slopes. In doing this, you are evaluating a = dv/dt at these points. Plot these slopes above the times at which you measured the slopes. Your graph should closely resemble the graph shown below.
Motion in One Dimension
111
-15
-10
-5
0
5
10
15
0 0.5 1 1.5
t
a
122 •• Picture the Problem Because the acceleration of the rocket varies with time, it is not constant and integration of this function is required to determine the rocket’s velocity and position as functions of time. The conditions on x and v at t = 0 are known as initial conditions. (a) Integrate a(t) to find v(t):
∫ ∫ +=== C )()( 221 btdttbdttatv
where C, the constant of integration, can be determined from the initial conditions.
Integrate v(t) to find x(t):
( ) ( ) [ ] DC
C3
61
221
++=
+== ∫ ∫tbt
dtbtdttvtx
where D is a second constant of integration.
Using the initial conditions, find the constants C and D:
( )
( ) 0D00and
0C00
=⇒=
=⇒=
x
v
( ) 361 bttx =∴
(b) Evaluate v(5 s) and x(5 s) with C = D = 0 and b = 3 m/s2: ( ) ( )( ) m/s5.37s5m/s3
21s5 22 ==v
and
( ) ( )( ) m5.62s5m/s361s5 32 ==x
123 •• Picture the Problem The acceleration is a function of time; therefore it is not constant. The instantaneous velocity can be determined by integration of the acceleration and the average velocity from the displacement of the particle during the given time interval.
Chapter 2
112
(a) Because the acceleration is the derivative of the velocity, integrate the acceleration to find the instantaneous velocity v(t).
( ) ( )( )
( )∫∫==
==⇒=t
t
tv
v
dttadvtvdtdvta
00 00
'''
Calculate the instantaneous velocity using the acceleration given. ( ) ( ) ∫
=
=t
t
dtttv0
3
0
''m/s2.0
and
( ) ( ) 23m/s1.0 ttv =
(b) To calculate the average velocity, we need the displacement: ( ) ( )
( )
( )∫∫==
==⇒≡t
t
tx
x
dttvdxtxdtdxtv
00 00
'''
Because the velocity is the derivative of the displacement, integrate the velocity to find ∆x.
( ) ( ) ( )3
m/s1.0''m/s1.03
3
0
23
0
tdtttxt
t
== ∫=
and
( ) ( )
m 11.2
3s) 2(s 7m/s1.0
s) (2s) (733
3
=
⎥⎦
⎤⎢⎣
⎡ −=
−=∆ xxx
Using the definition of the average velocity, calculate vav. m/s 23.2
s 5m 11.2
av ==∆∆
=txv
124 • Determine the Concept Because the acceleration is a function of time, it is not constant. Hence we’ll need to integrate the acceleration function to find the velocity as a function of time and integrate the velocity function to find the position as a function of time. The important concepts here are the definitions of velocity, acceleration, and average velocity. (a) Starting from to = 0, integrate the instantaneous acceleration to obtain the instantaneous velocity as a function of time:
( )∫∫ +=
=
tv
v
dtbtadv
dtdva
00 '''
thatfollowsit
From
0
and 2
21
00 bttavv ++=
(b) Now integrate the instantaneous velocity to obtain the position as a function of time:
dtdxv =From
it follows that
Motion in One Dimension
113
( )
∫
∫∫
⎟⎠⎞
⎜⎝⎛ ++=
==
t
t
t
t
x
x
dttbtav
dttvdx
0
00
''2
'
'''
209
0
and 3
612
021
00 bttatvxx +++=
(c) The definition of the average velocity is the ratio of the displacement to the total time elapsed:
tbttatv
ttxx
txv
3612
021
0
0
0av
++=
−−
=∆∆
≡
and 2
61
021
0av bttavv ++=
Note that vav is not the same as that due to constant acceleration: ( )
( )
av
241
021
0
221
000
0avonacceleraticonstant
2
2
vbttav
bttavv
vvv
≠++=
+++=
+=
General Problems 125 ••• Picture the Problem The acceleration of the marble is constant. Because the motion is downward, choose a coordinate system with downward as the positive direction. The equation gexp = (1 m)/(∆t)2 originates in the constant-acceleration equation
( )221
0 tatvx ∆+∆=∆ . Because the motion starts from rest, the displacement of the marble is 1 m, the acceleration is the experimental value gexp, and the equation simplifies to gexp = (1 m)/(∆t)2. Express the percent difference between the accepted and experimental values for the acceleration due to gravity:
accepted
expaccepteddifference%g
gg −=
Using a constant-acceleration equation, express the velocity of the marble in terms of its initial velocity, acceleration, and displacement:
yavv ∆+= 220
2f
or, because v0 = 0 and a = g, ygv ∆= 22
f
Solve for vf:
ygv ∆= 2f
Let v1 be the velocity the ball has reached when it has fallen 0.5 cm, ( )( ) m/s313.0m005.0m/s81.92 2
1 ==v
Chapter 2
114
and v2 be the velocity the ball has reached when it has fallen 0.5 m to obtain.
and
( )( ) m/s13.3m5.0m/s81.92 22 ==v
Using a constant-acceleration equation, express v2 in terms of v1, g and ∆t:
tgvv ∆+= 12
Solve for ∆t: g
vvt 12 −=∆
Substitute numerical values and evaluate ∆t: s2872.0
m/s81.9m/s313.0m/s13.3
2 =−
=∆t
Calculate the experimental value of the acceleration due to gravity from gexp = (1 m)/(∆t)2:
( )2
2exp m/s13.12s2872.0
m1==g
Finally, calculate the percent difference between this experimental result and the value accepted for g at sea level. %6.23
m/s81.9m/s13.12m/s81.9
difference% 2
22
=
−=
*126 ••• Picture the Problem We can obtain an average velocity, vav = ∆x/∆t, over fixed time intervals. The instantaneous velocity, v = dx/dt can only be obtained by differentiation. (a) The graph of x versus t is shown below:
-6
-4
-2
0
2
4
6
8
0 5 10 15 20 25 30 35
t (s)
v (m
/s)
(b) Draw a tangent line at the origin and measure its rise and run. Use this ratio to obtain an approximate value for the slope at the origin:
The tangent line appears to, at least approximately, pass through the point (5, 4). Using the origin as the second point,
Motion in One Dimension
115
∆x = 4 cm – 0 = 4 cm and
∆t = 5 s – 0 = 5 s
Therefore, the slope of the tangent line and the velocity of the body as it passes through the origin is approximately:
( ) cm/s800.0s5
cm4runrise0 ==
∆∆
==txv
(c) Calculate the average velocity for the series of time intervals given by completing the table shown below:
t0 t ∆t x0 x ∆x vav=∆x/∆t (s) (s) (s) (cm) (cm) (cm) (m/s) 0 6 6 0 4.34 4.34 0.723 0 3 3 0 2.51 2.51 0.835 0 2 2 0 1.71 1.71 0.857 0 1 1 0 0.871 0.871 0.871 0 0.5 0.5 0 0.437 0.437 0.874 0 0.25 0.25 0 0.219 0.219 0.875
(d) Express the time derivative of the position: tA
dtdx ωω cos=
Substitute numerical values and
evaluate dtdx
at t = 0:
( )( )
cm/s0.875
s0.175m0.05
0cos
1
=
=
==
−
ωω AAdtdx
(e) Compare the average velocities from part (c) with the instantaneous velocity from part (d):
As ∆t, and thus ∆x, becomes small, the value for the average velocity approaches that for the instantaneous velocity obtained in part (d). For ∆t = 0.25 s, they agree to three significant figures.
127 ••• Determine the Concept Because the velocity varies nonlinearly with time, the acceleration of the object is not constant. We can find the acceleration of the object by differentiating its velocity with respect to time and its position function by integrating the velocity function. The important concepts here are the definitions of acceleration and velocity. (a) The acceleration of the object is the derivative of its velocity with respect to time:
( )[ ]
( )tv
tvdtd
dtdva
ωω
ω
cos
sin
max
max
=
==
Chapter 2
116
constant. isit with timely sinusoidal varies Because
nota
(b) Integrate the velocity with respect to time from 0 to t to obtain the change in position of the body:
( )[ ]∫∫ =t
t
x
x
dttvdx00
''sin' max ω
and
( )
( )ω
ωω
ωω
maxmax
0
max0
cos
'cos
vtv
tvxxt
+−
=
⎥⎦⎤
⎢⎣⎡−
=−
or
( )[ ]tvxx ωω
cos1max0 −+=
Note that, as given in the problem statement, x(0 s) = x0.
128 ••• Picture the Problem Because the acceleration of the particle is a function of its position, it is not constant. Changing the variable of integration in the definition of acceleration will allow us to determine its velocity and position as functions of position. (a) Because a = dv/dt, we must integrate to find v(t). Because a is given as a function of x, we’ll need to change variables in order to carry out the integration. Once we’ve changed variables, we’ll separate them with v on the left side of the equation and x on the right:
( )xdxdvv
dtdx
dxdv
dtdva 2s2 −====
or, upon separating variables,
( )xdxvdv 2s2 −=
Integrate from xo and vo to x and v: ( )∫∫ −
=
=x
x
v
v
dxxdvv00
''s2'' 2
0
and ( )( )2
0222
02 s2 xxvv −=− −
Solve for v to obtain: ( )( )2
0222
0 s2 xxvv −+= −
Now set vo = 0, xo = 1 m, x = 3 m, b =2 s–2 and evaluate the speed: ( ) ( ) ( )[ ]222 m1m3s2 −±= −v
and m/s00.4=v
Motion in One Dimension
117
(b) Using the definition of v, separate the variables, and integrate to get an expression for t:
( )dtdxxv =
and
( )∫∫ ′′
=x
x
t
xvxddt
00
'
To evaluate this integral we first must find v(x). Show that the acceleration is always positive and use this to find the sign of v(x).
a = (2 s–2 )x and x0 = 1 m. x0 is positive, so a0 is also positive. v0 is zero and a0 is positive, so the object moves in the direction of increasing x. As x increases the acceleration remains positive, so the velocity also remains positive. Thus,
( )( )20
22s2 xxv −= − .
Substitute ( )( )20
22s2 xx −− for v and evaluate the integral. (It can be found in standard integral tables.)
( )
( )( )
( )
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −+=
−′
′=
−′
′=
′′
==
−
−
−
∫
∫
∫∫
0
20
2
2
20
22
20
22
0
lns21
s21
s2
0
0
0
xxxx
xxxd
xxxd
xvxddt't
x
x
x
x
x
x
t
Evaluate this expression with xo = 1 m and x = 3 m to obtain:
s25.1=t
129 ••• Picture the Problem The acceleration of this particle is not constant. Separating variables and integrating will allow us to express the particle’s position as a function of time and the differentiation of this expression will give us the acceleration of the particle as a function of time. (a) Write the definition of velocity: dt
dxv =
We are given that x = bv, where b = 1 s. Substitute for v and separate variables to obtain:
xdxbdt
bx
dtdx
=⇒=
Integrate and solve for x(t): ( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛=−⇒
′′
= ∫∫0
0 ln '00
xxbtt
xxdbdt
x
x
t
t
and
Chapter 2
118
( ) ( ) bttextx
/
00−
=
(b) Differentiate twice to obtain v(t) and a(t):
( ) bttex
bdtdxv
/
001 −
==
and ( ) btt
exbdt
dva/
02
01 −==
Substitute the result in part (a) to obtain the desired results:
)(1)(
and
)(1)(
2 txb
ta
txb
tv
=
=
so
)(1)(1)( 2 txb
tvb
ta ==
in time.instant each at same theare and of valuesnumerical theone, is units, SIin expressed , of valuenumerical theBecause
xv,a,b
130 ••• Picture the Problem Because the acceleration of the rock is a function of time, it is not constant. Choose a coordinate system in which downward is positive and the origin at the point of release of the rock. Separate variables in a(t) = dv/dt = ge−bt to obtain:
dtgedv bt−=
Integrate from to = 0, vo = 0 to some later time t and velocity v: [ ]
( ) ( )btbt
tbtt
btv
evebg
eb
gdtgedvv
−−
−−
−=−=
−=== ∫∫
11
''
term
0'
0
'
0
where
bgv =term
Separate variables in
( )btevdtdyv −−== 1term to obtain:
( )dtevdy bt−−= 1term
Integrate from to = 0, yo = 0 to some later time t and position y: ( )∫∫ −−=
tbt
y
tevdy0
'term
0
'd1'
Motion in One Dimension
119
( )bt
tbt
eb
vtv
eb
tvy
−
−
−−=
⎥⎦⎤
⎢⎣⎡ +=
1
1'
termterm
0
'term
This last result is very interesting. It says that throughout its free-fall, the object experiences drag; therefore it has not fallen as far at any given time as it would have if it were falling at the constant velocity, vterm. On the other hand, just as the velocity of the object asymptotically approaches vterm, the distance it has covered during its free-fall as a function of time asymptotically approaches the distance it would have fallen if it had fallen with vterm throughout its motion.
( ) tvbvtvty termtermlarge →−→
This should not be surprising because in the expression above, the first term grows linearly with time while the second term approaches a constant and therefore becomes less important with time.
*131 ••• Picture the Problem Because the acceleration of the rock is a function of its velocity, it is not constant. Choose a coordinate system in which downward is positive and the origin is at the point of release of the rock. Rewrite a = g – bv explicitly as a differential equation:
bvgdtdv
−=
Separate the variables, v on the left, t on the right:
dtbvg
dv=
−
Integrate the left-hand side of this equation from 0 to v and the right-hand side from 0 to t:
∫∫ =−
t
00
''
' dtbvg
dvv
and
tg
bvgb
=⎟⎟⎠
⎞⎜⎜⎝
⎛ −− ln1
Solve this expression for v.
( )bte
bgv −−= 1
Finally, differentiate this expression with respect to time to obtain an expression for the acceleration and
btgedtdva −==
Chapter 2
120
complete the proof.
132 ••• Picture the Problem The skydiver’s acceleration is a function of her velocity; therefore it is not constant. Expressing her acceleration as the derivative of her velocity, separating the variables, and then integrating will give her velocity as a function of time. (a) Rewrite a = g – cv2 explicitly as a differential equation:
2cvgdtdv
−=
Separate the variables, with v on the left, and t on the right:
dtcvg
dv=
− 2
Eliminate c by using 2Tvgc = :
gdt
vv
dv
dt
vvg
dv
vvgg
dv
T
TT
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
=−
2
22
2
1
or
1
Integrate the left-hand side of this equation from 0 to v and the right-hand side from 0 to t:
gtdtg
Tvv
dv tv
==
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
∫∫00
2 ''1
'
The integral can be found in integral tables:
( ) ( )tvgvv
gtvvv
TT
TT
//tanhor
)/(tanh
1-
1
=
=−
Solve this equation for v to obtain: t
vgvvT
T ⎟⎟⎠
⎞⎜⎜⎝
⎛= tanh
Because c has units of m−1, and g has units of m/s2, (cg)−1/2 will have units of time. Let’s represent this expression with the time-scale factor T:
i.e., T = (cg)−1/2
Motion in One Dimension
121
The skydiver falls with her terminal velocity when a = 0. Using this definition, relate her terminal velocity to the acceleration due to gravity and the constant c in the acceleration equation:
20 Tcvg −= and
cgvT =
Convince yourself that T is also equal to vT/g and use this relationship to eliminate g and vT in the solution to the differential equation:
( ) ⎟⎠⎞
⎜⎝⎛=
Ttvtv T tanh
(b) The following table was generated using a spreadsheet and the equation we derived in part (a) for v(t). The cell formulas and their algebraic forms are:
Cell Content/Formula Algebraic FormD1 56 vT D2 5.71 T B7 B6 + 0.25 t + 0.25 C7 $B$1*TANH(B7/$B$2)
⎟⎠⎞
⎜⎝⎛
TtvT tanh
A B C D E 1 vT = 56 m/s 2 T= 5.71 s 3 4 5 time (s) v (m/s) 6 0.00 0.00 7 0.25 2.45 8 0.50 4.89 9 0.75 7.32
10 1.00 9.71
54 12.00 54.35 55 12.25 54.49 56 12.50 54.61 57 12.75 54.73 58 13.00 54.83 59 13.25 54.93
Chapter 2
122
0
10
20
30
40
50
60
0 2 4 6 8 10 12 14
t (s)
v (m
/s)
Note that the velocity increases linearly over time (i.e., with constant acceleration) for about time T, but then it approaches the terminal velocity as the acceleration decreases.
123
Chapter 3 Motion in Two and Three Dimensions Conceptual Problems *1 • Determine the Concept The distance traveled along a path can be represented as a sequence of displacements.
Suppose we take a trip along some path and consider the trip as a sequence of many very small displacements. The net displacement is the vector sum of the very small displacements, and the total distance traveled is the sum of the magnitudes of the very small displacements. That is,
total distance = NN 1,3,22,11,0 ... −∆++∆+∆+∆ rrrrrrrr
where N is the number of very small displacements. (For this to be exactly true we have to take the limit as N goes to infinity and each displacement magnitude goes to zero.) Now, using ″the shortest distance between two points is a straight line,″ we have
NNN 1,3,22,11,0,0 ... −∆++∆+∆+∆≤∆ rrrrrrrrrr
,
where N,0r
r∆ is the magnitude of the net displacement.
Hence, we have shown that the magnitude of the displacement of a particle is less than or equal to the distance it travels along its path.
2 • Determine the Concept The displacement of an object is its final position vector minus its initial position vector ( if rrr rrr
−=∆ ). The displacement can be less but never more than the distance traveled. Suppose the path is one complete trip around the earth at the equator. Then, the displacement is 0 but the distance traveled is 2πRe.
Chapter 3
124
3 • Determine the Concept The important distinction here is that average velocity is being requested, as opposed to average speed. The average velocity is defined as the displacement divided by the elapsed time.
00av =
∆=
∆∆
=tt
rvr
r
circuit. completeeach of end theat zero always is velocity average thes,car travel race fast the howmatter no that see weThus zero. is track thearound any tripfor nt displaceme The
What is the correct answer if we were asked for average speed? The average speed is defined as the distance traveled divided by the elapsed time.
tv
∆≡
distancetotalav
zero.not is average theand zeroan greater thbe will traveleddistance total theany track, ofcircuit complete oneFor
4 • False. Vectors are quantities with magnitude and direction that can be added and subtracted like displacements. Consider two vectors that are equal in magnitude and oppositely directed. Their sum is zero, showing by counterexample that the statement is false. 5 • Determine the Concept We can answer this question by expressing the relationship between the magnitude of vector A
r and its
component AS and then using properties of the cosine function.
Express AS in terms of A and θ : AS = A cosθ
Take the absolute value of both sides of this expression:
⎜AS⎜ = ⎜A cosθ ⎜ = A⎜cosθ ⎜
and
⎜cosθ ⎜=AAS
Motion in One and Two Dimensions
125
Using the fact that 0 < ⎟cosθ ⎜≤ 1, substitute for⎟cosθ ⎜to obtain: 10 S ≤<
AA
or AA ≤< S0
vector. theof magnitude theto equalor than less bemust vector a ofcomponent a of magnitude The No.
equal. arecomponent its and vector theof magnitude the then,180 of multiplesor 0 toequal is figure in theshown angle theIf °°θ
*6 • Determine the Concept The diagram shows a vector A
rand its components Ax
and Ay. We can relate the magnitude of Ar
is related to the lengths of its components through the Pythagorean theorem.
Suppose that A
ris equal to zero. Then .0222 =+= yx AAA
But .0022 ==⇒=+ yxyx AAAA
too.zero bemust components its ofeach zero, toequal is vector a If No.
7 • Determine the Concept No. Consider the special case in which AB
rr−= .
If 0then,0 =≠−= CABrrr
and the magnitudes of the components of BArr
and are
larger than the components of .Cr
*8 • Determine the Concept The instantaneous acceleration is the limiting value, as ∆t approaches zero, of .t∆∆vr Thus, the acceleration vector is in the same direction as .vr∆ False. Consider a ball that has been thrown upward near the surface of the earth and is slowing down. The direction of its motion is upward. The diagram shows the ball’s velocity vectors at two instants of time and the determination of .vr∆ Note that because vr∆ is downward so is the acceleration of the ball.
Chapter 3
126
9 • Determine the Concept The instantaneous acceleration is the limiting value, as ∆t approaches zero, of t∆∆vr and is in the same direction as .vr∆ Other than through the definition of ,ar the instantaneous velocity and acceleration vectors are unrelated. Knowing the direction of the velocity at one instant tells one nothing about how the velocity is changing at that instant. correct. is )(e
10 • Determine the Concept The changing velocity of the golf ball during its flight can be understood by recognizing that it has both horizontal and vertical components. The nature of its acceleration near the highest point of its flight can be understood by analyzing the vertical components of its velocity on either side of this point. At the highest point of its flight, the ball is still traveling horizontally even though its vertical velocity is momentarily zero. The figure to the right shows the vertical components of the ball’s velocity just before and just after it has reached its highest point. The change in velocity during this short interval is a non-zero, downward-pointing vector. Because the acceleration is proportional to the change in velocity, it must also be nonzero.
correct. is )(d
Remarks: Note that vx is nonzero and vy is zero, while ax is zero and ay is nonzero. 11 • Determine the Concept The change in the velocity is in the same direction as the acceleration. Choose an x-y coordinate system with east being the positive x direction and north the positive y direction.
Given our choice of coordinate system, the x component of a
ris negative and so vr will
decrease. The y component of ar is positive and so vr will increase toward the north. correct. is )(c
*12 • Determine the Concept The average velocity of a particle, ,avvr is the ratio of the particle’s displacement to the time required for the displacement. (a) We can calculate r
r∆ from the given information and ∆t is known. correct. is )(a
(b) We do not have enough information to calculate vr∆ and cannot compute the
Motion in One and Two Dimensions
127
particle’s average acceleration. (c) We would need to know how the particle’s velocity varies with time in order to compute its instantaneous velocity. (d) We would need to know how the particle’s velocity varies with time in order to compute its instantaneous acceleration. 13 •• Determine the Concept The velocity vector is always in the direction of motion and, thus, tangent to the path.
(a) path. theo tangent tis motion,
ofdirection in the being always of econsequenc a as vector, velocity The
(b) A sketch showing two velocity vectors for a particle moving along a path is shown to the right.
14 • Determine the Concept An object experiences acceleration whenever either its speed changes or it changes direction. The acceleration of a car moving in a straight path at constant speed is zero. In the other examples, either the magnitude or the direction of the velocity vector is changing and, hence, the car is accelerated. correct. is )(b
*15 • Determine the Concept The velocity vector is defined by ,/ dtdrv
rr= while the
acceleration vector is defined by ./ dtdva rr=
(a) A car moving along a straight road while braking.
(b) A car moving along a straight road while speeding up.
(c) A particle moving around a circular track at constant speed.
16 • Determine the Concept A particle experiences accelerated motion when either its speed or direction of motion changes. A particle moving at constant speed in a circular path is accelerating because the
Chapter 3
128
direction of its velocity vector is changing. If a particle is moving at constant velocity, it is not accelerating. 17 •• Determine the Concept The acceleration vector is in the same direction as the change in velocity vector, .vr∆
(a) The sketch for the dart thrown upward is shown to the right. The acceleration vector is in the direction of the change in the velocity vector .vr∆
(b) The sketch for the falling dart is shown to the right. Again, the acceleration vector is in the direction of the change in the velocity vector .vr∆
(c) The acceleration vector is in the direction of the change in the velocity vector … and hence is downward as shown the right:
*18 •• Determine the Concept The acceleration vector is in the same direction as the change in velocity vector, .vr∆
The drawing is shown to the right.
19 •• Determine the Concept The acceleration vector is in the same direction as the change in velocity vector, .vr∆ The sketch is shown to the right.
Motion in One and Two Dimensions
129
20 • Determine the Concept We can decide what the pilot should do by considering the speeds of the boat and of the current. Give up. The speed of the stream is equal to the maximum speed of the boat in still water. The best the boat can do is, while facing directly upstream, maintain its position relative to the bank. correct. is )(d
*21 • Determine the Concept True. In the absence of air resistance, both projectiles experience the same downward acceleration. Because both projectiles have initial vertical velocities of zero, their vertical motions must be identical. 22 • Determine the Concept In the absence of air resistance, the horizontal component of the projectile’s velocity is constant for the duration of its flight.
At the highest point, the speed is the horizontal component of the initial velocity. The vertical component is zero at the highest point. correct. is )(e
23 • Determine the Concept In the absence of air resistance, the acceleration of the ball depends only on the change in its velocity and is independent of its velocity. As the ball moves along its trajectory between points A and C, the vertical component of its velocity decreases and the change in its velocity is a downward pointing vector. Between points C and E, the vertical component of its velocity increases and the change in its velocity is also a downward pointing vector. There is no change in the horizontal component of the velocity. correct. is )(d
24 • Determine the Concept In the absence of air resistance, the horizontal component of the velocity remains constant throughout the flight. The vertical component has its maximum values at launch and impact. (a) The speed is greatest at A and E. (b) The speed is least at point C. (c) The speed is the same at A and E. The horizontal components are equal at these points but the vertical components are oppositely directed. 25 • Determine the Concept Speed is a scalar quantity, whereas acceleration, equal to the rate of change of velocity, is a vector quantity.
(a) False. Consider a ball on the end of a string. The ball can move with constant speed
Chapter 3
130
(a scalar) even though its acceleration (a vector) is always changing direction.
(b) True. From its definition, if the acceleration is zero, the velocity must be constant and so, therefore, must be the speed.
26 • Determine the Concept The average acceleration vector is defined by ./av t∆∆= va rr
The direction of avar is that of
,if vvv rrr−=∆ as shown to the right.
27 • Determine the Concept The velocity of B relative to A is .ABBA vvv rrr
−= The direction of ABBA vvv rrr
−= is shown to the right.
*28 •• (a) The vectors ( )tA
r and ( )tt ∆+A
r are of equal length but point in slightly different
directions. Ar
∆ is shown in the diagram below. Note that Ar
∆ is nearly perpendicular to ( )tAr
. For very small time intervals, Ar
∆ and ( )tAr
are perpendicular to one another.
Therefore, dtd /Ar
is perpendicular to .Ar
(b) If A
rrepresents the position of a particle, the particle must be undergoing circular
motion (i.e., it is at a constant distance from some origin). The velocity vector is tangent to the particle’s trajectory; in the case of a circle, it is perpendicular to the circle’s radius.
(c) Yes, it could in the case of uniform circular motion. The speed of the particle is constant, but its heading is changing constantly. The acceleration vector in this case is
Motion in One and Two Dimensions
131
always perpendicular to the velocity vector. 29 •• Determine the Concept The velocity vector is in the same direction as the change in the position vector while the acceleration vector is in the same direction as the change in the velocity vector. Choose a coordinate system in which the y direction is north and the x direction is east. (a)
Path Direction of velocity vector
AB north BC northeast CD east DE southeast EF south
(b) Path Direction of acceleration
vector AB north BC southeast CD 0 DE southwest EF north
(c) ere.smaller th ispath
theof radius thesince DEfor larger but ,comparable are magnitudes The
*30 •• Determine the Concept We’ll assume that the cannons are identical and use a constant-acceleration equation to express the displacement of each cannonball as a function of time. Having done so, we can then establish the condition under which they will have the same vertical position at a given time and, hence, collide. The modified diagram shown below shows the displacements of both cannonballs.
Express the displacement of the cannonball from cannon A at any time t after being fired and before any collision:
221
0 tt gvr rrr+=∆
Express the displacement of the cannonball from cannon A at any time t′ after being fired and before any collision:
221
0 tt ′+′′=′∆ gvr rrr
Chapter 3
132
usly.simultaneoguns thefire should they Therefore, times.allat sight of line thebelow
distance same theare balls theand ' usly,simultaneo fired are guns theIf2
21 gt
tt =
Remarks: This is the ″monkey and hunter″ problem in disguise. If you imagine a monkey in the position shown below, and the two guns are fired simultaneously, and the monkey begins to fall when the guns are fired, then the monkey and the two cannonballs will all reach point P at the same time.
31 •• Determine the Concept The droplet leaving the bottle has the same horizontal velocity as the ship. During the time the droplet is in the air, it is also moving horizontally with the same velocity as the rest of the ship. Because of this, it falls into the vessel, which has the same horizontal velocity. Because you have the same horizontal velocity as the ship does, you see the same thing as if the ship were standing still. 32 • Determine the Concept
(a) stone. theof velocity therepresent
could themofeither stone, theofpath theo tangent tare and Because DArr
(b)
( ) ( )
( ) ( )
stone. theofon accelerati therepresent could vector only the
and lar toperpendicu is / Therefore, another. one lar toperpendicu
are and intervals, timesmallFor very . lar toperpendicu
nearly is that Note above. diagram in theshown are and vectors
twoThese circle. thearound moves stone theas directionsdifferent slightlyin point but length equal of be and vectorsLet the
E
AA
AAA
AA
BA
r
rr
rrr
rr
rr
dtd
tt
ttt
∆
∆∆
∆+
Motion in One and Two Dimensions
133
33 • Determine the Concept True. An object accelerates when its velocity changes; that is, when either its speed or its direction changes. When an object moves in a circle the direction of its motion is continually changing. 34 •• Picture the Problem In the diagram, (a) shows the pendulum just before it reverses direction and (b) shows the pendulum just after it has reversed its direction. The acceleration of the bob is in the direction of the change in the velocity if vvv rrr
−=∆ and is tangent to the pendulum trajectory at the point of reversal of direction. This makes sense because, at an extremum of motion, v = 0, so there is no centripetal acceleration. However, because the velocity is reversing direction, the tangential acceleration is nonzero.
35 • Determine the Concept The principle reason is aerodynamic drag. When moving through a fluid, such as the atmosphere, the ball's acceleration will depend strongly on its velocity. Estimation and Approximation *36 •• Picture the Problem During the flight of the ball the acceleration is constant and equal to 9.81 m/s2 directed downward. We can find the flight time from the vertical part of the motion, and then use the horizontal part of the motion to find the horizontal distance. We’ll assume that the release point of the ball is 2 m above your feet. Make a sketch of the motion. Include coordinate axes, initial and final positions, and initial velocity components:
Obviously, how far you throw the ball will depend on how fast you can throw it. A major league baseball pitcher can throw a fastball at 90 mi/h or so. Assume that you can throw a ball at two-thirds that speed to obtain:
m/s8.26mi/h1
m/s0.447mi/h600 =×=v
Chapter 3
134
There is no acceleration in the x direction, so the horizontal motion is one of constant velocity. Express the horizontal position of the ball as a function of time:
tvx x0= (1)
Assuming that the release point of the ball is a distance h above the ground, express the vertical position of the ball as a function of time:
221
0 tatvhy yy ++= (2)
(a) For θ = 0 we have:
( )m/s8.26
0cosm/s8.26cos 000
=°== θvv x
and ( ) 00sinm/s8.26sin 000 =°== θvv y
Substitute in equations (1) and (2) to obtain:
( )tx m/s8.26= and
( ) 2221 m/s81.9m2 ty −+=
Eliminate t between these equations to obtain: ( )
22
2
m/s8.26m/s4.91m2 xy −=
At impact, y = 0 and x = R: ( )
22
2
m/s8.26m/s4.91m20 R−=
Solve for R to obtain:
m1.17=R
(b) Using trigonometry, solve for v0x and v0y:
( )m/s0.19
45cosm/s8.26cos 000
=°== θvv x
and ( )
m/s0.19
45sinm/s8.26sin 000
=
°== θvv y
Substitute in equations (1) and (2) to obtain:
( )tx m/s0.19= and
( ) ( ) 2221 m/s81.9m/s19.0m2 tty −++=
Eliminate t between these equations to obtain: ( )
22
2
m/s0.19m/s4.905m2 xxy −+=
Motion in One and Two Dimensions
135
At impact, y = 0 and x = R. Hence: ( )
22
2
m/s0.19m/s4.905m20 RR −+=
or ( ) 0m2.147m60.73 22 =−− RR
Solve for R (you can use the ″solver″ or ″graph″ functions of your calculator) to obtain:
m6.75=R
(c) Solve for v0x and v0y: m/s8.2600 == vv x and
00 =yv
Substitute in equations (1) and (2) to obtain:
( )tx m/s8.26= and
( ) 2221 m/s81.9m14 ty −+=
Eliminate t between these equations to obtain: ( )
22
2
m/s8.26m/s4.905m14 xy −=
At impact, y = 0 and x = R: ( )
22
2
m/s8.26m/s4.905m140 R−=
Solve for R to obtain:
m3.45=R
(d) Using trigonometry, solve for v0x and v0y:
v0x = v0 y = 19.0 m / s
Substitute in equations (1) and (2) to obtain:
( )tx m/s0.19= and
( ) ( ) 2221 m/s81.9m/s19.0m14 tty −++=
Eliminate t between these equations to obtain: ( )
22
2
m/s0.19m/s4.905m14 xxy −+=
At impact, y = 0 and x = R: ( )
22
2
m/s0.19m/s4.905m140 RR −+=
Solve for R (you can use the ″solver″ or ″graph″ function of your calculator) to obtain:
m6.85=R
Chapter 3
136
37 •• Picture the Problem We’ll ignore the height of Geoff’s release point above the ground and assume that he launched the brick at an angle of 45°. Because the velocity of the brick at the highest point of its flight is equal to the horizontal component of its initial velocity, we can use constant-acceleration equations to relate this velocity to the brick’s x and y coordinates at impact. The diagram shows an appropriate coordinate system and the brick when it is at point P with coordinates (x, y).
Using a constant-acceleration equation, express the x coordinate of the brick as a function of time:
221
00 tatvxx xx ++= or, because x0 = 0 and ax = 0,
tvx x0=
Express the y coordinate of the brick as a function of time:
221
00 tatvyy yy ++=
or, because y0 = 0 and ay = −g, 2
21
0 gttvy y −=
Eliminate the parameter t to obtain: ( ) 220
0 2tan x
vgxy
x
−= θ
Use the brick’s coordinates when it strikes the ground to obtain: ( ) 2
20
0 2tan0 R
vgR
x
−= θ
where R is the range of the brick. Solve for v0x to obtain:
00 tan2 θ
gRv x =
Substitute numerical values and evaluate v0x:
( )( ) m/s8.1445tan2
m44.5m/s9.81 2
0 =°
=xv
Note that, at the brick’s highest point, vy = 0.
Vectors, Vector Addition, and Coordinate Systems 38 • Picture the Problem Let the positive y direction be straight up, the positive x direction be to the right, and A
rand B
rbe the position vectors for the minute and hour hands. The
Motion in One and Two Dimensions
137
pictorial representation below shows the orientation of the hands of the clock for parts (a) through (d).
(a) The position vector for the minute hand at12:00 is:
( ) jA ˆm5.000:12 =r
The position vector for the hour hand at 12:00 is:
( ) jB ˆm25.000:12 =r
(b) At 3:30, the minute hand is positioned along the −y axis, while the hour hand is at an angle of (3.5 h)/12 h × 360° = 105°, measured clockwise from the top. The position vector for the minute hand is:
( ) jA ˆm5.030:3 −=r
Find the x-component of the vector representing the hour hand:
( ) m241.0105sinm25.0 =°=xB
Find the y-component of the vector representing the hour hand:
( ) m0647.0105cosm25.0 −=°=yB
The position vector for the hour hand is:
( ) ( ) jiB ˆm0647.0ˆm241.030:3 −=r
(c) At 6:30, the minute hand is positioned along the −y axis, while the hour hand is at an angle of (6.5 h)/12 h × 360° = 195°, measured clockwise from the top. The position vector for the minute hand is:
( ) jA ˆm5.030:6 −=r
Find the x-component of the vector representing the hour hand:
( ) m0647.0195sinm25.0 −=°=xB
Find the y-component of the vector representing the hour hand:
( ) m241.0195cosm25.0 −=°=yB
The position vector for the hour hand is:
( ) ( ) jiB ˆm241.0ˆm0647.030:6 −−=r
(d) At 7:15, the minute hand is positioned along the +x axis, while the hour hand is at an angle of (7.25 h)/12 h × 360° = 218°, measured clockwise from the top.
Chapter 3
138
The position vector for the minute hand is:
( )iA ˆm5.015:7 =r
Find the x-component of the vector representing the hour hand:
( ) m154.0218sinm25.0 −=°=xB
Find the y-component of the vector representing the hour hand:
( ) m197.0218cosm25.0 −=°=yB
The position vector for the hour hand is:
( ) ( ) jiB ˆm197.0ˆm154.015:7 −−=r
(e) Find A
r − B
r at 12:00: ( ) ( )
( ) j
jjBAˆm25.0
ˆm25.0ˆm5.0
=
−=−rr
Find A
r − B
r at 3:30: ( )
( ) ( )[ ]( ) ( ) ji
ji
jBA
ˆm435.0ˆm241.0
ˆm0647.0ˆm241.0
ˆm5.0
−−=
−−
−=−rr
Find A
r − B
r at 6:30: ( )
( ) ( )[ ]( ) ( ) ji
ji
jBA
ˆm259.0ˆm0647.0
ˆm241.0ˆm0647.0
ˆm5.0
−−=
−−
−=−rr
Find A
r − Br
at 7:15: ( )( ) ( )[ ]
( ) ( ) ji
ji
jBA
ˆm697.0ˆm152.0
ˆm197.0ˆm152.0
ˆm5.0
+=
−−−
=−rr
*39 • Picture the Problem The resultant displacement is the vector sum of the individual displacements. The two displacements of the bear and its resultant displacement are shown to the right:
Motion in One and Two Dimensions
139
Using the law of cosines, solve for the resultant displacement:
( ) ( )( )( ) °−
+=cos135m12m122
m12m12 222R
and m2.22=R
Using the law of sines, solve for α:
m2.22135sin
m12sin °
=α
∴ α = 22.5° and the angle with the horizontal is 45° − 22.5° = °5.22
40 • Picture the Problem The resultant displacement is the vector sum of the individual displacements. (a) Using the endpoint coordinates for her initial and final positions, draw the student’s initial and final position vectors and construct her displacement vector.
Find the magnitude of her displacement and the angle this displacement makes with the positive x-axis:
.135 @ m 25 isnt displacemeHer °
(b) .135 @ 25 also
isnt displaceme his so ),(in as same theare positions final and initial His
°
a
*41 • Picture the Problem Use the standard rules for vector addition. Remember that changing the sign of a vector reverses its direction.
(a)
(b)
Chapter 3
140
(c)
(d)
(e)
42 • Picture the Problem The figure shows the paths walked by the Scout. The length of path A is 2.4 km; the length of path B is 2.4 km; and the length of path C is 1.5 km:
(a) Express the distance from the campsite to the end of path C:
2.4 km – 1.5 km = km9.0
(b) Determine the angle θ subtended by the arc at the origin (campsite):
°==
==
57.3rad1km2.4km2.4
radiuslengtharc
radiansθ
east.ofnorth rad 1 iscamp fromdirectionHis
(c) Express the total distance as the sum of the three parts of his walk:
dtot = deast + darc + dtoward camp
Substitute the given distances to find the total:
dtot = 2.4 km + 2.4 km + 1.5 km = 6.3 km
Motion in One and Two Dimensions
141
Express the ratio of the magnitude of his displacement to the total distance he walked and substitute to obtain a numerical value for this ratio: 7
1
km6.3km0.9
walkeddistance Totalntdisplaceme his of Magnitude
=
=
43 • Picture the Problem The direction of a vector is determined by its components.
°−=⎟⎟⎠
⎞⎜⎜⎝
⎛ −= − 5.32
m/s5.5m/s3.5tan 1θ
The vector is in the fourth quadrant and
correct. is )(b
44 • Picture the Problem The components of the resultant vector can be obtained from the components of the vectors being added. The magnitude of the resultant vector can then be found by using the Pythagorean Theorem. A table such as the one shown to the right is useful in organizing the information in this problem. Let D
r
be the sum of vectors ,Ar
,Br
and .Cr
Vector x-component y-component
Ar
6 −3
Br
−3 4
Cr
2 5
Dr
Determine the components of Dr
by adding the components of ,A
r,Br
and .C
r
Dx = 5 and Dy = 6
Use the Pythagorean Theorem to calculate the magnitude of D
r:
( ) ( ) 81.765 2222 =+=+= yx DDD
and correct. is )(d
45 • Picture the Problem The components of the given vector can be determined using right-triangle trigonometry. Use the trigonometric relationships between the magnitude of a vector and its components to calculate the x- and y-components of each vector.
A θ Ax Ay (a) 10 m 30° 8.66 m 5 m (b) 5 m 45° 3.54 m 3.54 m (c) 7 km 60° 3.50 km 6.06 km (d) 5 km 90° 0 5 km
Chapter 3
142
(e) 15 km/s 150° −13.0 km/s 7.50 km/s (f) 10 m/s 240° −5.00 m/s −8.66 m/s (g) 8 m/s2 270° 0 −8.00 m/s2
*46 • Picture the Problem Vectors can be added and subtracted by adding and subtracting their components. Write A
rin component form:
Ax = (8 m) cos 37° = 6.4 m Ay = (8 m) sin 37° = 4.8 m ∴ ( ) ( ) jiA ˆm8.4ˆm4.6 +=r
(a), (b), (c) Add (or subtract) x- and y-components:
( ) ( )
( ) ( )
( ) ( ) jiF
jiE
jiD
ˆm8.23ˆm6.17
ˆm8.9ˆm4.3
ˆm8.7ˆm4.0
+−=
−−=
+=
r
r
r
(d) Solve for G
rand add
components to obtain: ( )
( ) ( ) ji
CBAG
ˆm9.2ˆm3.1
221
−=
++−=rrrr
47 •• Picture the Problem The magnitude of each vector can be found from the Pythagorean theorem and their directions found using the inverse tangent function. (a) jiA ˆ3ˆ5 +=
r
83.522 =+= yx AAA
and, because Ar
is in the 1st quadrant,
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 0.31tan 1
x
y
AA
θ
(b) jiB ˆ7ˆ10 −=
r
2.1222 =+= yx BBB
and, because Br
is in the 4th quadrant,
°−=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 0.35tan 1
x
y
BB
θ
(c) kjiC ˆ4ˆ3ˆ2 +−−=
r 39.5222 =++= zyx CCCC
°=⎟⎠⎞
⎜⎝⎛= − 1.42cos 1
CCzθ
where θ is the polar angle measured from the positive z-axis and
Motion in One and Two Dimensions
143
°=⎟⎠
⎞⎜⎝
⎛ −=⎟
⎠⎞
⎜⎝⎛= −− 112
292coscos 11
CCxφ
48 • Picture the Problem The magnitude and direction of a two-dimensional vector can be found by using the Pythagorean Theorem and the definition of the tangent function.
(a) jiA ˆ7ˆ4 −−=r
06.822 =+= yx AAA
and, because Ar
is in the 3rd quadrant,
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 240tan 1
x
y
AA
θ
jiB ˆ2ˆ3 −=
r 61.322 =+= yx BBB
and, because Br
is in the 4th quadrant,
°−=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 7.33tan 1
x
y
BB
θ
jiBAC ˆ9ˆ −−=+=
rrr 06.922 =+= yx CCC
and, because Cr
is in the 3rd quadrant,
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 264tan 1
x
y
CC
θ
(b) Follow the same steps as in (a). A = 12.4 ; θ = °− 0.76
B = 6.32 ; θ = °71.6
C = 3.61 ; θ = °33.7
49 • Picture the Problem The components of these vectors are related to the magnitude of each vector through the Pythagorean Theorem and trigonometric functions. In parts (a) and (b), calculate the rectangular components of each vector and then express the vector in rectangular form.
Chapter 3
144
(a) Express vr
in rectangular form: jiv ˆˆyx vv +=
r
Evaluate vx and vy: vx = (10 m/s) cos 60° = 5 m/s
and vy = (10 m/s) sin 60° = 8.66 m/s
Substitute to obtain: jiv ˆ)m/s66.8(ˆ)m/s5( +=r
(b) Express vr in rectangular form:
jiA ˆˆyx AA +=
r
Evaluate Ax and Ay:
Ax = (5 m) cos 225° = −3.54 m and Ay = (5 m) sin 225° = −3.54 m
Substitute to obtain: ( ) ( ) jiA ˆm54.3ˆm54.3 −+−=r
(c) There is nothing to calculate as we are given the rectangular components:
( ) ( ) jir ˆm6ˆm14 −=r
50 • Picture the Problem While there are infinitely many vectors B that can be constructed such that A = B, the simplest are those which lie along the coordinate axes. Determine the magnitude of :A
r
543 2222 =+=+= yx AAA
Write three vectors of the same magnitude as :A
r
jBiBiB ˆ5andˆ5ˆ5 321 =−==
rrr,,
The vectors are shown to the right:
Motion in One and Two Dimensions
145
*51 •• Picture the Problem While there are several walking routes the fly could take to get from the origin to point C, its displacement will be the same for all of them. One possible route is shown in the figure.
Express the fly’s displacement D
rduring its trip from
the origin to point C and find its magnitude:
( ) ( ) ( )kji
CBADˆm3ˆm3ˆm3 ++=
++=rrrr
and
( ) ( ) ( )m20.5
m3m3m3 222
=
++=D
*52 • Picture the Problem The diagram shows the locations of the transmitters relative to the ship and defines the distances separating the transmitters from each other and from the ship. We can find the distance between the ship and transmitter B using trigonometry.
Relate the distance between A and B to the distance from the ship to A and the angle θ:
SB
ABtanDD
=θ
Solve for and evaluate the distance from the ship to transmitter B:
km17330tankm100
tanAB
SB =°
==θ
DD
Chapter 3
146
Velocity and Acceleration Vectors 53 • Picture the Problem For constant speed and direction, the instantaneous velocity is identical to the average velocity. Take the origin to be the location of the stationary radar and construct a pictorial representation.
Express the average velocity: t
r∆∆
=r
ravv
Determine the position vectors:
( )
( ) ( ) jir
jr
ˆkm1.14ˆkm1.14
and
ˆkm10
2
1
−+=
−=
r
r
Find the displacement vector: ( ) ( ) ji
rrrˆkm1.4ˆkm1.14
12
−+=
−=∆rrr
Substitute for r
r∆ and ∆t to find the
average velocity. ( ) ( )
( ) ( ) ji
jiv
ˆkm/h1.4ˆkm/h1.14
h1
ˆkm1.4ˆkm1.14av
−+=
−+=
r
54 • Picture the Problem The average velocity is the change in position divided by the elapsed time. (a) The average velocity is: t
rv∆∆
=av
Find the position vectors and the displacement vector:
( ) ( ) jir ˆm3ˆm20 +=r
( ) ( ) jir ˆm7ˆm62 +=r
and
( ) ( ) jirrr ˆm4ˆm412 +=−=∆rrr
Find the magnitude of the displacement vector for the interval between t = 0 and t = 2 s:
( ) ( ) m66.5m4m4 2202 =+=∆r
Motion in One and Two Dimensions
147
Substitute to determine vav: m/s83.2
s2m66.5
av ==v
and
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 0.45
m4m4tan 1θ measured
from the positive x axis.
(b) Repeat (a), this time using the displacement between t = 0 and t = 5 s to obtain:
( ) ( ) jir ˆm14ˆm135 +=r
,
( ) ( ) jirrr ˆm11ˆm110505 +=−=∆rrr
,
( ) ( ) m15.6m11m11 2205 =+=∆r ,
m/s3.11s5m15.6
av ==v ,
and
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 0.45
m11m11tan 1θ measured
from the positive x axis. *55 • Picture the Problem The magnitude of the velocity vector at the end of the 2 s of acceleration will give us its speed at that instant. This is a constant-acceleration problem.
Find the final velocity vector of the particle:
( ) ( )( )( ) ( ) ji
ji
jijiv
ˆm/s0.6ˆm/s0.4
ˆs0.2m/s0.3ˆm/s0.4
ˆˆˆˆ
2
0
+=
+=
+=+= tavvv yxyxr
Find the magnitude of :vr ( ) ( ) m/s7.21m/s6.0m/s4.0 22 =+=v
and correct. is )(b
56 • Picture the Problem Choose a coordinate system in which north coincides with the positive y direction and east with the positive x direction. Expressing the west and north velocity vectors is the first step in determining vr∆ and avar . (a) The magnitudes of
NW and vv rrare 40 m/s and 30 m/s,
respectively. The change in the magnitude of the particle’s velocity during this time is:
m/s10WN
−=
−=∆ vvv
Chapter 3
148
(b) The change in the direction of the velocity is from west to north.
The change in direction is °90
(c) The change in velocity is:
( ) ( )( ) ( ) ji
ijvvvˆm/s30ˆm/s40
ˆm/s40ˆm/s30WN
+=
−−=−=∆rrr
Calculate the magnitude and direction of :vr∆ ( ) ( ) m/s50m/s30m/s40 22 =+=∆v
and
°== −+ 9.36
m/s40m/s30tan 1
axisxθ
(d) Find the average acceleration during this interval:
( ) ( )
( ) ( )ji
jiva
ˆm/s6ˆm/s8
s5
ˆm/s30ˆm/s40
22
av
+=
+=∆∆≡ t
rr
The magnitude of this vector is: ( ) ( ) 22222
av m/s10m/s6m/s8 =+=a
and its direction is
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 9.36
m/s8m/s6tan 2
21θ measured
from the positive x axis. 57 • Picture the Problem The initial and final positions and velocities of the particle are given. We can find the average velocity and average acceleration using their definitions by first calculating the given displacement and velocities using unit vectors .ˆ and ji (a) The average velocity is: t∆∆≡ rv rr
av
The displacement of the particle during this interval of time is:
( ) ( ) jir ˆm80ˆm100 +=∆r
Substitute to find the average velocity:
( ) ( )
( ) ( ) ji
jiv
ˆm/s7.26ˆm/s3.33
s3
ˆm80ˆm100av
+=
+=
r
(b) The average acceleration is:
t∆∆= va rrav
Motion in One and Two Dimensions
149
Find vvv vrr∆and,, 21 :
( ) ( )
( ) ( )( ) ( ) jiv
jiv
jiv
ˆm/s30.5ˆm/s00.9
ˆm/s0.23ˆm/s3.19
and
ˆm/s3.28ˆm/s3.28
2
1
−+−=∆∴
+=
+=
r
r
r
Using ∆t = 3 s, find the average acceleration: ( ) ( )jia ˆm/s77.1ˆm/s00.3 22
av −+−=r
*58 •• Picture the Problem The acceleration is constant so we can use the constant-acceleration equations in vector form to find the velocity at t = 2 s and the position vector at t = 4 s. (a) The velocity of the particle, as a function of time, is given by:
tavv rrr+= 0
Substitute to find the velocity at t = 2 s: [ ]( )
ji
ji
jiv
ˆm/s) 3( m/s) (10
s2ˆ)m/s (3ˆ)m/s (4
ˆm/s) 9(ˆm/s) (222
−+=
++
−+=r
(b) Express the position vector as a function of time:
221
00 tt avrrrrrr
++=
Substitute and simplify: [ ]( )
[ ]( )ji
ji
ji
jir
ˆm) 9( m) (44
s4ˆ)m/s (3 )m/s (4
s4ˆm/s) (-9 m/s) (2
ˆm) (3 m) (4
22221
−+=
++
++
+=r
Find the magnitude and direction of rr
at t = 4 s: ( ) ( ) m9.44m9m44s) (4 22 =−+=r
and, because rr
is in the 4th quadrant,
°−=⎟⎟⎠
⎞⎜⎜⎝
⎛ −= − 6.11
m44m9tan 1θ
59 •• Picture the Problem The velocity vector is the time-derivative of the position vector and the acceleration vector is the time-derivative of the velocity vector. Differentiate r
rwith respect to time: ( ) ( )[ ]
( ) ji
jirv
ˆ1040ˆ30
ˆ540ˆ30 2
t
tttdtd
dtd
−+=
−+==r
r
Chapter 3
150
where vr has units of m/s if t is in seconds.
Differentiate vr with respect to time: ( )[ ]( )j
jia
ˆm/s10
ˆ1040ˆ30
2−=
−+== tdtd
dtvdrr
60 •• Picture the Problem We can use the constant-acceleration equations in vector form to solve the first part of the problem. In the second part, we can eliminate the parameter t from the constant-acceleration equations and express y as a function of x. (a) Use 0with 00 =+= vavv rrrr t to find :vr
( ) ( )[ ]tjiv ˆm/s4ˆm/s6 22 +=r
Use 221
00 tt avrrrrrr
++= with ( )ir ˆm100 =r
to find :rr
( ) ( )[ ] ( )[ ]jir ˆm/s2ˆm/s3m10 2222 tt ++=r
(b) Obtain the x and y components of the path from the vector equation in (a):
( ) 22m/s3m10 tx += and
( ) 22m/s2 ty = Eliminate the parameter t from these equations and solve for y to obtain:
m3
20 32 −= xy
Use this equation to plot the graph shown below. Note that the path in the xy plane is a straight line.
0
2
4
6
8
10
12
14
16
18
20
0 10 20 30 40
x (m)
y (m
)
Motion in One and Two Dimensions
151
61 ••• Picture the Problem The displacements of the boat are shown in the figure. We need to determine each of the displacements in order to calculate the average velocity of the boat during the 30-s trip.
(a) Express the average velocity of the boat:
t∆∆
=rvr
rav
Express its total displacement: ( ) ( )ij
rrrˆˆ
WW2
NN21
WN
−∆+∆=
∆+∆=∆
tvta
rrr
To calculate the displacement we first have to find the speed after the first 20 s:
vW = vN, f = aN∆tN = 60 m/s
so ( ) ( )
( ) ( )ij
ijrˆm600 ˆm600
ˆm/s 60ˆW
2NN2
1
−=
∆−∆=∆ ttar
Substitute to find the average velocity:
( )( )
( )( )ji
jirv
ˆˆm/s20
s30
ˆˆm600av
+−=
+−=
∆∆
=t
rr
(b) The average acceleration is given by:
( ) ( )ii
vvra
ˆm/s2s30
0ˆm/s60 2
ifav
−=−−
=
∆−
=∆∆
=tt
rrrr
(c) The displacement of the boat from the dock at the end of the 30-s trip was one of the intermediate results we obtained in part (a).
( ) ( )( )( )ji
ijrˆˆm600
ˆm600ˆm600
+−=
−+=∆r
Chapter 3
152
*62 ••• Picture the Problem Choose a coordinate system with the origin at Petoskey, the positive x direction to the east, and the positive y direction to the north. Let t = 0 at 9:00 a.m. and θ be the angle between Robert’s velocity vector and the easterly direction and let ″M″ and ″R″ denote Mary and Robert, respectively. You can express the positions of Mary and Robert as functions of time and then equate their north (y) and east (x) coordinates at the time they rendezvous.
Express Mary’s position as a function of time:
( ) jjr ˆ8ˆMM ttv ==
r
where Mrr
is in miles if t is in hours.
Note that Robert’s initial position coordinates (xi, yi) are:
(xi, yi) = (−13 mi, 22.5 mi)
Express Robert’s position as a function of time:
ji
jirˆ}]sin)1(6{5.22[ˆ}]cos)1(6{13[
ˆ)]1)(sin([ˆ1)]))(cos( [ RiRiR
θθ
θθ
−++−+−=
−++−+=
tt
tvytvxr
where Rrr
is in miles if t is in hours. When Mary and Robert rendezvous, their coordinates will be the same. Equating their north and east coordinates yields:
East: –13 + 6t cosθ – 6 cosθ = 0 (1) North: 22.5 + 6t sinθ – 6 sinθ = 8t (2)
Solve equation (1) for cosθ : ( )16
13cos−
=t
θ (3)
Solve equation (2) for sinθ : ( )16
5.228sin−
−=
ttθ (4)
Square and add equations (3) and (4) to obtain:
( ) ( )
2222
1613
165.2281cossin ⎥
⎦
⎤⎢⎣
⎡−
+⎥⎦
⎤⎢⎣
⎡−
−==+
tttθθ
Simplify to obtain a quadratic equation in t:
063928828 2 =+− tt
Solve (you could use your calculator’s ″solver″ function) this
min15h3h24.3 ==t
Motion in One and Two Dimensions
153
equation for the smallest value of t (both roots are positive) to obtain: Now you can find the distance traveled due north by Mary:
( )( ) mi9.25h24.3mi/h8MM === tvr
Finally, solving equation (3) for θ and substituting 3.24 h for t yields:
( ) ( ) °=⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡−
= −− 7.14124.36
13cos16
13cos 11
tθ
and so Robert should head east. ofnorth 7.14 °
Remarks: Another solution that does not depend on the components of the vectors utilizes the law of cosines to find the time t at which Mary and Robert meet and then uses the law of sines to find the direction that Robert must head in order to rendezvous with Mary. Relative Velocity 63 •• Picture the Problem Choose a coordinate system in which north is the positive y direction and east is the positive x direction. Let θ be the angle between north and the direction of the plane’s heading. The velocity of the plane relative to the ground, PGvr , is the sum of the velocity of the plane relative to the air,
PAvr , and the velocity of the air relative to the ground, AGvr . i.e.,
AGPAPG vvv rrr+=
The pilot must head in such a direction that the east-west component of PGvr is zero in order to make the plane fly due north.
(a) From the diagram one can see that:
vAG cos 45° = vPA sinθ
Solve for and evaluate θ :
north of west 1.13
km/h250km/h6.56sin 1
°=
⎟⎟⎠
⎞⎜⎜⎝
⎛= −θ
(b) Because the plane is headed due north, add the north components of PGvr = (250 km/h) cos 13.1°
+ (80 km/h) sin 45°
Chapter 3
154
AGPA and vv rr to determine the
plane’s ground speed: = km/h300
64 •• Picture the Problem Let SBvr represent the velocity of the swimmer relative to the bank; SWvr the velocity of the swimmer relative to the water; and WBvr the velocity of the water relative to the shore; i.e.,
SBvr = SWvr + WBvr The current of the river causes the swimmer to drift downstream. (a) The triangles shown in the figure are similar right triangles. Set up a proportion between their sides and solve for the speed of the water relative to the bank:
m80m40
SW
WB =vv
and ( ) m/s800.0m/s6.12
1WB ==v
(b) Use the Pythagorean Theorem to solve for the swimmer’s speed relative to the shore:
( ) ( )m/s79.1
m/s8.0m/s6.1 22
2WS
2SWSB
=
+=
+= vvv
(c) The swimmer should head in a direction such that the upstream component of her velocity is equal to the speed of the water relative to the shore:
Use a trigonometric function to evaluate θ: °=⎟⎟
⎠
⎞⎜⎜⎝
⎛= − 0.30
m/s1.6m/s0.8sin 1θ
Motion in One and Two Dimensions
155
*65 ••
Use the diagram to express the condition relating the eastward component of AGvr and the westward component of .PAvr This must be satisfied if the plane is to stay on its northerly course. [Note: this is equivalent to equating the x-components of equation (1).]
(50 km/h) cos 45° = (240 km/h) sinθ
Now solve for θ to obtain:
( )°=⎥
⎦
⎤⎢⎣
⎡ °= − 47.8
km/h240cos45km/h50sin 1θ
Add the north components of PAvr and AGvr to find the velocity of the plane relative to the ground:
vPG + vAGsin45° = vPAcos8.47° and vPG = (240 km/h)cos 8.47° − (50 km/h)sin 45° = 202 km/h
Finally, find the time of flight:
h57.2km/h202km520
travelleddistance
PGflight
==
=v
t
Picture the Problem Let the velocity of the plane relative to the ground be represented by ;PGvr the velocity of the plane relative to the air by ,PAvr and the velocity of the air relative to the ground by
.AGvr Then
AGPAPG vvv rrr+= (1)
Choose a coordinate system with the origin at point A, the positive x direction to the east, and the positive y direction to the north. θ is the angle between north and the direction of the plane’s heading. The pilot must head so that the east-west component of PGvr is zero in order to make the plane fly due north.
Chapter 3
156
66 •• Picture the Problem Let BSvr be the velocity of the boat relative to the shore;
BWvr be the velocity of the boat relative to the water; and WSvr represent the velocity of the water relative to the shore. Independently of whether the boat is going upstream or downstream:
BSvr = BWvr + WSvr
Going upstream, the speed of the boat relative to the shore is reduced by the speed of the water relative to the shore. Going downstream, the speed of the boat relative to the shore is increased by the same amount.
For the upstream leg of the trip:
vBS = vBW − vWS
For the downstream leg of the trip:
vBS = vBW + vWS
Express the total time for the trip in terms of the times for its upstream and downstream legs: WSBWWSBW
downstreamupstreamtotal
vvL
vvL
ttt
++
−=
+=
Multiply both sides of the equation by (vBW − vWS)(vBW + vWS) (the product of the denominators) and rearrange the terms to obtain:
02 2WSBW
total
2BW =−− vv
tLv
Solve the quadratic equation for vBW. (Only the positive root is physically meaningful.)
vBW = km/h18.5
67 •• Picture the Problem Let pgvr be the velocity of the plane relative to the ground;
agvr be the velocity of the air relative to the
ground; and pavr the velocity of the plane
relative to the air. Then, pgvr
= pavr
+
.agvr The wind will affect the flight times differently along these two paths.
Motion in One and Two Dimensions
157
The velocity of the plane, relative to the ground, on its eastbound leg is equal to its velocity on its westbound leg. Using the diagram, find the velocity of the plane relative to the ground for both directions:
( ) ( ) m/s1.14m/s5m/s15 22
2ag
2papg
=−=
−= vvv
Express the time for the east-west roundtrip in terms of the distances and velocities for the two legs:
s141m/s14.1
m102
circle the of radius
circletheofradius
3
westboundpg,
eastboundpg,
westboundeastboundEWtrip,round
=×
=
+
=
+=
v
v
ttt
Use the distances and velocities for the two legs to express and evaluate the time for the north-south roundtrip:
s150m/s) (5m/s) (15
m10m/s) (5m/s) (15
m10
circletheofradiuscircletheofradius
33
southboundpg,northboundpg,southboundnorthboundNStrip,round
=+
+−
=
+=+=vv
ttt
wind. theacross planeyour fly shouldyou , Because NStrip,roundEWroundtrip, tt <
68 • Picture the Problem This is a relative velocity problem. The given quantities are the direction of the velocity of the plane relative to the ground and the velocity (magnitude and direction) of the air relative to the ground. Asked for is the direction of the velocity of the air relative to the ground. Using ,AGPAPG vvv rrr
+= draw a vector addition diagram and solve for the unknown quantity.
Calculate the heading the pilot must take: °== − 5.11
kts150kts30sin 1θ
Because this is also the angle of the plane's heading clockwise from north, it is also its azimuth or the required true heading:
Az = (011.5°)
Chapter 3
158
*69 •• Picture the Problem The position of B relative to A is the vector from A to B; i.e.,
ABAB rrr rrr−=
The velocity of B relative to A is
dtd ABAB rv rr=
and the acceleration of B relative to A is
dtd ABAB va rr=
Choose a coordinate system with the origin at the intersection, the positive x direction to the east, and the positive y direction to the north.
(a) Find ABAB and,, rrr rrr
:
( )[ ]( )[ ]ir
jrˆm/s20
ˆm/s2m40
A
2221
B
t
t
=
−=r
r
and
( )[ ]( )[ ]j
i
r
ˆm/s2m40
ˆm/s2022
21
ABAB
t
t
rr
−+
−=
−=rrr
Evaluate ABrr at t = 6 s:
jir ˆ m) (4 m) (120)s6(AB +=r
(b) Find dtd ABAB rv rr= :
( ){ }[( ){ } ]
ji
j
irv
ˆ)m/s 2(ˆm/s) 20(
ˆm/s 2m 40
m/s 20
2
2221
ABAB
t
t
tdtd
dtd
−+−=
−+
−==rr
r
Evaluate ABvr at t = 6 s:
( ) ( ) ( ) jiv ˆm/s12ˆm/s20s6AB −−=r
(c) Find dtd ABAB va rr= :
[ ]( )j
jia
ˆm/s2
ˆ )m/s 2( m/s) 20(
2
2AB
−=
−+−= tdtdr
Note that ABar is independent of time. *70 ••• Picture the Problem Let h and h′ represent the heights from which the ball is dropped and to which it rebounds, respectively. Let v and v′ represent the speeds with which the ball strikes the racket and rebounds from it. We can use a constant-acceleration equation to relate the pre- and post-collision speeds of the ball to its drop and rebound heights.
Motion in One and Two Dimensions
159
(a) Using a constant-acceleration equation, relate the impact speed of the ball to the distance it has fallen:
ghvv 220
2 += or, because v0 = 0,
ghv 2=
Relate the rebound speed of the ball to the height to which it rebounds:
gh'v'v 222 −= or because v = 0,
gh'v' 2=
Divide the second of these equations by the first to obtain:
hh'
ghgh'
vv'
==22
Substitute for h′ and evaluate the ratio of the speeds:
8.064.0==
hh
vv'
⇒ vv' 8.0=
(b) Call the speed of the racket V. In a reference frame where the racket is unmoving, the ball initially has speed V, moving toward the racket. After it "bounces" from the racket, it will have speed 0.8 V, moving away from the racket. In the reference frame where the racket is moving and the ball initially unmoving, we need to add the speed of the racket to the speed of the ball in the racket's rest frame. Therefore, the ball's speed is:
mi/h100
m/s548.18.0
≈
==+= VVVv'
This speed is close to that of a tennis pro’s serve. Note that this result tells us that the ball is moving significantly faster than the racket.
(c) racket. theas
fast as e than twicmore movenever can ball the),(part in result theFrom b
Circular Motion and Centripetal Acceleration 71 • Picture the Problem We can use the definition of centripetal acceleration to express ac in terms of the speed of the tip of the minute hand. We can find the tangential speed of the tip of the minute hand by using the distance it travels each revolution and the time it takes to complete each revolution. Express the acceleration of the tip of the minute hand of the clock as a function of the length of the hand and the speed of its tip:
Rva
2
c =
Use the distance the minute hand travels every hour to express its speed:
TRv π2
=
Chapter 3
160
Substitute to obtain: 2
2
c4
TRa π
=
Substitute numerical values and evaluate ac:
( )( )
262
2
c m/s1052.1s3600m5.04 −×==
πa
Express the ratio of ac to g:
72
26c 1055.1
m/s9.81m/s101.52 −
−
×=×
=ga
72 • Picture the Problem The diagram shows the centripetal and tangential accelerations experienced by the test tube. The tangential acceleration will be zero when the centrifuge reaches its maximum speed. The centripetal acceleration increases as the tangential speed of the centrifuge increases. We can use the definition of centripetal acceleration to express ac in terms of the speed of the test tube. We can find the tangential speed of the test tube by using the distance it travels each revolution and the time it takes to complete each revolution. The tangential acceleration can be found from the change in the tangential speed as the centrifuge is spinning up.
(a) Express the acceleration of the centrifuge arm as a function of the length of its arm and the speed of the test tube:
Rva
2
c =
Use the distance the test tube travels every revolution to express its speed:
TRv π2
=
Substitute to obtain: 2
2
c4
TRa π
=
Substitute numerical values and evaluate ac:
( )
25
2
2
c
m/s1070.3
mins60
rev15000min1
m15.04
×=
⎟⎟⎠
⎞⎜⎜⎝
⎛×
=πa
Motion in One and Two Dimensions
161
(b) Express the tangential acceleration in terms of the difference between the final and initial tangential speeds:
tTR
tT
R
tvva
∆=
∆
−=
∆−
=π
π202
ift
Substitute numerical values and evaluate aT:
( )
( )
2
t
m/s14.3
s75min
s60rev15000
min1m15.02
=
⎟⎟⎠
⎞⎜⎜⎝
⎛×
=πa
73 • Picture the Problem The diagram includes a pictorial representation of the earth in its orbit about the sun and a force diagram showing the force on an object at the equator that is due to the earth’s rotation,
,RFr
and the force on the object due to the
orbital motion of the earth about the sun, .oF
rBecause these are centripetal forces,
we can calculate the accelerations they require from the speeds and radii associated with the two circular motions.
Express the radial acceleration due to the rotation of the earth: R
va2R
R =
Express the speed of the object on the equator in terms of the radius of the earth R and the period of the earth’s rotation TR:
RR
2T
Rv π=
Substitute for vR in the expression for aR to obtain: 2
R
2
R4
TRa π
=
Substitute numerical values and evaluate aR:
( )( )
g
a
3
22
2
32
R
1044.3
m/s1037.3
h1s3600h24
m1063704
−
−
×=
×=
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛
×=
π
Note that this effect gives rise to the well-known latitude correction for g.
Chapter 3
162
Express the radial acceleration due to the orbital motion of the earth: r
va2o
o =
Express the speed of the object on the equator in terms of the earth-sun distance r and the period of the earth’s motion about the sun To:
oo
2T
rv π=
Substitute for vo in the expression for ao to obtain: 2
o
2
o4T
ra π=
Substitute numerical values and evaluate ao:
( )( )
g
a
423
2
112
o
1007.6m/s1095.5
h1s3600
d1h24d365
m10.514
−− ×=×=
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
×=
π
74 •• Picture the Problem We can relate the acceleration of the moon toward the earth to its orbital speed and distance from the earth. Its orbital speed can be expressed in terms of its distance from the earth and its orbital period. From tables of astronomical data, we find that the sidereal period of the moon is 27.3 d and that its mean distance from the earth is 3.84×108 m. Express the centripetal acceleration of the moon: r
va2
c =
Express the orbital speed of the moon: T
rv π2=
Substitute to obtain:
2
2
c4T
ra π=
Substitute numerical values and evaluate ac:
( )
g
πa
4
23
2
82
c
1078.2
m/s1072.2h
s3600d
h24d27.3
m103.844
−
−
×=
×=
⎟⎠⎞
⎜⎝⎛ ××
×=
Motion in One and Two Dimensions
163
Remarks: Note that moon to earth fromdistance
earthofradiusgac = (ac is just the acceleration
due to the earth’s gravity evaluated at the moon’s position). This is Newton’s famous ″falling apple″ observation. 75 • Picture the Problem We can find the number of revolutions the ball makes in a given period of time from its speed and the radius of the circle along which it moves. Because the ball’s centripetal acceleration is related to its speed, we can use this relationship to express its speed. Express the number of revolutions per minute made by the ball in terms of the circumference c of the circle and the distance x the ball travels in time t:
cxn = (1)
Relate the centripetal acceleration of the ball to its speed and the radius of its circular path:
Rvga
2
c ==
Solve for the speed of the ball:
Rgv =
Express the distance x traveled in time t at speed v:
vtx =
Substitute to obtain:
tRgx =
The distance traveled per revolution is the circumference c of the circle:
Rc π2=
Substitute in equation (1) to obtain: t
Rg
RtRg
nππ 21
2==
Substitute numerical values and evaluate n: ( ) 1
2
min33.4s60m0.8
m/s9.8121 −==π
n
Remarks: The ball will oscillate at the end of this string as a simple pendulum with a period equal to 1/n. Projectile Motion and Projectile Range 76 • Picture the Problem Neglecting air resistance, the accelerations of the ball are constant and the horizontal and vertical motions of the ball are independent of each other. We can use the horizontal motion to determine the time-of-flight and then use this information to determine the distance the ball drops. Choose a coordinate system in which the origin is at the point of release of the ball, downward is the positive y direction, and the horizontal
Chapter 3
164
direction is the positive x direction. Express the vertical displacement of the ball:
( )221
0 tatvy yy ∆+∆=∆ or, because v0y = 0 and ay = g,
( )221 tgy ∆=∆
Find the time of flight from vx = ∆x/∆t: ( )( )
( )( ) s0.473m/km1000km/h140s/h3600m18.4
==
∆=∆
xvxt
Substitute to find the vertical displacement in 0.473 s:
( )( ) m1.10s0.473m/s9.81 2221 ==∆y
77 • Picture the Problem In the absence of air resistance, the maximum height achieved by a projectile depends on the vertical component of its initial velocity. The vertical component of the projectile’s initial velocity is:
v0y = v0 sinθ0
Use the constant-acceleration equation:
v y2 = v0 y
2 + 2ay∆y
Set vy = 0, a = −g, and ∆y = h to obtain:
( )g
vh2
sin 200 θ
=
*78 •• Picture the Problem Choose the coordinate system shown to the right. Because, in the absence of air resistance, the horizontal and vertical speeds are independent of each other, we can use constant-acceleration equations to relate the impact speed of the projectile to its components.
The horizontal and vertical velocity components are:
v0x = vx= v0cosθ and v0y = v0sinθ
Using a constant-acceleration equation, relate the vertical
yavv yyy ∆+= 220
2
or, because ay = −g and ∆y = −h,
Motion in One and Two Dimensions
165
component of the velocity to the vertical displacement of the projectile:
( ) ghvvy 2sin 20
2 += θ
Express the relationship between the magnitude of a velocity vector and its components, substitute for the components, and simplify to obtain:
( )( )
ghv
ghv
vvvvv yyx
2
2cossin
cos
20
2220
220
222
+=
++=
+=+=
θθ
θ
Substitute for v: ( ) ghvv 22.1 2
02
0 +=
Set v = 1.2 v0, h = 40 m and solve for v0:
m/s2.420 =v
Remarks: Note that v is independent of θ. This will be more obvious once conservation of energy has been studied. 79 •• Picture the Problem Example 3-12 shows that the dart will hit the monkey unless the dart hits the ground before reaching the monkey’s line of fall. What initial speed does the dart need in order to just reach the monkey’s line of fall? First, we will calculate the fall time of the monkey, and then we will calculate the horizontal component of the dart’s velocity. Using a constant-acceleration equation, relate the monkey’s fall distance to the fall time:
221 gth =
Solve for the time for the monkey to fall to the ground: g
ht 2=
Substitute numerical values and evaluate t:
( ) s51.1m/s9.81
m2.1122 ==t
Let θ be the angle the barrel of the dart gun makes with the horizontal. Then:
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 3.11
m50m10tan 1θ
Use the fact that the horizontal velocity is constant to determine v0:
( ) m/s8.33cos11.3
s1.51m50 cos0 =
°==
θxvv
Chapter 3
166
80 •• Picture the Problem Choose the coordinate system shown in the figure to the right. In the absence of air resistance, the projectile experiences constant acceleration in both the x and y directions. We can use the constant-acceleration equations to express the x and y coordinates of the projectile along its trajectory as functions of time. The elimination of the parameter t will yield an expression for y as a function of x that we can evaluate at (R, 0) and (R/2, h). Solving these equations simultaneously will yield an expression for θ.
Express the position coordinates of the projectile along its flight path in terms of the parameter t:
( )tvx θcos0= and
( ) 221
0 sin gttvy −= θ
Eliminate the parameter t to obtain:
( ) 222
0 cos2tan x
vgxy
θθ −= (1)
Evaluate equation (1) at (R, 0) to
obtain: g
vR θθ cossin2 20=
Evaluate equation (1) at (R/2, h) to obtain:
( )g
vh2sin 2
0 θ=
Equate R and h and solve the resulting equation for θ :
( ) °== − 0.764tan 1θ
Remarks: Note that this result is independent of v0. 81 •• Picture the Problem In the absence of air resistance, the motion of the ball is uniformly accelerated and its horizontal and vertical motions are independent of each other. Choose the coordinate system shown in the figure to the right and use constant-acceleration equations to relate the x and y components of the ball’s initial velocity. Use the components of v0 to express θ in terms of v0x and v0y: x
y
vv
0
01tan−=θ (1)
Motion in One and Two Dimensions
167
Use the Pythagorean relationship between the velocity and its components to express v0:
20
200 yx vvv += (2)
Using a constant-acceleration equation, express the vertical speed of the projectile as a function of its initial upward speed and time into the flight:
vy= v0y+ ay t
Because vy = 0 halfway through the flight (at maximum elevation):
v0y = (9.81 m/s2)(1.22 s) = 12.0 m/s
Determine v0x: m/s4.16
s2.44m40
0x ==∆∆
=txv
Substitute in equation (2) and evaluate v0:
( ) ( )m/s3.20
m/s0.12m/s4.16 220
=
+=v
Substitute in equation (1) and evaluate θ :
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 2.36
m/s16.4m/s12.0tan 1θ
*82 •• Picture the Problem In the absence of friction, the acceleration of the ball is constant and we can use the constant- acceleration equations to describe its motion. The figure shows the launch conditions and an appropriate coordinate system. The speeds v, vx, and vy are related through the Pythagorean Theorem.
The squares of the vertical and horizontal components of the object’s velocity are: θ
θ
220
2
220
2
cosand
2sin
vv
ghvv
x
y
=
−=
The relationship between these variables is:
222yx vvv +=
Substitute and simplify to obtain: ghvv 220
2 −=
Note that v is independent of θ ... as was to be shown.
Chapter 3
168
83 •• Picture the Problem In the absence of air resistance, the projectile experiences constant acceleration during its flight and we can use constant-acceleration equations to relate the speeds at half the maximum height and at the maximum height to the launch angle θ of the projectile.
The angle the initial velocity makes with the horizontal is related to the initial velocity components. x
y
vv
0
0tan =θ
Write the equation y,avv yy ∆+= 22
02 for ∆y = h and
vy = 0:
20 20 ghvhy y −=⇒=∆ (1)
Write the equation y,avv yy ∆+= 22
02 for ∆y = h/2:
2
2 2
20
2 hgvvhy yy −=⇒=∆ (2)
We are given vy = (3/4)v0. Square both sides and express this using the components of the velocity. The x component of the velocity remains constant.
( ) 43 2
020
222
0 yxyx vvvv +⎟⎠⎞
⎜⎝⎛=+ (3)
where we have used vx = v0x .
(Equations 1, 2, and 3 constitute three equations and four unknowns v0x, v0y, vy, and h. To solve for any of these unknowns, we first need a fourth equation. However, to solve for the ratio (v0y/v0x) of two of the unknowns, the three equations are sufficient. That is because dividing both sides of each equation by v0x
2 gives three equations and three unknowns vy/v0x, v0y/v0x, and h/ .2
x0v Solve equation 2 for gh and substitute in equation 1: ( )22
020 2 hyy vvv −= ⇒
2
202 y
y
vv =
Substitute for vy2 in equation 3: ( )2
020
220
20 4
321
yxyx vvvv +⎟⎠⎞
⎜⎝⎛=+
Motion in One and Two Dimensions
169
Divide both sides by v0x2 and solve
for v0y/v0x to obtain:
7
and
1169
211
0
0
20
20
20
20
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=+
x
y
x
y
x
y
vv
vv
vv
Using tan θ = v0y/v0x, solve for θ : ( ) °==⎟⎟⎠
⎞⎜⎜⎝
⎛= −− 3.697tantan 1
0
01
x
y
vv
θ
84 • Picture the Problem The horizontal speed of the crate, in the absence of air resistance, is constant and equal to the speed of the cargo plane. Choose a coordinate system in which the direction the plane is moving is the positive x direction and downward is the positive y direction and apply the constant-acceleration equations to describe the crate’s displacements at any time during its flight.
(a) Using a constant-acceleration equation, relate the vertical displacement of the crate ∆y to the time of fall ∆t:
( )221
0 tgtvy y ∆+∆=∆ or, because v0y = 0,
( )221 tgy ∆=∆
Solve for ∆t:
gyt ∆
=∆2
Substitute numerical values and evaluate ∆t:
( ) s5.49m/s81.9
m101222
3
=×
=∆t
(b) The horizontal distance traveled in 49.5 s is:
( ) ( )
km4.12
s5.49s3600
h1km/h900
0
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
∆=∆= tvxR x
(c) Because the velocity of the plane is constant, it will be directly over the crate when it hits the ground; i.e., the distance to the aircraft will be the elevation of the aircraft.
km0.12=∆y
Chapter 3
170
*85 •• Picture the Problem In the absence of air resistance, the accelerations of both Wiley Coyote and the Roadrunner are constant and we can use constant-acceleration equations to express their coordinates at any time during their leaps across the gorge. By eliminating the parameter t between these equations, we can obtain an expression that relates their y coordinates to their x coordinates and that we can solve for their launch angles.
(a) Using constant-acceleration equations, express the x coordinate of the Roadrunner while it is in flight across the gorge:
221
00 tatvxx xx ++= or, because x0 = 0, ax = 0 and v0x = v0 cosθ0,
( )tvx 00 cosθ=
Using constant-acceleration equations, express the y coordinate of the Roadrunner while it is in flight across the gorge:
221
00 tatvyy yy ++= or, because y0 = 0, ay =−g and v0y = v0 sinθ0,
( ) 221
00 sin gttvy −= θ
Eliminate the parameter t to obtain: ( ) 2
022
00 cos2
tan xv
gxyθ
θ −= (1)
Letting R represent the Roadrunner’s range and using the trigonometric identity sin2θ = 2sinθ cosθ, solve for and evaluate its launch speed:
( )( )
m/s0.18
30sinm/s9.81m5.16
2sin
2
00
=
°==
θRgv
(b) Letting R represent Wiley’s range, solve equation (1) for his launch angle:
⎟⎟⎠
⎞⎜⎜⎝
⎛= −
20
10 sin
21
vRgθ
Substitute numerical values and evaluate θ0:
( )( )( )
°=
⎥⎦
⎤⎢⎣
⎡= −
0.13
m/s.081m/s9.81m14.5sin
21
2
21
0θ
Motion in One and Two Dimensions
171
86 • Picture the Problem Because, in the absence of air resistance, the vertical and horizontal accelerations of the cannonball are constant, we can use constant-acceleration equations to express the ball’s position and velocity as functions of time and acceleration. The maximum height of the ball and its time-of-flight are related to the components of its launch velocity.
(a) Using a constant-acceleration equation, relate h to the initial and final speeds of the cannonball:
yavv yy ∆+= 220
2 or, because v = 0 and ay = −g,
ygv y ∆−= 20 20
Find the vertical component of the firing speed:
v0y = v0sinθ = (300 m/s)sin 45° = 212 m/s
Solve for and evaluate h: ( )( ) km29.2
m/s81.92m/s212
2 2
220 ===g
vh y
(b) The total flight time is: ( ) s2.43
m/s9.81m/s21222
2
20
updnup
===
=+=∆
gv
tttt
y
(c) Express the x coordinate of the ball as a function of time:
( ) tvtvx x ∆=∆= θcos00
Evaluate x (= R) when ∆t = 43.2 s: ( )[ ]( )km9.16
s43.2cos45m/s300
=
°=x
87 •• Picture the Problem Choose a coordinate system in which the origin is at the base of the tower and the x- and y-axes are as shown in the figure to the right. In the absence of air resistance, the horizontal speed of the stone will remain constant during its fall and a constant-acceleration equation can be used to determine the time of fall. The final velocity of the stone will be the vector sum of its x and y components.
Chapter 3
172
(a) Using a constant-acceleration equation, express the vertical displacement of the stone (the height of the tower) as a function of the fall time:
( )221
0 tatvy yy ∆+∆=∆
or, because v0y = 0 and a = −g, ( )2
21 tgy ∆−=∆
Solve for and evaluate the time of fall:
( ) s21.2m/s81.9
m24222 =
−−=
∆−=∆
gyt
Use the definition of average velocity to find the velocity with which the stone was thrown from the tower:
s/m14.8s2.21
m180 ==
∆∆
≡=txvv xx
(b) Find the y component of the stone’s velocity after 2.21 s:
m/s 21.7 s) m/s2)(2.21 (9.81 0
0
−=−=
−= gtvv yy
Express v in terms of its components:
22yx vvv +=
Substitute numerical values and evaluate v: ( ) ( )
m/s2.23
m/s7.21m/s14.8 22
=
−+=v
88 •• Picture the Problem In the absence of air resistance, the acceleration of the projectile is constant and its horizontal and vertical motions are independent of each other. We can use constant-acceleration equations to express the horizontal and vertical displacements of the projectile in terms of its time-of-flight. Using a constant-acceleration equation, express the horizontal displacement of the projectile as a function of time:
( )221
0 tatvx xx ∆+∆=∆
or, because v0x = v0cosθ and ax = 0, ( ) tvx ∆=∆ θcos0
Using a constant-acceleration equation, express the vertical displacement of the projectile as a function of time:
( )221
0 tatvy yy ∆+∆=∆ or, because v0y = v0sinθ and ay = −g,
( ) ( )221
0 cos tgtvy ∆−∆=∆ θ
Substitute numerical values to obtain the quadratic equation:
( )( )( )( )22
21 m/s81.9
60sinm/s60m200
t
t
∆−
∆°=−
Solve for ∆t:
∆t = 13.6 s
Motion in One and Two Dimensions
173
Substitute for ∆t and evaluate the horizontal distance traveled by the projectile:
∆x = (60 m/s)(cos60°)(13.6 s) = m408
89 •• Picture the Problem In the absence of air resistance, the acceleration of the cannonball is constant and its horizontal and vertical motions are independent of each other. Choose the origin of the coordinate system to be at the base of the cliff and the axes directed as shown and use constant- acceleration equations to describe both the horizontal and vertical displacements of the cannonball.
Express the direction of the velocity vector when the projectile strikes the ground:
⎟⎟⎠
⎞⎜⎜⎝
⎛= −
x
y
vv1tanθ
Express the vertical displacement using a constant-acceleration equation:
( )221
0 tatvy yy ∆+∆=∆ or, because v0y = 0 and ay = −g,
( )221 tgy ∆−=∆
Set ∆x = −∆y (R = −h) to obtain: ( )221 tgtvx x ∆=∆=∆
Solve for vx: tg
txvx ∆=
∆∆
= 21
Find the y component of the projectile as it hits the ground:
xyy vtgtavv 20 −=∆−=∆+=
Substitute and evaluate θ : ( ) °−=−=⎟⎟
⎠
⎞⎜⎜⎝
⎛= −− 4.632tantan 11
x
y
vv
θ
90 • Picture the Problem In the absence of air resistance, the vertical and horizontal motions of the projectile experience constant accelerations and are independent of each other. Use a coordinate system in which up is the positive y direction and horizontal is the positive x direction and use constant-acceleration equations to describe the horizontal and vertical displacements of the projectile as functions of the time into the flight.
Chapter 3
174
(a) Use a constant-acceleration equation to express the horizontal displacement of the projectile as a function of time:
( ) tvtvx x
∆=∆=∆
θcos0
0
Evaluate this expression when ∆t = 6 s:
( )( )( ) m900s6cos60m/s300 =°=∆x
(b) Use a constant-acceleration equation to express the vertical displacement of the projectile as a function of time:
( ) ( )221
0 sin tgtvy ∆−∆=∆ θ
Evaluate this expression when ∆t = 6 s:
( )( )( ) ( )( ) km38.1s6m/s9.81s6sin60m/s300 2221 =−°=∆y
91 •• Picture the Problem In the absence of air resistance, the acceleration of the projectile is constant and the horizontal and vertical motions are independent of each other. Choose the coordinate system shown in the figure with the origin at the base of the cliff and the axes oriented as shown and use constant-acceleration equations to find the range of the cannonball. Using a constant-acceleration equation, express the horizontal displacement of the cannonball as a function of time:
( )221
0 tatvx xx ∆+∆=∆
or, because v0x = v0cosθ and ax = 0, ( ) tvx ∆=∆ θcos0
Using a constant-acceleration equation, express the vertical displacement of the cannonball as a function of time:
( )221
0 tatvy yy ∆+∆=∆ or, because y = −40 m, a = −g, and v0y = v0sinθ,
( )( )( )( )22
21 m/s81.9
30sinm/s42.2m04
t
t
∆−
∆°=−
Solve the quadratic equation for ∆t:
∆t = 5.73 s
Calculate the range: ( )( )( )m209
s5.73cos30m/s42.2
=
°=∆= xR
Motion in One and Two Dimensions
175
*92 •• Picture the Problem Choose a coordinate system in which the origin is at ground level. Let the positive x direction be to the right and the positive y direction be upward. We can apply constant-acceleration equations to obtain parametric equations in time that relate the range to the initial horizontal speed and the height h to the initial upward speed. Eliminating the parameter will leave us with a quadratic equation in R, the solution to which will give us the range of the arrow. In (b), we’ll find the launch speed and angle as viewed by an observer who is at rest on the ground and then use these results to find the arrow’s range when the horse is moving at 12 m/s.
(a) Use constant-acceleration equations to express the horizontal and vertical coordinates of the arrow’s motion:
tvxxxR x00 =−=∆= and
( ) 221
0 tgtvhy y −++= where
θcos00 vv x = and θsin00 vv y =
Solve the x-component equation for time:
θcos00 vR
vRt
x
==
Eliminate time from the y-component equation:
2
000 2
1⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
xxy v
RgvRvhy
and, at (R, 0),
( ) 222
0 cos2tan0 R
vgRh
θθ −+=
Solve for the range to obtain:
⎟⎟⎠
⎞⎜⎜⎝
⎛++=
θθ 22
0
20
sin2112sin
2 vgh
gvR
Substitute numerical values and evaluate R:
( )( )
( )( )( ) ( ) m6.81
10sinm/s45m25.2m/s81.921120sin
m/s81.92m/s45
22
2
2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛
°++°=R
Chapter 3
176
(b) Express the speed of the arrow in the horizontal direction:
( )m/s56.3
m/s12cos10m/s45archerarrow
=+°=
+= vvvx
Express the vertical speed of the arrow:
( ) m/s7.81sin10m/s45 =°=yv
Express the angle of elevation from the perspective of someone on the ground:
°=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛= −− 90.7
m/s56.3m/s7.81tantan 11
x
y
vv
θ
Express the arrow’s speed relative to the ground: ( ) ( )
m/s56.8m/s7.81m/s56.3 22
220
=
+=
+= yx vvv
Substitute numerical values and evaluate R:
( )( )
( )( )( ) ( ) m104
9.7sinm/s8.65m2.25m/s9.812115.81sin
m/s9.812m/s8.65
22
2
2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛
°++°=R
Remarks: An alternative solution for part (b) is to solve for the range in the reference frame of the archer and then add to it the distance the frame travels, relative to the earth, during the time of flight. 93 • Picture the Problem In the absence of air resistance, the horizontal and vertical motions are independent of each other. Choose a coordinate system oriented as shown in the figure to the right and apply constant-acceleration equations to find the time-of-flight and the range of the spud-plug.
(a) Using a constant-acceleration equation, express the vertical displacement of the plug:
( )221
0 tatvy yy ∆+∆=∆ or, because v0y = 0 and ay = −g,
( )221 tgy ∆−=∆
Solve for and evaluate the flight time ∆t:
( )
s452.0
m/s9.81m00.122
2
=
−−=
∆−=∆
gyt
Motion in One and Two Dimensions
177
(b) Using a constant-acceleration equation, express the horizontal displacement of the plug:
( )221
0 tatvx xx ∆+∆=∆ or, because ax = 0 and v0x = v0,
tvx ∆=∆ 0
Substitute numerical values and evaluate R:
( )( ) m22.6s0.452m/s50 ===∆ Rx
94 •• Picture the Problem An extreme value (i.e., a maximum or a minimum) of a function is determined by setting the appropriate derivative equal to zero. Whether the extremum is a maximum or a minimum can be determined by evaluating the second derivative at the point determined by the first derivative. Evaluate dR/dθ0:
( )[ ] ( )0
20
00
20
0
2cos22sin θθθθ g
vdd
gv
ddR
==
Set dR/dθ0= 0 for extrema and solve for θ0:
( ) 02cos20
20 =θ
gv
and °== − 450cos 1
21
0θ
Determine whether 45° is a maximum or a minimum: ( )[ ]
0
2sin4 45020
452
0
2
0
0
<
−= °=
°=
θθ
θθ
gvd
Rd
∴ R is a maximum at θ0 = 45° 95 • Picture the Problem We can use constant-acceleration equations to express the x and y coordinates of a bullet in flight on the moon as a function of t. Eliminating this parameter will yield an expression for y as a function of x that we can use to find the range of the bullet. The necessity that the centripetal acceleration of an object in orbit at the surface of a body equal the acceleration due to gravity at the surface will allow us to determine the required muzzle velocity for orbital motion.
(a) Using a constant-acceleration equation, express the x coordinate of a bullet in flight on the moon:
221
00 tatvxx xx ++= or, because x0 = 0, ax = 0 and v0x = v0cosθ0,
( )tvx 00 cosθ=
Chapter 3
178
Using a constant-acceleration equation, express the y coordinate of a bullet in flight on the moon:
221
00 tatvyy yy ++= or, because y0 = 0, ay = −gmoon and v0y = v0sinθ0,
( ) 2moon2
100 sin tgtvy −= θ
Eliminate the parameter t to obtain: ( ) 2
022
0
moon0 cos2
tan xv
gxyθ
θ −=
When y = 0 and x = R: ( ) 2
022
0
moon0 cos2
tan0 Rv
gRθ
θ −=
and
0moon
20 2sin θ
gvR =
Substitute numerical values and evaluate R:
( )
km485
m104.85sin90m/s1.67m/s900 5
2
2
=
×=°=R
This result is probably not very accurate
because it is about 28% of the moon’s radius (1740 km). This being the case, we can no longer assume that the ground is ″flat″ because of the curvature of the moon.
(b) Express the condition that the centripetal acceleration must satisfy for an object in orbit at the surface of the moon:
rv
ga2moonc
=
=
Solve for and evaluate v: ( )( )km/s1.70
m101.74m/s1.67 62moon
=
×== rgv
96 ••• Picture the Problem We can show that ∆R/R = –∆g/g by differentiating R with respect to g and then using a differential approximation. Differentiate the range equation with respect to g:
gR
gv
gv
dgd
dgdR
−=
−=⎟⎟⎠
⎞⎜⎜⎝
⎛= 02
20
0
20 2sin2sin θθ
Motion in One and Two Dimensions
179
Approximate dR/dg by ∆R/∆g: g
RgR
−=∆∆
Separate the variables to obtain: g
gRR ∆
−=∆
i.e., for small changes in gravity ( ggg ∆±≈ ), the fractional change in R is linearly opposite to the fractional change in g.
Remarks: This tells us that as gravity increases, the range will decrease, and vice versa. This is as it must be because R is inversely proportional to g. 97 ••• Picture the Problem We can show that ∆R/R = 2∆v0/ v0 by differentiating R with respect to v0 and then using a differential approximation. Differentiate the range equation with respect to v0:
0
00
0
20
00
2
2sin22sin
vR
gv
gv
dvd
dvdR
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛= θθ
Approximate dR/dv0 by ∆R/∆v0:
00
2vR
vR
=∆∆
Separate the variables to obtain:
0
02vv
RR ∆
=∆
i.e., for small changes in the launch velocity ( 000 vvv ∆±≈ ), the fractional change in R is twice the fractional change in v0.
Remarks: This tells us that as launch velocity increases, the range will increase twice as fast, and vice versa. 98 ••• Picture the Problem Choose a coordinate system in which the origin is at the base of the surface from which the projectile is launched. Let the positive x direction be to the right and the positive y direction be upward. We can apply constant-acceleration equations to obtain parametric equations in time that relate the range to the initial horizontal speed and the height h to the initial upward speed. Eliminating the parameter will leave us with a quadratic equation in R, the solution to which is the result we are required to establish. Write the constant-acceleration equations for the horizontal and vertical parts of the projectile’s
tvx x0= and
Chapter 3
180
motion:
( ) 221
0 tgtvhy y −++= where
θcos00 vv x = and θsin00 vv y =
Solve the x-component equation for time: θcos00 v
xvxt
x
==
Using the x-component equation, eliminate time from the y-component equation to obtain:
( ) 222
0 cos2tan x
vgxhy
θθ −+=
When the projectile strikes the ground its coordinates are (R, 0) and our equation becomes:
( ) 222
0 cos2tan0 R
vgRh
θθ −+=
Using the plus sign in the quadratic formula to ensure a physically meaningful root (one that is positive), solve for the range to obtain:
0
20
022
0
2sin2sin
211 θθ g
vv
ghR ⎟⎟⎠
⎞⎜⎜⎝
⎛++=
*99 •• Picture the Problem We can use trigonometry to relate the maximum height of the projectile to its range and the sighting angle at maximum elevation and the range equation to express the range as a function of the launch speed and angle. We can use a constant-acceleration equation to express the maximum height reached by the projectile in terms of its launch angle and speed. Combining these relationships will allow us to conclude that θφ tantan 2
1= . Referring to the figure, relate the maximum height of the projectile to its range and the sighting angle φ:
2tan
Rh
=φ
Express the range of the rocket and use the trigonometric identity
θθθ cossin22sin = to rewrite the expression as:
θθθ cossin2)2sin(22
gv
gvR ==
Using a constant-acceleration equation, relate the maximum height of a projectile to the vertical component of its launch speed:
ghvv yy 220
2 −= or, because vy = 0 and v0y = v0sinθ,
ghv 2sin220 =θ
Solve for the maximum height h: θ2
2
sin2gvh =
Motion in One and Two Dimensions
181
Substitute for R and h and simplify to obtain:
θθθ
θφ tan
cossin2
sin2
2tan 2
12
22
==
gv
gv
100 • Picture the Problem In the absence of air resistance, the horizontal and vertical displacements of the projectile are independent of each other and describable by constant-acceleration equations. Choose the origin at the firing location and with the coordinate axes as shown in the figure and use constant-acceleration equations to relate the vertical displacement to vertical component of the initial velocity and the horizontal velocity to the horizontal displacement and the time of flight.
(a) Using a constant-acceleration equation, express the vertical displacement of the projectile as a function of its time of flight:
( )221
0 tatvy yy ∆+∆=∆ or, because ay = −g,
( )221
0 tgtvy y ∆−∆=∆
Solve for v0y: ( )t
tgyv y ∆∆+∆
=2
21
0
Substitute numerical values and evaluate v0y:
( )( )
m/s121
s20s20m/s9.81m450 22
21
0
=
+=yv
(b) The horizontal velocity remains constant, so: m/s150
s20m3000
0 ==∆∆
==tx
vv xx
*101 •• Picture the Problem In the absence of air resistance, the acceleration of the stone is constant and the horizontal and vertical motions are independent of each other. Choose a coordinate system with the origin at the throwing location and the axes oriented as shown in the figure and use constant- acceleration equations to express the x and y coordinates of the stone while it is in flight.
Chapter 3
182
Using a constant-acceleration equation, express the x coordinate of the stone in flight:
221
00 tatvxx xx ++= or, because x0 = 0, v0x = v0 and ax = 0,
tvx 0=
Using a constant-acceleration equation, express the y coordinate of the stone in flight:
221
00 tatvyy yy ++= or, because y0 = 0, v0y = 0 and ay = g,
221 gty =
Referring to the diagram, express the relationship between θ, y and x at impact:
xy
=θtan
Substitute for x and y and solve for the time to impact:
tvg
tvgt
00
2
22tan ==θ
Solve for t to obtain:
θtan2 0
gvt =
Referring to the diagram, express the relationship between θ, L, y and x at impact:
θθ
tancos yLx ==
Substitute for y to obtain: θcos
2
2
Lg
gt=
Substitute for t and solve for L to obtain: θ
θcostan2 2
0
gvL =
102 ••• Picture the Problem The equation of a particle’s trajectory is derived in the text so we’ll use it as our starting point in this derivation. We can relate the coordinates of the point of impact (x, y) to the angle φ and use this relationship to eliminate y from the equation for the cannonball’s trajectory. We can then solve the resulting equation for x and relate the horizontal component of the point of impact to the cannonball’s range. The equation of the cannonball’s trajectory is given in the text:
2
022
00 cos2)(tan)( x
vgxxy ⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
θθ
Relate the x and y components of a point on the ground to the angle φ:
y(x) = (tan φ)x
Express the condition that the cannonball hits the ground:
( ) 2
022
00 cos2)(tantan x
vgxx ⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
θθφ
Motion in One and Two Dimensions
183
Solve for x to obtain:
gvx )tan(tancos2 00
220 φθθ −
=
Relate the range of the cannonball’s flight R to the horizontal distance x:
φcosRx =
Substitute to obtain:
gvR )tan(tancos2cos 00
220 φθθφ −
=
Solve for R:
φφθθ
cos)tan(tancos2 00
220
gvR −
=
103 •• Picture the Problem In the absence of air resistance, the acceleration of the rock is constant and the horizontal and vertical motions are independent of each other. Choose the coordinate system shown in the figure with the origin at the base of the building and the axes oriented as shown and apply constant-acceleration equations to relate the horizontal and vertical displacements of the rock to its time of flight. Find the horizontal and vertical components of v0:
v0x = v0 cos53° = 0.602v0 v0y = v0 sin53° = 0.799v0
Using a constant-acceleration equation, express the horizontal displacement of the projectile:
( ) tvtvx x ∆=∆==∆ 00 602.0m20
Using a constant-acceleration equation, express the vertical displacement of the projectile:
( )( ) ( )2
21
0
221
0
799.0
m20
tgtv
tgtvy y
∆−∆=
∆−∆=−=∆
Solve the x-displacement equation for ∆t: 0602.0
m20v
t =∆
Substitute ∆t into the expression for ∆y:
( ) ( )( )220 m/s91.4799.0m20 ttv ∆−∆=−
Solve for v0 to obtain: m/s8.100 =v
Find ∆t at impact: ( ) s08.3
cos53m/s10.8m20
=°
=∆t
Chapter 3
184
Using constant-acceleration equations, find vy and vx at impact:
m/s50.60 == xx vv and
m/s210 −=∆−= tgvv yy
Express the velocity at impact in vector form: jiv ˆm/s) 21.6( m/s) (6.50 −+=
r
104 •• Picture the Problem The ball experiences constant acceleration, except during its collision with the wall, so we can use the constant-acceleration equations in the analysis of its motion. Choose a coordinate system with the origin at the point of release, the positive x axis to the right, and the positive y axis upward. Using a constant-acceleration equation, express the vertical displacement of the ball as a function of ∆t:
( )221
0 tgtvy y ∆−∆=∆
When the ball hits the ground, ∆y = −2 m:
( )( )( )22
21 m/s81.9
m/s10m2
t
t
∆−
∆=−
Solve for the time of flight:
t flight = ∆t = 2.22 s
Find the horizontal distance traveled in this time:
∆x = (10 m/s) (2.22 s) = 22.2 m
The distance from the wall is:
∆x – 4 m = m2.18
Hitting Targets and Related Problems 105 • Picture the Problem In the absence of air resistance, the acceleration of the pebble is constant. Choose the coordinate system shown in the diagram and use constant-acceleration equations to express the coordinates of the pebble in terms of the time into its flight. We can eliminate the parameter t between these equations and solve for the launch velocity of the pebble. We can determine the launch angle from the sighting information and, once the range is known, the time of flight can be found using the horizontal component of the initial velocity.
Motion in One and Two Dimensions
185
Referring to the diagram, express θ in terms of the given distances:
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 91.6
m40m4.85
tan 1θ
Use a constant-acceleration equation to express the horizontal position of the pebble as a function of time:
221
00 tatvxx xx ++= or, because x0 = 0, v0x = v0cosθ, and ax = 0,
( )tvx θcos0= (1)
Use a constant-acceleration equation to express the vertical position of the pebble as a function of time:
221
00 tatvyy yy ++= or, because y0 = 0, v0y = v0sinθ, and ay = −g,
( ) 221
0 sin gttvy −= θ
Eliminate the parameter t to obtain: ( ) 2
220 cos2
tan xv
gxyθ
θ −=
At impact, y = 0 and x = R: ( ) 2
220 cos2
tan0 Rv
gRθ
θ −=
Solve for v0 to obtain:
θ2sin0Rgv =
Substitute numerical values and evaluate v0: m/s6.40
8.13sin)m/s m)(9.81 40( 2
0 =°
=v
Substitute in equation (1) to relate R to tflight:
( ) flight0 cos tvR θ=
Solve for and evaluate the time of flight:
( ) s0.992cos6.91m/s40.6m40
flight =°
=t
*106 •• Picture the Problem The acceleration of the ball is constant (zero horizontally and –g vertically) and the vertical and horizontal components are independent of each other. Choose the coordinate system shown in the figure and assume that v and t are unchanged by throwing the ball slightly downward.
Express the horizontal displacement of the ball as a function of time:
( )221
0 tatvx xx ∆+∆=∆
Chapter 3
186
or, because ax = 0, tvx x∆=∆ 0
Solve for the time of flight if the ball were thrown horizontally:
s0.491m/s37.5m18.4
0
==∆
=∆xvx
t
Using a constant-acceleration equation, express the distance the ball would drop (vertical displacement) if it were thrown horizontally:
( )221
0 tatvy yy ∆+∆=∆ or, because v0y = 0 and ay = −g,
( )221 tgy ∆−=∆
Substitute numerical values and evaluate ∆y:
( )( ) m1.18s0.491m/s9.81 2221 −=−=∆y
The ball must drop an additional 0.62 m before it gets to home plate.
y = (2.5 – 1.18) m = 1.32 m above ground
Calculate the initial downward speed the ball must have to drop 0.62 m in 0.491 s:
m1.26s0.491m0.62
−=−
=yv
Find the angle with horizontal:
°−=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛= −−
92.1
m/s37.5m/s1.26tantan 11
x
y
vv
θ
Remarks: One can readily show that 22yx vv + = 37.5 m/s to within 1%; so the
assumption that v and t are unchanged by throwing the ball downward at an angle of 1.93° is justified. 107 •• Picture the Problem The acceleration of the puck is constant (zero horizontally and –g vertically) and the vertical and horizontal components are independent of each other. Choose a coordinate system with the origin at the point of contact with the puck and the coordinate axes as shown in the figure and use constant-acceleration equations to relate the variables v0y, the time t to reach the wall, v0x, v0, and θ0.
Using a constant-acceleration equation for the motion in the y direction, express v0y as a function of the puck’s displacement ∆y:
yavv yyy ∆+= 220
2
or, because vy= 0 and ay = −g, ygv y ∆−= 20 2
0
Motion in One and Two Dimensions
187
Solve for and evaluate v0y:
( )( )m/s41.7
m/s81.9m80.222 20
=
=∆= ygv y
Find t from the initial velocity in the y direction:
s0.756m/s9.81m/s7.41
20 ===g
vt y
Use the definition of average velocity to find v0x:
m/s15.9s0.756
m12.00 ==
∆==
txvv xx
Substitute numerical values and evaluate v0: ( ) ( )
m/s5.17
m/s41.7m/s9.15 22
20
200
=
+=
+= yx vvv
Substitute numerical values and evaluate θ :
°=
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛= −−
0.25
m/s15.9m/s7.41tantan 1
0
01
x
y
vv
θ
108 •• Picture the Problem In the absence of air resistance, the acceleration of Carlos and his bike is constant and we can use constant-acceleration equations to express his x and y coordinates as functions of time. Eliminating the parameter t between these equations will yield y as a function of x … an equation we can use to decide whether he can jump the creek bed as well as to find the minimum speed required to make the jump.
(a) Use a constant-acceleration equation to express Carlos’ horizontal position as a function of time:
221
00 tatvxx xx ++= or, because x0 = 0, v0x = v0cosθ, and ax = 0,
( )tvx θcos0=
Use a constant-acceleration equation to express Carlos’ vertical position as a function of time:
221
00 tatvyy yy ++=
or, because y0 = 0, v0y = v0sinθ, and ay = −g,
( ) 221
0 sin gttvy −= θ
Chapter 3
188
Eliminate the parameter t to obtain: ( ) 2
220 cos2
tan xv
gxyθ
θ −=
Substitute y = 0 and x = R to obtain: ( ) 2
220 cos2
tan0 Rv
gRθ
θ −=
Solve for and evaluate R: ( ) ( )
m30.4
20sinm/s81.9m/s1.112sin 2
2
0
20
=
°== θgvR
brakes! apply the should He
(b) Solve the equation we used in the previous step for v0,min: ( )0
min,0 2sin θRgv =
Letting R = 7 m, evaluate v0,min: ( )( )
km/h51.0m/s2.14sin20
m/s81.9m7 2
min,0
==
°=v
109 ••• Picture the Problem In the absence of air resistance, the bullet experiences constant acceleration along its parabolic trajectory. Choose a coordinate system with the origin at the end of the barrel and the coordinate axes oriented as shown in the figure and use constant-acceleration equations to express the x and y coordinates of the bullet as functions of time along its flight path.
Use a constant-acceleration equation to express the bullet’s horizontal position as a function of time:
221
00 tatvxx xx ++= or, because x0 = 0, v0x = v0cosθ, and ax = 0,
( )tvx θcos0=
Use a constant-acceleration equation to express the bullet’s vertical position as a function of time:
221
00 tatvyy yy ++= or, because y0 = 0, v0y = v0sinθ, and ay = −g,
( ) 221
0 sin gttvy −= θ
Eliminate the parameter t to obtain: ( ) 2
220 cos2
tan xv
gxyθ
θ −=
Motion in One and Two Dimensions
189
Let y = 0 when x = R to obtain: ( ) 222
0 cos2tan0 R
vgR
θθ −=
Solve for the angle above the horizontal that the rifle must be fired to hit the target:
⎟⎟⎠
⎞⎜⎜⎝
⎛= −
20
121
0 sinvRgθ
Substitute numerical values and evaluate θ0:
( )( )( )
°=
⎥⎦
⎤⎢⎣
⎡= −
450.0m/s250
m/s81.9m100sin 2
21
21
0θ
Note: A second value for θ0, 89.6° is physically unreasonable.
Referring to the diagram, relate h to θ0 and solve for and evaluate h: m100
tan 0h
=θ
and ( ) ( ) m785.0450.0tanm100 =°=h
General Problems 110 • Picture the Problem The sum and difference of two vectors can be found from the components of the two vectors. The magnitude and direction of a vector can be found from its components.
(a) The table to the right summarizes the components of A
r
and Br
.
Vector x component y component
(m) (m) Ar
0.707 0.707
Br
0.866 −0.500 (b) The table to the right shows the components of S
r.
Vector x component y component
(m) (m) Ar
0.707 0.707
Br
0.866 −0.500
Sr
1.57 0.207
Determine the magnitude and direction of S
rfrom its components:
m59.122 =+= yx SSS
and, because Sr
is in the 1st
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 50.7tan 1
x
yS S
Sθ
Chapter 3
190
(c) The table to the right shows the components of :D
r
Vector x component y component
(m) (m) Ar
0.707 0.707
Br
0.866 −0.500
Dr
−0.159 1.21
Determine the magnitude and direction of D
rfrom its components:
m22.122 =+= yx DDD
and, because Dr
is in the 2nd quadrant,
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 5.97tan 1
x
yD D
Dθ
*111 • Picture the Problem A vector quantity can be resolved into its components relative to any coordinate system. In this example, the axes are orthogonal and the components of the vector can be found using trigonometric functions. The x and y components of g
rare
related to g through the sine and cosine functions:
gx = gsin30° = 2m/s91.4
and gy = gcos30° = 2m/s50.8
112 • Picture the Problem The figure shows two arbitrary, co-planar vectors that (as drawn) do not satisfy the condition that A/B = Ax/Bx. Because Ax AA θcos= and
Bx BB θcos= , 1coscos
=B
A
θθ
for the
condition to be satisfied.
∴ A/B = Ax/Bx if and only if A
rand B
r are parallel (θA = θB) or on opposite sides of the
x-axis (θA = –θB). 113 • Picture the Problem We can plot the path of the particle by substituting values for t and evaluating rx and ry coordinates of .rr The velocity vector is the time derivative of the position vector. (a) We can assign values to t in the parametric equations x = (5 m/s)t and y = (10 m/s)t to obtain ordered pairs (x, y) that lie on the path of the particle. The path is shown in the following graph:
Motion in One and Two Dimensions
191
0
5
10
15
20
25
0 2 4 6 8 10 12
x (m)
y (m
)
(b) Evaluate dtdrr :
( ) ( )[ ]( ) ( ) ji
jirv
ˆm/s10ˆm/s5
ˆm/s10ˆm/s5
+=
+== ttdtd
dtdrr
Use its components to find the magnitude of vr :
m/s2.1122 =+= yx vvv
114 •• Picture the Problem In the absence of air resistance, the hammer experiences constant acceleration as it falls. Choose a coordinate system with the origin and coordinate axes as shown in the figure and use constant-acceleration equations to describe the x and y coordinates of the hammer along its trajectory. We’ll use the equation describing the vertical motion to find the time of flight of the hammer and the equation describing the horizontal motion to determine its range. Using a constant-acceleration equation, express the x coordinate of the hammer as a function of time:
221
00 tatvxx xx ++= or, because x0 = 0, v0x = v0cosθ0, and ax = 0,
( )tvx 00 cosθ=
Using a constant-acceleration equation, express the y coordinate of the hammer as a function of time:
221
00 tatvyy yy ++= or, because y0 = h, v0y = v0sinθ, and ay = −g,
( ) 221
0 sin gttvhy −+= θ
Chapter 3
192
Substitute numerical values to obtain:
( )( )( ) 22
21 m/s81.9
30sinm/s4m10t
ty−
°+=
Substitute the conditions that exist when the hammer hits the ground:
( )( ) 22
21 m/s81.9
30sinm/s4m100t
t−
°−=
Solve for the time of fall to obtain:
t = 1.24 s
Use the x-coordinate equation to find the horizontal distance traveled by the hammer in 1.24 s:
( )( )( )m4.29
s1.24cos30m/s4
=
°=R
115 •• Picture the Problem We’ll model Zacchini’s flight as though there is no air resistance and, hence, the acceleration is constant. Then we can use constant- acceleration equations to express the x and y coordinates of Zacchini’s motion as functions of time. Eliminating the parameter t between these equations will leave us with an equation we can solve forθ. Because the maximum height along a parabolic trajectory occurs (assuming equal launch and landing elevations) occurs at half range, we can use this same expression for y as a function of x to find h.
Use a constant-acceleration equation to express Zacchini’s horizontal position as a function of time:
221
00 tatvxx xx ++= or, because x0 = 0, v0x = v0cosθ, and ax = 0,
( )tvx θcos0=
Use a constant-acceleration equation to express Zacchini’s vertical position as a function of time:
221
00 tatvyy yy ++= or, because y0 = 0, v0y = v0sinθ, and ay = −g,
( ) 221
0 sin gttvy −= θ
Eliminate the parameter t to obtain: ( ) 2
220 cos2
tan xv
gxyθ
θ −=
Use Zacchini’s coordinates when he lands in a safety net to obtain: ( ) 2
220 cos2
tan0 Rv
gRθ
θ −=
Motion in One and Two Dimensions
193
Solve for his launch angle θ :
⎟⎟⎠
⎞⎜⎜⎝
⎛= −
20
121 sin
vRgθ
Substitute numerical values and evaluate θ :
( )( )( )
°=⎥⎦
⎤⎢⎣
⎡= − 3.31
m/s2.42m/s81.9m53sin 2
21
21θ
Use the fact that his maximum height was attained when he was halfway through his flight to obtain:
( )2
220 2cos22
tan ⎟⎠⎞
⎜⎝⎛−=
Rv
gRhθ
θ
Substitute numerical values and evaluate h:
( )( )
m06.82m53
3.31cosm/s2.242m/s81.9
2m533.31tan
2
22
2
=⎟⎠⎞
⎜⎝⎛
°−°=h
116 •• Picture the Problem Because the acceleration is constant; we can use the constant-acceleration equations in vector form and the definitions of average velocity and average (instantaneous) acceleration to solve this problem. (a) The average velocity is given by:
ji
rrrv
ˆm/s) 2.5(ˆm/s) (3
12av
−+=∆−
=∆∆
=tt
rrrr
The average velocity can also be expressed as: 2
21av
vvvrr
r +=
and 2av1 2 vvv rrr
−=
Substitute numerical values to obtain: jiv ˆm/s) 1(ˆm/s) (11 +=
r
(b) The acceleration of the particle is given by: ji
vvva
ˆ)m/s 3.5(ˆ)m/s 2( 22
12
−+=
∆−
=∆∆
=tt
rrrr
(c) The velocity of the particle as a function of time is:
( ) jiavv ˆ])m/s 3.5(m/s) [(1ˆ])m/s (2m/s) 1[( 221 tttt −+++=+=
rrr
(d) Express the position vector as a function of time:
221
11)( ttt avrr rrrr++=
Chapter 3
194
Substitute numerical values and evaluate ( )trr :
( ) jir ˆ]m/s 1.75m/s) (1m) (3[ˆ])m/s (1m/s) (1m) [(4)( 2222 ttttt −+++++=r
*117 •• Picture the Problem In the absence of air resistance, the steel ball will experience constant acceleration. Choose a coordinate system with its origin at the initial position of the ball, the x direction to the right, and the y direction downward. In this coordinate system y0 = 0 and a = g. Letting (x, y) be a point on the path of the ball, we can use constant-acceleration equations to express both x and y as functions of time and, using the geometry of the staircase, find an expression for the time of flight of the ball. Knowing its time of flight, we can find its range and identify the step it strikes first. The angle of the steps, with respect to the horizontal, is:
°=⎟⎟⎠
⎞⎜⎜⎝
⎛= − 0.31
m0.3m0.18tan 1θ
Using a constant-acceleration equation, express the x coordinate of the steel ball in its flight:
221
00 tatvxx y++= or, because x0 = 0 and ay = 0,
tvx 0=
Using a constant-acceleration equation, express the y coordinate of the steel ball in its flight:
221
00 tatvyy yy ++= or, because y0 = 0, v0y = 0, and ay = g,
221 gty =
The equation of the dashed line in the figure is: 02
tanvgt
xy
== θ
Solve for the flight time: θtan2 0
gvt =
Find the x coordinate of the landing position:
θθ
tan2tan
20
gvyx ==
Substitute the angle determined in the first step:
( ) m1.10tan31m/s9.81
m/s322
2
=°=x
step.4th theis m 1.10 with stepfirst The >x
Motion in One and Two Dimensions
195
118 •• Picture the Problem Ignoring the influence of air resistance, the acceleration of the ball is constant once it has left your hand and we can use constant-acceleration equations to express the x and y coordinates of the ball. Elimination of the parameter t will yield an equation from which we can determine v0. We can then use the y equation to express the time of flight of the ball and the x equation to express its range in terms of x0, v0,θ and the time of flight.
Use a constant-acceleration equation to express the ball’s horizontal position as a function of time:
221
00 tatvxx xx ++= or, because x0 = 0, v0x = v0cosθ, and ax = 0,
( )tvx θcos0= (1)
Use a constant-acceleration equation to express the ball’s vertical position as a function of time:
221
00 tatvyy yy ++= or, because y0 = x0, v0y = v0sinθ, and ay = −g,
( ) 221
00 sin gttvxy −+= θ (2)
Eliminate the parameter t to obtain: ( ) 2
220
0 cos2tan x
vgxxy
θθ −+=
For the throw while standing on level ground we have: ( ) 2
0220
0 cos2tan0 x
vgx
θθ −=
and
( )gv
gv
gvx
20
20
20
0 452sin2sin =°== θ
Solve for v0: 00 gxv =
At impact equation (2) becomes: ( ) 2flight2
1flight00 sin0 gttgxx −+= θ
Solve for the time of flight: ( )2sinsin 20flight ++= θθ
gxt
Chapter 3
196
Substitute in equation (1) to express the range of the ball when thrown from an elevation x0 at an angle θ with the horizontal:
( )( ) ( )
( )2sinsincos
2sinsincos
cos
20
200
flight0
++=
++=
=
θθθ
θθθ
θ
x
gxgx
tgxR
Substitute θ = 0°, 30°, and 45°: ( ) 041.10 xx =°
( ) 073.130 xx =°
and
( ) 062.145 xx =°
119 ••• Picture the Problem Choose a coordinate system with its origin at the point where the motorcycle becomes airborne and with the positive x direction to the right and the positive y direction upward. With this choice of coordinate system we can relate the x and y coordinates of the motorcycle (which we’re treating as a particle) using Equation 3-21. (a) The path of the motorcycle is given by:
y(x) = (tanθ)x −g
2v02 cos2 θ
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ x
2
For the jump to be successful, h < y(x). Solving for v0, we find: )tan(2cosmin hx
gxv−
>θθ
(b) Use the values given to obtain: vmin > mph58.0orm/s26.0
(c) In order for our expression for vmin to be real valued; i.e., to predict values for vmin that are physically meaningful, x tanθ − h > 0.
∴ hmax < x tanθ The interpretation is that the bike "falls away" from traveling on a straight-line path due to the free-fall acceleration downwards. No matter what the initial speed of the bike, it must fall a little bit before reaching the other side of the pit.
Motion in One and Two Dimensions
197
120 ••• Picture the Problem Let the origin be at the position of the boat when it was engulfed by the fog. Take the x and y directions to be east and north, respectively. Let BWvr be the velocity of the boat relative to the water, BSvr be the velocity of the boat relative to the shore, and WSvr be the velocity of the water with respect to the shore. Then
BSvr = BWvr + WSvr .
θ is the angle of WSvr with respect to the x (east) direction.
(a) Find the position vector for the boat at t = 3 h:
( )( ){ }( )( ){ }
( ){ }( ){ }j
i
j
ir
ˆkm4km6.22
ˆkm6.22
ˆkm 4 sin135km32
ˆ135coskm32boat
−+
−=
−°+
°=
t
t
t
tr
Find the coordinates of the boat at t = 3 h:
( )[ ]( )h3cos135coskm/h10 WS θvrx +°= and
( )[ ]( )h3sin135sinkm/h10 WS θvry +°=
Simplify the expressions involving rx and ry and equate these simplified expressions to the x and y components of the position vector of the boat:
3vWS cosθ = −1.41 km/h and 3vWS sinθ = −2.586 km/h
Divide the second of these equations by the first to obtain: km41.1
km586.2tan−
−=θ
or
°°=⎟⎟⎠
⎞⎜⎜⎝
⎛−
−= − 241.4or 4.61
km1.41km2.586tan 1θ
Because the boat has drifted south, use θ = 241.4° to obtain: ( )
°==
°
−==
4.241atkm/h982.0
4.241cos3km/h41.1
cosWS
θ
θxvv
Chapter 3
198
(b) Letting φ be the angle between east and the proper heading for the boat, express the components of the velocity of the boat with respect to the shore:
vBS,x = (10 km/h) cosφ + (0.982 km/h) cos(241.3°) vBS,y = (10 km/h) sinφ + (0.982 km/h) sin(241.3°)
For the boat to travel northwest:
vBS,x = –vBS,y
Substitute the velocity components, square both sides of the equation, and simplify the expression to obtain the equations:
sinφ + cosφ = 0.133, sin2φ + cos2φ + 2 sinφ cosφ = 0.0177, and 1 + sin(2φ) = 0.0177
Solve for φ:
φ = 129.6° or 140.4°
Because the current pushes south, the boat must head more northerly than 135°:
Using 129.6°, the correct heading is northofwest6.39 ° .
(c) Find vBS:
vBS,x = –6.84 km/h and vBS = vBx /cos135° = 9.68 km/h
To find the time to travel 32 km, divide the distance by the boat’s actual speed:
t = (32 km)/(9.68 km/h) = min18h3h31.3 =
*121 •• Picture the Problem In the absence of air resistance, the acceleration of the projectile is constant and the equation of a projectile for equal initial and final elevations, which was derived from the constant-acceleration equations, is applicable. We can use the equation giving the range of a projectile for equal initial and final elevations to evaluate the ranges of launches that exceed or fall short of 45° by the same amount. Express the range of the projectile as a function of its initial speed and angle of launch:
0
20 2sin θgvR =
Let θ0= 45° ± θ: ( )
( )θ
θ
2cos
290sin
20
20
±=
±°=
gvgvR
Because cos(–θ) = cos(+θ) (the cosine function is an even function):
)(45)(45 θθ −°=+° RR
Motion in One and Two Dimensions
199
122 •• Picture the Problem In the absence of air resistance, the acceleration of both balls is that due to gravity and the horizontal and vertical motions are independent of each other. Choose a coordinate system with the origin at the base of the cliff and the coordinate axes oriented as shown and use constant-acceleration equations to relate the x and y components of the ball’s speed.
Independently of whether a ball is thrown upward at the angle α or downward at β, the vertical motion is described by:
ghv
yavv
y
yy
2
220
20
2
−=
∆+=
The horizontal component of the motion is given by:
vx = v0x
Find v at impact from its components: ghv
ghvvvvv yxyx
2
2
20
20
20
22
−=
−+=+=
Chapter 3
200
201
Chapter 4 Newton’s Laws Conceptual Problems *1 •• Determine the Concept A reference frame in which the law of inertia holds is called an inertial reference frame. If an object with no net force acting on it is at rest or is moving with a constant speed in a straight line (i.e., with constant velocity) relative to the reference frame, then the reference frame is an inertial reference frame. Consider sitting at rest in an accelerating train or plane. The train or plane is not an inertial reference frame even though you are at rest relative to it. In an inertial frame, a dropped ball lands at your feet. You are in a noninertial frame when the driver of the car in which you are riding steps on the gas and you are pushed back into your seat. 2 •• Determine the Concept A reference frame in which the law of inertia holds is called an inertial reference frame. A reference frame with acceleration a relative to the initial frame, and with any velocity relative to the initial frame, is inertial. 3 • Determine the Concept No. If the net force acting on an object is zero, its acceleration is zero. The only conclusion one can draw is that the net force acting on the object is zero. *4 • Determine the Concept An object accelerates when a net force acts on it. The fact that an object is accelerating tells us nothing about its velocity other than that it is always changing. Yes, the object must have an acceleration relative to the inertial frame of reference. According to Newton’s 1st and 2nd laws, an object must accelerate, relative to any inertial reference frame, in the direction of the net force. If there is ″only a single nonzero force,″ then this force is the net force. Yes, the object’s velocity may be momentarily zero. During the period in which the force is acting, the object may be momentarily at rest, but its velocity cannot remain zero because it must continue to accelerate. Thus, its velocity is always changing. 5 • Determine the Concept No. Predicting the direction of the subsequent motion correctly requires knowledge of the initial velocity as well as the acceleration. While the acceleration can be obtained from the net force through Newton’s 2nd law, the velocity can only be obtained by integrating the acceleration. 6 • Determine the Concept An object in an inertial reference frame accelerates if there is a net force acting on it. Because the object is moving at constant velocity, the net force acting on it is zero. correct. is )(c
Chapter 4
202
7 • Determine the Concept The mass of an object is an intrinsic property of the object whereas the weight of an object depends directly on the local gravitational field. Therefore, the mass of the object would not change and localgrav mgw = . Note that if the gravitational field is zero then the gravitational force is also zero. *8 • Determine the Concept If there is a force on her in addition to the gravitational force, she will experience an additional acceleration relative to her space vehicle that is proportional to the net force required producing that acceleration and inversely proportional to her mass. She could do an experiment in which she uses her legs to push off from the wall of her space vehicle and measures her acceleration and the force exerted by the wall. She could calculate her mass from the ratio of the force exerted by the wall to the acceleration it produced. *9 • Determine the Concept One’s apparent weight is the reading of a scale in one’s reference frame. Imagine yourself standing on a scale that, in turn, is on a platform accelerating upward with an acceleration a. The free-body diagram shows the force the gravitational field exerts on you, ,mg
r and
the force the scale exerts on you, .appwr The scale reading (the force the scale exerts on you) is your apparent weight.
Choose the coordinate system shown in the free-body diagram and apply
∑ = aF rrm to the scale:
∑ =−= yy mamgwF app or
ymamgw +=app
So, your apparent weight would be greater than your true weight when observed from a reference frame that is accelerating upward. That is, when the surface on which you are standing has an acceleration a such that ay is positive: 0>ya .
10 •• Determine the Concept Newton's 2nd law tells us that forces produce changes in the velocity of a body. If two observers pass each other, each traveling at a constant velocity, each will experience no net force acting on them, and so each will feel as if he or she is standing still. 11 • Determine the Concept Neither block is accelerating so the net force on each block is zero. Newton’s 3rd law states that objects exert equal and opposite forces on each other.
Newton’s Laws
203
(a) and (b) Draw the free-body diagram for the forces acting on the block of mass m1:
Apply ∑ = aF rr
m to the block 1:
111n21 amgmFFy =−=∑ or, because a1 = 0,
01n21 =− gmF
Therefore, the magnitude of the force that block 2 exerts on block 1 is given by:
gmF 1n21 =
From Newton’s 3rd law of motion we know that the force that block 1 exerts on block 2 is equal to, but opposite in direction, the force that block 2 exerts on block 1.
gmF 1n12n12n21 =⇒−= FFrr
(c) and (d) Draw the free-body diagram for the forces acting on block 2:
Apply ∑ = aF rrm to block 2:
222n12nT22 amgmFFF y =−−=∑
or, because a2 = 0,
( )gmmgmgmgmFF
21
212nT2nT2
+=+=+=
and the normal force that the table exerts on body 2 is
( )gmmF 21nT2 +=
From Newton’s 3rd law of motion we know that the force that block 2 exerts on the table is equal to, but opposite in direction, the force that the table exerts on block 2.
gmmF )( 21n2Tn2TnT2 +=⇒−= FFrr
Chapter 4
204
*12 • (a) True. By definition, action-reaction force pairs cannot act on the same object. (b) False. Action equals reaction independent of any motion of the two objects.
13 • Determine the Concept Newton’s 3rd law of motion describes the interaction between the man and his less massive son. According to the 3rd law description of the interaction of two objects, these are action-reaction forces and therefore must be equal in magnitude.
correct. is )(b
14 • Determine the Concept According to Newton’s 3rd law the reaction force to a force exerted by object A on object B is the force exerted by object B on object A. The bird’s weight is a gravitational field force exerted by the earth on the bird. Its reaction force is the gravitational force the bird exerts on the earth. correct. is )(b
15 • Determine the Concept We know from Newton’s 3rd law of motion that the reaction to the force that the bat exerts on the ball is the force the ball exerts on the bat and is equal in magnitude but oppositely directed. The action-reaction pair consists of the force with which the bat hits the ball and the force the ball exerts on the bat. These forces are equal in magnitude, act in opposite directions. correct. is )(c
16 • Determine the Concept The statement of Newton’s 3rd law given in the problem is not complete. It is important to remember that the action and reaction forces act on different bodies. The reaction force does not cancel out because it does not act on the same body as the external force.
*17 • Determine the Concept The force diagrams will need to include the ceiling, string, object, and earth if we are to show all of the reaction forces as well as the forces acting on the object. (a) The forces acting on the 2.5-kg object are its weight ,W
rand the
tension ,1Tr
in the string. The reaction
forces are 'Wr
acting on the earth and '1T
racting on the string.
Newton’s Laws
205
(b) The forces acting on the string are its weight, the weight of the object, and ,F
r the force exerted by the
ceiling. The reaction forces are
1Tr
acting on the string and 'Fr
acting on the ceiling.
18 • Determine the Concept Identify the objects in the block’s environment that are exerting forces on the block and then decide in what directions those forces must be acting if the block is sliding down the inclined plane. Because the incline is frictionless, the force the incline exerts on the block must be normal to the surface. The second object capable of exerting a force on the block is the earth and its force; the weight of the block acts directly downward. The magnitude of the normal force is less than that of the weight because it supports only a portion of the weight. .conditions esesatisfy th )( FBDin shown forces The c
19 • Determine the Concept In considering these statements, one needs to decide whether they are consistent with Newton’s laws of motion. A good strategy is to try to think of a counterexample that would render the statement false. (a) True. If there are no forces acting on an object, the net force acting on it must be zero and, hence, the acceleration must be zero. (b) False. Consider an object moving with constant velocity on a frictionless horizontal surface. While the net force acting on it is zero (it is not accelerating), gravitational and normal forces are acting on it. (c) False. Consider an object that has been thrown vertically upward. While it is still rising, the direction of the gravitational force acting on it is downward. (d) False. The mass of an object is an intrinsic property that is independent of its location (the gravitational field in which it happens to be situated). 20 • Determine the Concept In considering these alternatives, one needs to decide which alternatives are consistent with Newton’s 3rd law of motion. According to Newton’s 3rd law, the magnitude of the gravitational force exerted by her body on the earth is equal and opposite to the force exerted by the earth on her. correct. is )(a
Chapter 4
206
*21 • Determine the Concept In considering these statements, one needs to decide whether they are consistent with Newton’s laws of motion. In the absence of a net force, an object moves with constant velocity. correct. is )(d
22 • Determine the Concept Draw the free-body diagram for the towel. Because the towel is hung at the center of the line, the magnitudes of 1T
rand 2T
rare the same.
No. To support the towel, the tension in the line must have a vertical component equal to the towel’s weight. Thus θ > 0. 23 • Determine the Concept The free-body diagram shows the forces acting on a person in a descending elevator. The upward force exerted by the scale on the person, ,appwr is the person’s apparent weight.
Apply yy maF =∑ to the person and solve for wapp:
wapp – mg = may or wapp = mg + may = m(g + ay)
weight.apparent sperson' on theeffect no
haselevator theof velocity the
, oft independen is Because app vw
Remarks: Note that a nonconstant velocity will alter the apparent weight. Estimation and Approximation 24 •• Picture the Problem Assuming a stopping distance of 25 m and a mass of 80 kg, use Newton’s 2nd law to determine the force exerted by the seat belt. The force the seat belt exerts on the driver is given by:
Fnet = ma, where m is the mass of the driver.
Newton’s Laws
207
Using a constant-acceleration equation, relate the velocity of the car to its stopping distance and acceleration:
xavv ∆+= 220
2 or, because v = 0,
xav ∆=− 220
Solve for a:
xva∆−
=2
20
Substitute numerical values and evaluate a:
( )2
23
m/s5.12m252
kmm10
s3600h1
hkm90
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛××
−=a
Substitute for a and evaluate Fnet: ( )( )
kN00.1
m/s12.5kg80 2net
−=
−=F
Fnet is negative because it is opposite the direction of motion.
*25 ••• Picture the Problem The free-body diagram shows the forces acting on you and your bicycle as you are either ascending or descending the grade. The magnitude of the normal force acting on you and your bicycle is equal to the component of your weight in the y direction and the magnitude of the tangential force is the x component of your weight. Assume a combined mass (you plus your bicycle) of 80 kg.
(a) Apply ∑ = yy maF to you and your bicycle and solve for Fn:
Fn – mg cosθ = 0, because there is no acceleration in the y direction. ∴ Fn = mg cosθ
Determine θ from the information concerning the grade:
tanθ = 0.08 and θ = tan−1(0.08) = 4.57°
Substitute to determine Fn: Fn = (80 kg)(9.81 m/s2) cos4.57° = N782
Apply ∑ = xx maF to you and your bicycle and solve for Ft, the tangential force exerted by the road on the wheels:
Ft – mg sinθ = 0, because there is no acceleration in the x direction.
Chapter 4
208
Evaluate Ft: Ft = (80 kg)(9.81 m/s2) sin4.57° = N6.62
(b) incline. down the going and
up going same theare forces theon,accelerati no is thereBecause
Newton’s First and Second Laws: Mass, Inertia, and Force 26 • Picture the Problem The acceleration of the particle can be found from the stopping distance by using a constant-acceleration equation. The mass of the particle and its acceleration are related to the net force through Newton’s second law of motion. Choose a coordinate system in which the direction the particle is moving is the positive x direction and apply .net aF rv
m=
Use Newton’s 2nd law to relate the mass of the particle to the net force acting on it and its acceleration:
xaFm net=
Because the force is constant, use a constant-acceleration equation with vx = 0 to determine a:
xavv xxx ∆+= 220
2 and
xva x
x ∆−
=2
20
Substitute to obtain: 2
0
net2
xvxFm ∆
=
Substitute numerical values and evaluate m: ( )
kg00.3m/s 25.0
N) (15.0m) (62.522 ==m
and correct. is )(b
27 • Picture the Problem The acceleration of the object is related to its mass and the net force acting on it by .0net maFF == (a) Use Newton’s 2nd law of motion to calculate the acceleration of the object: ( ) 22
0net
m/s6.00m/s32
2
==
==mF
mF
a
Newton’s Laws
209
(b) Let the subscripts 1 and 2 distinguish the two objects. The ratio of the two masses is found from Newton’s 2nd law:
31
m/s9m/s3
2
2
2
1
10
20
1
2 ====aa
aFaF
mm
(c) The acceleration of the two-mass system is the net force divided by the total mass m = m1 + m2:
214
3
1
12
10
21
0net
m/s25.2
3111
==
+=
+=
+==
a
amm
mFmm
Fm
Fa
28 • Picture the Problem The acceleration of an object is related to its mass and the net force acting on it by maF =net . Let m be the mass of the ship, a1 be the acceleration of the ship when the net force acting on it is F1, and a2 be its acceleration when the net force is F1 + F2. Using Newton’s 2nd law, express the net force acting on the ship when its acceleration is a1:
F1 = ma1
Express the net force acting on the ship when its acceleration is a2:
F1 + F2 = ma2
Divide the second of these equations by the first and solve for the ratio F2/F1:
2
1
1
21
mama
FFF
=+
and
11
2
1
2 −=aa
FF
Substitute for the accelerations to determine the ratio of the accelerating forces and solve for F2:
( ) ( )( ) ( ) 31
s10km/h4s10km/h16
1
2 =−=FF
or
12 3FF =
*29 •• Picture the Problem Because the deceleration of the bullet is constant, we can use a constant-acceleration equation to determine its acceleration and Newton’s 2nd law of motion to find the average resistive force that brings it to a stop. Apply aF rr
m=∑ to express the force exerted on the bullet by the wood:
Fwood = ma
Using a constant-acceleration xavv ∆+= 220
2
Chapter 4
210
equation, express the final velocity of the bullet in terms of its acceleration and solve for the acceleration:
and
xv
xvva
∆−
=∆−
=22
20
20
2
Substitute to obtain: x
mvF∆
−=2
20
wood
Substitute numerical values and evaluate Fwood:
( ) ( )( )
kN75.3
m06.02m/s500kg108.1 23
wood
−=
×−=
−
F
where the negative sign means that the direction of the force is opposite the velocity.
*30 •• Picture the Problem The pictorial representation summarizes what we know about the motion. We can find the acceleration of the cart by using a constant-acceleration equation.
The free-body diagram shows the forces acting on the cart as it accelerates along the air track. We can determine the net force acting on the cart using Newton’s 2nd law and our knowledge of its acceleration.
(a) Apply ∑ = xx maF to the cart to obtain an expression for the net force F:
maF =
Using a constant-acceleration equation, relate the displacement of the cart to its acceleration, initial speed, and travel time:
( ) 221
0 tatvx ∆+∆=∆ or, because v0 = 0,
Newton’s Laws
211
( ) 221 tax ∆=∆
Solve for a:
( )22
txa
∆∆
=
Substitute for a in the force equation to obtain: ( ) ( )22
22t
xmtxmF
∆∆
=∆∆
=
Substitute numerical values and evaluate F:
( )( )( )
N0514.0s4.55
m1.5kg0.35522 ==F
(b) Using a constant-acceleration equation, relate the displacement of the cart to its acceleration, initial speed, and travel time:
( ) ' 221
0 tatvx ∆+∆=∆ or, because v0 = 0,
( ) ' 221 tax ∆=∆
Solve for ∆t:
'2a
xt ∆=∆
If we assume that air resistance is negligible, the net force on the cart is still 0.0514 N and its acceleration is:
2m/s0713.0kg0.722N0514.0' ==a
Substitute numerical values and evaluate ∆t:
( ) s49.6m/s0.0713m1.52
2 ==∆t
31 • Picture the Problem The acceleration of an object is related to its mass and the net force acting on it according to .net aF rr
m= Let m be the mass of the object and choose a coordinate system in which the direction of 2F0 in (b) is the positive x and the direction of the left-most F0 in (a) is the positive y direction. Because both force and acceleration are vector quantities, find the resultant force in each case and then find the resultant acceleration. (a) Calculate the acceleration of the object from Newton’s 2nd law of motion:
mnetFa
rr=
Express the net force acting on the object:
jijiF ˆˆˆˆ00net FFFF yx +=+=
r
and
Find the magnitude and direction of this net force:
022
net 2FFFF yx =+=
and
Chapter 4
212
°=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛= −− 45tantan
0
011
FF
FF
x
yθ
Use this result to calculate the magnitude and direction of the acceleration: ( )
force.each from0.45@m/s24.4
m/s32
22
2
2
00net
°=
=
=== amF
mF a
(b) Calculate the acceleration of the object from Newton’s 2nd law of motion:
r a =
r F net /m
Express the net force acting on the object: ( )
( ) j
i
jiF
ˆ45cos2
ˆ45sin
ˆˆ
00
0
net
°++
°−=
+=
FF
F
FF yx
r
Find the magnitude and direction of this net force:
( ) ( ) 02
002
022
net 80.245cos245sin FFFFFFF yx =°++°−=+=
and
0
0
0011
2 from 6.14
4.7545sin
45cos2tantan
Fr
°=
°−=⎟⎟⎠
⎞⎜⎜⎝
⎛°−°+
=⎟⎟⎠
⎞⎜⎜⎝
⎛= −−
FFF
FF
x
yθ
Use this result to calculate the magnitude and direction of the acceleration: ( )
02
2
00net
2 from 6.14@m/s40.8
m/s380.2
80.280.2
Fr
°=
=
=== amF
mFa
32 • Picture the Problem The acceleration of an object is related to its mass and the net force acting on it according to .net mFa
rr=
Apply mnetFa
rr= to the object to
obtain:
( ) ( )
( ) ( )ji
jiFa
ˆm/s00.2ˆm/s00.4
kg5.1
ˆN3ˆN6
22
net
−=
−==
m
rr
Newton’s Laws
213
Find the magnitude of ar
:
( ) ( )2
2222
22
m/s47.4
m/s00.2m/s00.4
=
+=
+= yx aaa
33 • Picture the Problem The mass of the particle is related to its acceleration and the net force acting on it by Newton’s 2nd law of motion. Because the force is constant, we can use constant-acceleration formulas to calculate the acceleration. Choose a coordinate system in which the positive x direction is the direction of motion of the particle. The mass is related to the net force and the acceleration by Newton’s 2nd law:
x
x
aFm =∑=
aFr
r
Because the force is constant, the acceleration is constant. Use a constant-acceleration equation to find the acceleration:
( )
( )2
02
21
0
2so
,0where,
txa
vtatvx
x
xxx
∆∆
=
=∆+=∆
Substitute this result into the first equation and solve for and evaluate the mass m of the particle:
( ) ( )( )( )
kg0.12
m182s6N12
2
22
=
=∆∆
==xtF
aFm x
x
x
*34 • Picture the Problem The speed of either Al or Bert can be obtained from their accelerations; in turn, they can be obtained from Newtons 2nd law applied to each person. The free-body diagrams to the right show the forces acting on Al and Bert. The forces that Al and Bert exert on each other are action-and-reaction forces. (a) Apply xmaxF =∑ to Bert and solve for his acceleration:
BertBertBertonAl amF =−
2Bert
BertonAlBert
m/s200.0
kg100N20
−=
−=
−=
mF
a
Using a constant-acceleration equation, relate Bert’s speed to his initial speed, speed after 1.5 s, and acceleration and solve for his speed at the end of 1.5 s:
v = v0 + a∆t = 0 + (−0.200 m/s2)(1.5 s) = m/s300.0−
Chapter 4
214
(b) From Newton's 3rd law, an equal but oppositely directed force acts on Al while he pushes Bert. Because the ice is frictionless, Al speeds off in the opposite direction. Apply Newton’s 2nd law to the forces acting on Al and solve for his acceleration:
∑ == AlAlAlonBertAl, amFFx and
2Al
Alon Bert Al
m/s250.0kg08N20
=
==m
Fa
Using a constant-acceleration equation, relate Al’s speed to his initial speed, speed after 1.5 s, and acceleration; solve for his speed at the end of 1.5 s:
v = v0 + a∆t = 0 + (0.250 m/s2)(1.5 s) = m/s375.0
35 • Picture the Problem The free-body diagrams show the forces acting on the two blocks. We can apply Newton’s second law to the forces acting on the blocks and eliminate F to obtain a relationship between the masses. Additional applications of Newton’s 2nd law to the sum and difference of the masses will lead us to values for the accelerations of these combinations of mass.
(a) Apply ∑ = xx maF to the two blocks:
∑ == 111, amFFx and
∑ == 222, amFFx
Eliminate F between the two equations and solve for m2: 112
2
12
12 4
m/s3m/s12 mmm
aam ===
Express and evaluate the acceleration of an object whose mass is m2 – m1 when the net force acting on it is F: ( ) 22
31
131
11112
m/s00.4m/s12
34
===
=−
=−
=
a
mF
mmF
mmFa
(b) Express and evaluate the acceleration of an object whose mass is m2 + m1 when the net force acting on it is F: ( )
2
251
151
1
1112
m/s40.2
m/s125
4
=
===
+=
+=
amF
mmF
mmFa
Newton’s Laws
215
36 • Picture the Problem Because the velocity is constant, the net force acting on the log must be zero. Choose a coordinate system in which the positive x direction is the direction of motion of the log. The free-body diagram shows the forces acting on the log when it is accelerating in the positive x direction.
(a) Apply ∑ = xx maF to the log when it is moving at constant speed:
Fpull – Fres = max = 0
Solve for and evaluate Fres: Fres = Fpull = N250
(b) Apply ∑ = xx maF to the log when it is accelerating to the right:
Fpull – Fres = max
Solve for and evaluate Fpull: Fpull = Fres + max = 250 N + (75 kg) (2 m/s2) = N400
37 • Picture the Problem The acceleration can be found from Newton’s 2nd law. Because both forces are constant, the net force and the acceleration are constant; hence, we can use the constant-acceleration equations to answer questions concerning the motion of the object at various times. (a) Apply Newton’s 2nd law to the object to obtain:
( ) ( )
( ) ( )ji
ji
FFFa
ˆm/s50.3ˆm/s50.1
kg4
ˆN14ˆN6
22
21net
−+=
−+=
+==
mm
rrrr
(b) Using a constant-acceleration equation, express the velocity of the object as a function of time and solve for its velocity when t = 3 s:
( ) ( )[ ]( )( ) ( ) ji
ji
avv
ˆm/s5.10ˆm/s50.4
s3ˆm/s50.3ˆm/s50.10 22
0
−+=
−++=
+= trrr
(c) Express the position of the object in terms of its average velocity and evaluate this expression at t = 3 s:
( ) ( ) ji
vvr
ˆm8.15ˆm75.6
21
av
−+=
==
tt
r
rr
Chapter 4
216
Mass and Weight *38 • Picture the Problem The mass of the astronaut is independent of gravitational fields and will be the same on the moon or, for that matter, out in deep space. Express the mass of the astronaut in terms of his weight on earth and the gravitational field at the surface of the earth:
kg61.2N/kg9.81
N600
earth
earth ===gwm
and correct. is )(c
39 • Picture the Problem The weight of an object is related to its mass and the gravitational field through w = mg. (a) The weight of the girl is: ( )( )
N530
N/kg9.81kg54
=
== mgw
(b) Convert newtons to pounds:
lb119N/lb4.45N530
==w
40 • Picture the Problem The mass of an object is related to its weight and the gravitational field. Find the weight of the man in newtons:
( )( ) N734N/lb45.4lb165lb165 ==
Calculate the mass of the man from his weight and the gravitational field:
kg8.74N/kg9.81
N734===
gwm
Contact Forces
*41 • Picture the Problem Draw a free-body diagram showing the forces acting on the block. kF
ris the force exerted by the spring,
gW rrm= is the weight of the block, and nF
r
is the normal force exerted by the horizontal surface. Because the block is resting on a surface, Fk + Fn = W.
(a) Calculate the force exerted by the spring on the block:
( )( ) N0.60m1.0N/m600 === kxFx
Newton’s Laws
217
(b) Choosing the upward direction to be positive, sum the forces acting on the block and solve for Fn:
∑ =−+⇒= 00 WFF nkFr
and kn FWF −=
Substitute numerical values and evaluate Fn:
N7.57
N60)N/kg kg)(9.81 (12
=
−=nF
42 • Picture the Problem Let the positive x direction be the direction in which the spring is stretched. We can use Newton’s 2nd law and the expression for the force exerted by a stretched (or compressed) spring to express the acceleration of the box in terms of its mass m, the stiffness constant of the spring k, and the distance the spring is stretched x. Apply Newton’s 2nd law to the box to obtain: m
Fa ∑=
Express the force exerted on the box by the spring:
kxF −=
Substitute to obtain:
mkxa −
=
Substitute numerical values and evaluate a:
( )( )
2m/s33.5
kg6m0.04N/m800
−=
−=a
where the minus sign tells us that the box’s acceleration is toward its equilibrium position.
Free-Body Diagrams: Static Equilibrium
43 • Picture the Problem Because the traffic light is not accelerating, the net force acting on it must be zero; i.e., 1T
r+ 2T
r+ g
rm = 0.
Construct a free-body diagram showing the forces acting on the knot and choose the coordinate system shown:
Chapter 4
218
Apply ∑ = xx maF to the knot:
T1cos30° − T2cos60° = max = 0
Solve for T2 in terms of T1: 112 73.1
60cos30cos TTT =°°
=
12 an greater th is TT∴
44 • Picture the Problem Draw a free-body diagram showing the forces acting on the lamp and apply 0=∑ yF . From the FBD, it is clear that T1 supports the full weight mg = 418 N.
Apply 0=∑ yF to the lamp to obtain:
01 =−wT
Solve for T1: mgwT ==1
Substitute numerical values and evaluate T1:
( )( ) N418m/s81.9kg6.42 21 ==T
and correct. is )(b
*45 •• Picture the Problem The free-body diagrams for parts (a), (b), and (c) are shown below. In both cases, the block is in equilibrium under the influence of the forces and we can use Newton’s 2nd law of motion and geometry and trigonometry to obtain relationships between θ and the tensions. (a) and (b)
(c)
(a) Referring to the FBD for part (a), use trigonometry to determine θ :
°== − 9.36m0.625
m0.5cos 1θ
Newton’s Laws
219
(b) Noting that T = T′, apply
∑ = yy maF to the 0.500-kg block and solve for the tension T:
0 since0sin2 ==− amgT θ and
θsin2mgT =
Substitute numerical values and evaluate T:
( )( ) N08.42sin36.9
m/s9.81kg0.5 2
=°
=T
(c) The length of each segment is:
m0.4173
m1.25=
Find the distance d:
m0.291520.417mm1
=
−=d
Express θ in terms of d and solve for its value:
°=
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛= −−
7.45m0.417m2915.0cos
m0.417cos 11 dθ
Apply ∑ = yy maF to the 0.250-kg block and solve for the tension T3:
0. since0sin3 ==− amgT θ and
θsin3mgT =
Substitute numerical values and evaluate T3:
( )( ) N43.3sin45.7
m/s9.81kg0.25 2
3 =°
=T
Apply ∑ = xx maF to the 0.250-kg block and solve for the tension T2:
0. since0cos 23 ==− aTT θ and
θcos32 TT =
Substitute numerical values and evaluate T2:
( ) N40.27.45cosN43.32 =°=T
By symmetry: T1 = T3 = N43.3
Chapter 4
220
46 • Picture the Problem The suspended body is in equilibrium under the influence of the forces ,hT
r,45T
rand ;g
rm
i.e., hTr
+ 45Tr
+ gr
m = 0 Draw the free-body diagram of the forces acting on the knot just above the 100-N body. Choose a coordinate system with the positive x direction to the right and the positive y direction upward. Apply the conditions for translational equilibrium to determine the tension in the horizontal cord.
If the system is to remain in static equilibrium, the vertical component of T45 must be exactly balanced by, and therefore equal to, the tension in the string suspending the 100-N body:
Tv = T45 sin45° = mg
Express the horizontal component of T45:
Th = T45 cos45°
Because T45 sin45° = T45 cos45°: Th = mg = N100
47 • Picture the Problem The acceleration of any object is directly proportional to the net force acting on it. Choose a coordinate system in which the positive x direction is the same as that of 1F
rand the positive y direction is to the right. Add the two forces to
determine the net force and then use Newton’s 2nd law to find the acceleration of the object. If 3F
rbrings the system into equilibrium, it must be true that 3F
r+ 1F
r+ 2F
r= 0.
(a) Find the components of 1F
rand
:2Fr
{ }{ }
ji
j
iF
iF
ˆ)N26(ˆ)N15(
ˆ30cos)N30(
ˆ30sin)N30(
ˆ)N20(
2
1
+−=
°+
°−=
=r
r
Add 1F
r and 2F
r to find :totF
r
jiF ˆ)N26(ˆ)N5(tot +=
r
Newton’s Laws
221
Apply aF rrm=∑ to find the
acceleration of the object:
ji
Fa
ˆ)m/s60.2(ˆ)m/s500.0( 22
tot
+=
=m
rr
(b) Because the object is in equilibrium under the influence of the three forces, it must be true that:
r F 3 +
r F 1 +
r F 2 = 0
and ( )( ) ( ) ji
FFFˆN0.26ˆN00.5
213
−+−=
+−=rrr
*48 • Picture the Problem The acceleration of the object equals the net force, T
r− g
rm ,
divided by the mass. Choose a coordinate system in which upward is the positive y direction. Apply Newton’s 2nd law to the forces acting on this body to find the acceleration of the object as a function of T.
(a) Apply yy maF =∑ to the object:
T – w = T – mg = may
Solve this equation for a as a function of T: g
mTay −=
Substitute numerical values and evaluate ay:
22 m/s81.8m/s81.9kg5N5
−=−=ya
(b) Proceed as in (a) with T = 10 N: a = 2m/s81.7−
(c) Proceed as in (a) with T = 100 N: a = 2m/s2.10
49 •• Picture the Problem The picture is in equilibrium under the influence of the three forces shown in the figure. Due to the symmetry of the support system, the vectors T
rand
'Tr
have the same magnitude T. Choose a coordinate system in which the positive x direction is to the right and the positive y direction is upward. Apply the condition for translational equilibrium to obtain an expression for T as a function of θ and w. (a) Referring to Figure 4-37, apply the condition for translational equilibrium in the vertical direction and solve for T:
0sin2 =−=∑ wTFy θ and
θsin2wT =
Chapter 4
222
Tmin occurs when sinθ is a maximum:
°== − 901sin 1θ
Tmax occurs when sinθ is a minimum. Because the function is undefined when sinθ = 0, we can conclude that:
°→→ 0 as max θTT
(b) Substitute numerical values in the result in (a) and evaluate T:
( )( ) N6.192sin30
m/s81.9kg2 2
=°
=T
Remarks: θ = 90° requires wires of infinite length; therefore it is not possible. As θ gets small, T gets large without limit. *50 ••• Picture the Problem In part (a) we can apply Newton’s 2nd law to obtain the given expression for F. In (b) we can use a symmetry argument to find an expression for tan θ0. In (c) we can use our results obtained in (a) and (b) to express xi and yi. (a) Apply ∑ = 0yF to the balloon:
0sinsin 1i1iii =−+ −− θθ TTF
Solve for F to obtain: ii1i1i sinsin θθ TTF −= −−
(b) By symmetry, each support must balance half of the force acting on the entire arch. Therefore, the vertical component of the force on the support must be NF/2. The horizontal component of the tension must be TH. Express tanθ0 in terms of NF/2 and TH:
HH0 2
2tanT
NFT
NF==θ
By symmetry, θN+1 = − θ0. Therefore, because the tangent function is odd: H
1N0 2tantan
TNF
=−= +θθ
(c) Using TH = Ti cosθi = Ti−1cosθi−1, divide both sides of our result in (a) by TH and simplify to obtain: i1i
ii
ii
1i1i
1i1i
H
tantancossin
cossin
θθθθ
θθ
−=
−=
−
−−
−−
TT
TT
TF
Using this result, express tan θ1:
H01 tantan
TF
−= θθ
Substitute for tan θ0 from (a):
( )HHH
1 22
2tan
TFN
TF
TNF
−=−=θ
Newton’s Laws
223
Generalize this result to obtain: ( )
Hi 2
2tanTFiN −=θ
Express the length of rope between two balloons: 1balloonsbetween +
=N
Ll
Express the horizontal coordinate of the point on the rope where the ith balloon is attached, xi, in terms of xi−1 and the length of rope between two balloons:
1i1ii cos1 −− +
+= θN
Lxx
Sum over all the coordinates to obtain: ∑−
=+=
1i
0i cos
1 jjN
Lx θ
Proceed similarly for the vertical coordinates to obtain:
∑−
=+=
1i
0i sin
1 jjN
Ly θ
(d) A spreadsheet program is shown below. The formulas used to calculate the quantities in the columns are as follows:
Cell Content/Formula Algebraic Form C9 ($B$2-2*B9)/(2*$B$4) ( )
H2i2
TFN −
D9 SIN(ATAN(C9)) ( )i1tansin θ− E9 COS(ATAN(C9)) ( )i
1tancos θ− F10 F9+$B$1/($B$2+1)*E9
1i1i cos1 −− +
+ θN
Lx
G10 G9+$B$1/($B$2+1)*D91i1i cos
1 −− ++ θ
NLy
A B C D E F G 1 L = 10 m 2 N = 10 3 F = 1 N 4 TH= 3.72 N 5 6 7 8 I tan(thetai) sin(thetai) cos(thetai) xi yi 9 0 1.344 0.802 0.597 0.000 0.00010 1 1.075 0.732 0.681 0.543 0.72911 2 0.806 0.628 0.778 1.162 1.39512 3 0.538 0.474 0.881 1.869 1.96613 4 0.269 0.260 0.966 2.670 2.396
Chapter 4
224
14 5 0.000 0.000 1.000 3.548 2.63215 6 −0.269 −0.260 0.966 4.457 2.63216 7 −0.538 −0.474 0.881 5.335 2.39617 8 −0.806 −0.628 0.778 6.136 1.96618 9 −1.075 −0.732 0.681 6.843 1.39519 10 −1.344 −0.802 0.597 7.462 0.72920 11 8.005 0.000
(e) A horizontal component of tension 3.72 N gives a spacing of 8 m. At this spacing, the arch is 2.63 m high, tall enough for someone to walk through.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 1 2 3 4 5 6 7 8
x i
y i
51 •• Picture the Problem We know, because the speed of the load is changing in parts (a) and (c), that it is accelerating. We also know that, if the load is accelerating in a particular direction, there must be a net force in that direction. A free-body diagram for part (a) is shown to the right. We can apply Newton’s 2nd law of motion to each part of the problem to relate the tension in the cable to the acceleration of the load. Choose the upward direction to be the positive y direction.
(a) Apply yy maF =∑ to the load and solve for T:
T – mg = ma and
( )gammgmaT yy +=+= (1)
Substitute numerical values and evaluate T:
( )( )kN8.11
m/s81.9m/s2kg1000 22
=
+=T
(b) Because the crane is lifting the T = mg = kN81.9
Newton’s Laws
225
load at constant speed, a = 0: (c) Because the acceleration of the load is downward, a is negative. Apply yy maF =∑ to the load:
T – mg = may
Substitute numerical values in equation (1) and evaluate T:
T = (1000 kg)(9.81 m/s2 − 2 m/s2) = kN81.7
52 •• Picture the Problem Draw a free-body diagram for each of the depicted situations and use the conditions for translational equilibrium to find the unknown tensions.
(a)
ΣFx = T1cos60° − 30 N = 0 and T1 = (30 N)/cos60° = N0.60
ΣFy = T1sin60° − T2 = 0 and T2 = T1sin60° = N0.52
∴ m = T2/g = kg30.5
(b)
ΣFx = (80 N)cos60° − T1sin60° = 0 and T1 = (80 N)cos60°/sin60° = N2.46
ΣFy = (80 N)sin60° − T2 − T1cos60° = 0 T2 = (80 N)sin60° − (46.2 N)cos60° = N2.46
m = T2/g = kg .714
(c) ΣFx = −T1cos60° + T3cos60° = 0
and T1 = T3 ΣFy = 2T1sin60° − mg = 0 and T1 = T3 = (58.9 N)/(2sin60°) = N0.34
Chapter 4
226
∴ m = T1/g = kg .463
53 •• Picture the Problem Construct the free-body diagram for that point in the rope at which you exert the force F
rand choose the
coordinate system shown on the FBD. We can apply Newton’s 2nd law to the rope to relate the tension to F.
(a) Noting that T1 = T2 = T, apply
∑ = yy maF to the car: 2Tsinθ − F = may = 0 because the car’s acceleration is zero.
Solve for and evaluate T: kN82.3
3sin2N400
sin2=
°==
θFT
(b) Proceed as in part (a):
kN30.44sin2N600
=°
=T
Free-Body Diagrams: Inclined Planes and the Normal Force *54 • Picture the Problem The free-body diagram shows the forces acting on the box as the man pushes it across the frictionless floor. We can apply Newton’s 2nd law of motion to the box to find its acceleration.
Apply ∑ = xx maF to the box: xmaF =θcos
Solve for ax:
mFax
θcos=
Substitute numerical values and evaluate ax:
( ) 2m/s2.10kg20cos35N250
=°
=xa
Newton’s Laws
227
55 • Picture the Problem The free-body diagram shows the forces acting on the box as the man pushes it up the frictionless incline. We can apply Newton’s 2nd law of motion to the box to determine the smallest force that will move it up the incline at constant speed.
Apply xx maF =∑ to the box as it moves up the incline with constant speed:
( ) 0sin40cosmin =−−° θθ mgF
Solve for Fmin:
( )θθ−°
=40cossin
minmgF
Substitute numerical values and evaluate Fmin:
( )( ) N0.56cos25
m/s9.81kg20 2
min =°
=F
56 • Picture the Problem Forces always occur in equal and opposite pairs. If object A exerts a force, AB,F
ron object B, an equal but opposite force, ABBA ,, FF
rr−= is exerted by object B
on object A. The forces acting on the box are its weight,
,Wr
and the normal reaction force of the inclined plane on the box, .nF
rThe reaction
forces are the forces the box exerts on the inclined plane and the gravitational force the box exerts on the earth. The reaction forces are indicated with primes.
57 • Picture the Problem Because the block whose mass is m is in equilibrium, the sum of the forces nF
r, ,T
r and gmr
must be zero. Construct the free-body diagram for this object, use the coordinate system shown on the free-body diagram, and apply Newton’s 2nd law of motion.
Apply ∑ = xx maF to the block on the incline:
T – mgsin40° = max = 0 because the system is in equilibrium.
Chapter 4
228
Solve for m: °
=40sing
Tm
The tension must equal the weight of the 3.5-kg block because that block is also in equilibrium:
T = (3.5 kg)g and
°=
°=
40sinkg5.3
40sin)kg5.3(
ggm
Because this expression is not included in the list of solution candidates,
correct. is )(d
Remarks: Because the object whose mass is m does not hang vertically, its mass must be greater than 3.5 kg. *58 • Picture the Problem The balance(s) indicate the tension in the string(s). Draw free-body diagrams for each of these systems and apply the condition(s) for equilibrium.
(a)
0=−=∑ mgTFy and
( )( ) N1.98m/s81.9kg10 2 === mgT
(b)
0'=−=∑ TTFx or, because T ′= mg,
( )( ) N1.98m/s81.9kg10
'2 ==
== mgTT
(c)
02 =−=∑ mgTFy and
( )( ) N1.49m/s81.9kg10 221
21
==
= mgT
(d) 030sin =°−=∑ mgTFx and
( )( ) N1.4930sinm/s81.9kg10
30sin2 =°=
°= mgT
Newton’s Laws
229
Remarks: Note that (a) and (b) give the same answers … a rather surprising result until one has learned to draw FBDs and apply the conditions for translational equilibrium. 59 •• Picture the Problem Because the box is held in place (is in equilibrium) by the forces acting on it, we know that
Tr
+ nFr
+ Wr
= 0 Choose a coordinate system in which the positive x direction is in the direction of Tr
and the positive y direction is in the direction of .nF
r Apply Newton’s 2nd law
to the block to obtain expressions for Tr
and .nFr
(a) Apply xx maF =∑ to the box:
0sin =− θmgT
Solve for T:
θsinmgT =
Substitute numerical values and evaluate T:
( )( ) N42560sinm/s81.9kg50 2 =°=T
Apply yy maF =∑ to the box: 0cosn =− θmgF
Solve for Fn: θcosn mgF =
Substitute numerical values and evaluate Fn:
( )( )N245
60cosm/s81.9kg50 2n
=
°=F
(b) Using the result for the tension from part (a) to obtain:
mgmgT =°=° 90sin90
and 00sin0 =°=° mgT
Chapter 4
230
60 •• Picture the Problem Draw a free-body diagram for the box. Choose a coordinate system in which the positive x-axis is parallel to the inclined plane and the positive y-axis is in the direction of the normal force the incline exerts on the block. Apply Newton’s 2nd law of motion to both the x and y directions.
(a) Apply yy maF =∑ to the block: ( ) 025sinN10025cosn =°−°−mgF
Solve for Fn:
( ) °+°= 25sinN10025cosn mgF
Substitute numerical values and evaluate Fn:
( )( )( ) N14925sinN100
cos25m/s9.81kg12 2n
=°+
°=F
(b) Apply xx maF =∑ to the block:
( ) mamg =°−° 25sin25cosN100
Solve for a: ( )°−
°= 25sin25cosN100 g
ma
Substitute numerical values and evaluate a:
( ) ( )2
2
m/s41.3
25sinm/s81.9kg12
25cosN100
=
°−°
=a
*61 •• Picture the Problem The scale reading (the boy’s apparent weight) is the force the scale exerts on the boy. Draw a free-body diagram for the boy, choosing a coordinate system in which the positive x-axis is parallel to and down the inclined plane and the positive y-axis is in the direction of the normal force the incline exerts on the boy. Apply Newton’s 2nd law of motion in the y direction.
Apply yy maF =∑ to the boy to find Fn. Remember that there is no acceleration in the y direction:
030cosn =°−WF
Newton’s Laws
231
Substitute for W to obtain:
030cosn =°−mgF
Solve for Fn: °= 30cosn mgF
Substitute numerical values and evaluate Fn:
( )( )N552
30cosm/s81.9kg65 2n
=
°=F
62 •• Picture the Problem The free-body diagram for the block sliding up the incline is shown to the right. Applying Newton’s 2nd law to the forces acting in the x direction will lead us to an expression for ax. Using this expression in a constant-acceleration equation will allow us to express h as a function of v0 and g.
The height h is related to the distance ∆x traveled up the incline:
h = ∆xsinθ
Using a constant-acceleration equation, relate the final speed of the block to its initial speed, acceleration, and distance traveled:
xavv x∆+= 220
2 or, because v = 0,
xav x∆+= 20 20
Solve for ∆x to obtain:
xavx
2
20−
=∆
Apply ∑ = xx maF to the block and solve for its acceleration:
xmamg =− θsin and ax = −g sinθ
Substitute these results in the equation for h and simplify:
gv
gvxh
2
sinsin2
sin
20
20
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=∆= θ
θθ
which is independent of the ramp’s angle θ.
Chapter 4
232
Free-Body Diagrams: Elevators 63 • Picture the Problem Because the elevator is descending at constant speed, the object is in equilibrium and T
r+ g
rm = 0. Draw a
free-body diagram of the object and let the upward direction be the positive y direction. Apply Newton’s 2nd law with a = 0. Because the downward speed is constant, the acceleration is zero. Apply yy maF =∑ and solve for T:
T – mg = 0 ⇒ T = mg and
correct. is )(a
64 • Picture the Problem The sketch to the right shows a person standing on a scale in a descending elevator. To its right is a free-body diagram showing the forces acting on the person. The force exerted by the scale on the person, ,appwr is the person’s apparent weight. Because the elevator is slowing down while descending, the acceleration is directed upward.
Apply yy maF =∑ to the person:
ymamgw =−app
Solve for wapp: mgmamgw y >+=app
t.your weighan greater th bemust ) (your scale by theyou on exerted force normal the velocity,downward a slow"" to
required ison accelerati upwardan Because higher. be eight willapparent w The
weightapparent
*65 • Picture the Problem The sketch to the right shows a person standing on a scale in the elevator immediately after the cable breaks. To its right is the free-body diagram showing the forces acting on the person. The force exerted by the scale on the person, ,appwr is the person’s apparent weight.
Newton’s Laws
233
From the free-body diagram we can see that agw rrr mm =+app where gr
is the local gravitational field and ar is the acceleration of the reference frame (elevator). When the elevator goes into free fall ( ar = g
r), our equation becomes .app gagw rrrr mmm ==+ This
tells us that appwr = 0. correct. is )(e
66 • Picture the Problem The free-body diagram shows the forces acting on the 10-kg block as the elevator accelerates upward. Apply Newton’s 2nd law of motion to the block to find the minimum acceleration of the elevator required to break the cord. Apply yy maF =∑ to the block: T – mg = may
Solve for ay to determine the minimum breaking acceleration: g
mT
mmgTa −=
−=y
Substitute numerical values and evaluate ay:
22y m/s19.5m/s81.9
kg10N150
=−=a
67 •• Picture the Problem The free-body diagram shows the forces acting on the 2-kg block as the elevator ascends at a constant velocity. Because the acceleration of the elevator is zero, the block is in equilibrium under the influence of Tr
and .gr
m Apply Newton’s 2nd law of motion to the block to determine the scale reading.
(a) Apply yy maF =∑ to the block to obtain:
ymamgT =− (1)
For motion with constant velocity, ay = 0:
0=−mgT and mgT =
Substitute numerical values and evaluate T:
( )( ) N6.19m/s81.9kg2 2 ==T
(b) As in part (a), for constant velocity, a = 0:
ymamgT =− and
( )( ) N6.19m/s81.9kg2 2 ==T
Chapter 4
234
(c) Solve equation (1) for T and simplify to obtain:
( )yy agmmamgT +=+= (2)
Because the elevator is ascending and its speed is increasing, we have ay = 3 m/s2. Substitute numerical values and evaluate T:
( )( ) N6.25m/s3m/s81.9kg2 22 =+=T
(d) For 0 < t < 5 s: ay = 0 and
N 19.6s50 =→T
Using its definition, calculate a for 5 s < t < 9 s:
2m/s5.2s4m/s100
−=−
=∆∆
=tva
Substitute in equation (2) and evaluate T:
( )( )N6.14
m/s5.2m/s81.9kg2 22s9s5
=
−=→T
Free-Body Diagrams: Ropes, Tension, and Newton’s Third Law 68 • Picture the Problem Draw a free-body diagram for each object and apply Newton’s 2nd law of motion. Solve the resulting simultaneous equations for the ratio of T1 to T2. Draw the FBD for the box to the left and apply ∑ = xx maF :
T1 = m1a1
Draw the FBD for the box to the right and apply ∑ = xx maF :
T2 − T1 = m2a2
The two boxes have the same acceleration:
a1 = a2
Divide the second equation by the first:
2
1
12
1
mm
TTT
=−
Newton’s Laws
235
Solve for the ratio T1/T2 :
21
1
2
1
mmm
TT
+= and correct. is )(d
69 •• Picture the Problem Call the common acceleration of the boxes a. Assume that box 1 moves upward, box 2 to the right, and box 3 downward and take this direction to be the positive x direction. Draw free-body diagrams for each of the boxes, apply Newton’s 2nd law of motion, and solve the resulting equations simultaneously. (a)
(b)
(c)
(a) Apply ∑ = xx maF to the box whose mass is m1:
T1 – w1 = m1a
Apply ∑ = xx maF to the box whose mass is m2:
T2 – T1 = m2a
Noting that '22 TT = , apply
∑ = xx maF to the box whose mass is m3:
w3 – T2 = m3a
Add the three equations to obtain: w3 − w1 = (m1 + m2 + m3)a
Solve for a:
( )321
13
mmmgmma
++−
=
Substitute numerical values and evaluate a:
( )( )
2
2
m/s31.1
kg5.2kg5.3kg5.1m/s81.9kg5.1kg5.2
=
++−
=a
(b) Substitute for the acceleration in the equations obtained above to find the tensions:
T1 = N7.16 and T2 = N3.21
Chapter 4
236
*70 •• Picture the Problem Choose a coordinate system in which the positive x direction is to the right and the positive y direction is upward. Let 1,2F
rbe the contact force
exerted by m2 on m1 and 2,1Fr
be the force exerted by m1 on m2. These forces are equal and opposite so 1,2F
r= − .2,1F
r The free-
body diagrams for the blocks are shown to the right. Apply Newton’s 2nd law to each block separately and use the fact that their accelerations are equal.
(a) Apply ∑ = xx maF to the first block:
amamFF 1111,2 ==−
Apply ∑ = xx maF to the second block:
amamF 2222,1 == (1)
Add these equations to eliminate F2,1 and F1,2 and solve for a = a1 = a2: 21 mm
Fa+
=
Substitute your value for a into equation (1) and solve for F1,2:
21
21,2 mm
FmF+
=
(b) Substitute numerical values in the equations derived in part (a) and evaluate a and F1,2:
2m/s400.0kg6kg2
N2.3=
+=a
and ( )( ) N40.2
kg6kg2kg6N3.2
1,2 =+
=F
Remarks: Note that our results for the acceleration are the same as if the force F had acted on a single object whose mass is equal to the sum of the masses of the two blocks. In fact, because the two blocks have the same acceleration, we can consider them to be a single system with mass m1 + m2.
Newton’s Laws
237
71 • Picture the Problem Choose a coordinate system in which the positive x direction is to the right and the positive y direction is upward. Let 1,2F
rbe the contact force
exerted by m2 on m1 and 2,1Fr
be the force exerted by m1 on m2. These forces are equal and opposite so 1,2F
r= − .2,1F
r The free-
body diagrams for the blocks are shown. We can apply Newton’s 2nd law to each block separately and use the fact that their accelerations are equal.
(a) Apply ∑ = xx maF to the first block:
amamFF 2222,1 ==−
Apply ∑ = xx maF to the second block:
amamF 1111,2 == (1)
Add these equations to eliminate F2,1 and F1,2 and solve for a = a1 = a2: 21 mm
Fa+
=
Substitute your value for a into equation (1) and solve for F2,1:
21
12,1 mm
FmF+
=
(b) Substitute numerical values in the equations derived in part (a) and evaluate a and F2,1:
2m/s400.0kg6kg2
N2.3=
+=a
and ( )( ) N800.0
kg6kg2kg2N3.2
2,1 =+
=F
Remarks: Note that our results for the acceleration are the same as if the force F had acted on a single object whose mass is equal to the sum of the masses of the two blocks. In fact, because the two blocks have the same acceleration, we can consider them to be a single system with mass m1 + m2. 72 •• Picture the Problem The free-body diagrams for the boxes and the ropes are below. Because the vertical forces have no bearing on the problem they have not been included. Let the numeral 1 denote the 100-kg box to the left, the numeral 2 the rope connecting the boxes, the numeral 3 the box to the right and the numeral 4 the rope to which the force Fr
is applied. 4,3Fr
is the tension force exerted by m3 on m4, 3,4Fr
is the tension force
exerted by m4 on m3, 3,2Fr
is the tension force exerted by m2 on m3, 2,3Fr
is the tension
Chapter 4
238
force exerted by m3 on m2, 2,1Fr
is the tension force exerted by m1 on m2, and 1,2Fr
is the
tension force exerted by m2 on m1. The equal and opposite pairs of forces are 1,2Fr
=
− ,2,1Fr
2,3Fr
= − ,3,2Fr
and 3,4Fr
= − .4,3Fr
We can apply Newton’s 2nd law to each box and rope separately and use the fact that their accelerations are equal.
Apply ∑ = aF rr
m to the box whose mass is m1:
amamF 1111,2 == (1)
Apply ∑ = aF rrm to the rope
whose mass is m2:
amamFF 2222,12,3 ==− (2)
Apply ∑ = aF rrm to the box whose
mass is m3:
amamFF 3333,23,4 ==− (3)
Apply ∑ = aF rrm to the rope
whose mass is m4:
amamFF 4444,3 ==−
Add these equations to eliminate F2,1, F1,2, F3,2, F2,3, F4,3, and F3,4 and solve for F:
( )( )( ) N202m/s1.0kg202 2
4321
==
+++= ammmmF
Use equation (1) to find the tension at point A:
( )( ) N100m/s1.0kg100 21,2 ==F
Use equation (2) to find the tension at point B:
( )( )N101
m/s1.0kg1N100 2
22,12,3
=
+=
+= amFF
Use equation (3) to find the tension at point C:
( )( )N012
m/s1.0kg100N101 2
33,23,4
=
+=
+= amFF
Newton’s Laws
239
73 •• Picture the Problem Because the distribution of mass in the rope is uniform, we can express the mass m′ of a length x of the rope in terms of the total mass of the rope M and its length L. We can then express the total mass that the rope must support at a distance x above the block and use Newton’s 2nd law to find the tension as a function of x. Set up a proportion expressing the mass m′ of a length x of the rope as a function of M and L and solve for m′:
xLMm'
LM
xm'
=⇒=
Express the total mass that the rope must support at a distance x above the block:
xLMmm'm +=+
Apply ∑ = yy maF to the block and a length x of the rope:
axLMm
gxLMmTwT
⎟⎠⎞
⎜⎝⎛ +=
⎟⎠⎞
⎜⎝⎛ +−=−
Solve for T to obtain:
( ) ⎟⎠⎞
⎜⎝⎛ ++= x
LMmgaT
*74 •• Picture the Problem Choose a coordinate system with the positive y direction upward and denote the top link with the numeral 1, the second with the numeral 2, etc.. The free-body diagrams show the forces acting on links 1 and 2. We can apply Newton’s 2nd law to each link to obtain a system of simultaneous equations that we can solve for the force each link exerts on the link below it. Note that the net force on each link is the product of its mass and acceleration. (a) Apply ∑ = yy maF to the top link and solve for F:
mamgF 55 =− and
( )agmF += 5
Chapter 4
240
Substitute numerical values and evaluate F:
( )( )N16.6
m/s2.5m/s9.81kg0.15 22
=
+=F
(b) Apply ∑ = yy maF to a single link:
( )( )N250.0
m/s2.5kg0.1 2link1link1
=
== amF
(c) Apply ∑ = yy maF to the 1st through 5th links to obtain:
mamgFF =−− 2 , (1) mamgFF =−− 32 , (2) mamgFF =−− 43 , (3) mamgFF =−− 54 , and (4)
mamgF =−5 (5)
Add equations (2) through (5) to obtain:
mamgF 442 =−
Solve for F2 to obtain:
( )gammamgF +=+= 4442
Substitute numerical values and evaluate F2:
( )( )N92.4
m/s2.5m/s9.81kg0.14 222
=
+=F
Substitute for F2 to find F3, and then substitute for F3 to find F4:
N69.33 =F and N46.24 =F
Solve equation (5) for F5: ( )agmF +=5
Substitute numerical values and evaluate F5:
( )( )N23.1
m/s2.5m/s9.81kg0.1 225
=
+=F
75 • Picture the Problem A net force is required to accelerate the object. In this problem the net force is the difference between T
rand ).( gW rr
m= The free-body diagram of the object is shown to the right. Choose a coordinate system in which the upward direction is positive. Apply aF rr
m=∑ to the object to obtain:
Fnet = T – W = T – mg
Solve for the tension in the lower portion of the rope:
T = Fnet + mg = ma + mg = m(a + g)
Newton’s Laws
241
Using its definition, find the acceleration of the object:
a ≡ ∆v/∆t = (3.5 m/s)/(0.7 s) = 5.00 m/s2
Substitute numerical values and evaluate T:
T = (40 kg)(5.00 m/s2 + 9.81 m/s2) = 592 N and correct. is )(a
76 • Picture the Problem A net force in the downward direction is required to accelerate the truck downward. The net force is the difference between tW
rand .T
r
A free-body diagram showing these forces acting on the truck is shown to the right. Choose a coordinate system in which the downward direction is positive.
Apply yy maF =∑ to the truck to obtain:
yamgmT tt =−
Solve for the tension in the lower portion of the cable:
( )yy agmamgmT +=+= ttt
Substitute to find the tension in the rope:
( ) gmggmT tt 9.01.0 =−=
and correct. is )(c
77 •• Picture the Problem Because the string does not stretch or become slack, the two objects must have the same speed and therefore the magnitude of the acceleration is the same for each object. Choose a coordinate system in which up the incline is the positive x direction for the object of mass m1 and downward is the positive x direction for the object of mass m2. This idealized pulley acts like a piece of polished pipe; i.e., its only function is to change the direction the tension in the massless string acts. Draw a free-body diagram for each of the two objects, apply Newton’s 2nd law of motion to both objects, and solve the resulting equations simultaneously. (a) Draw the FBD for the object of mass m1:
Apply ∑ = xx maF to the object whose mass is m1:
T – m1gsinθ = m1a
Chapter 4
242
Draw the FBD for the object of mass m2:
Apply ∑ = xx maF to the object whose mass is m2:
m2g − T = m2a
Add the two equations and solve for a:
( )21
12 sinmm
mmga+−
=θ
Substitute for a in either of the equations containing the tension and solve for T:
( )21
21 sin1mm
mgmT++
=θ
(b) Substitute the given values into the expression for a:
2m/s45.2=a
Substitute the given data into the expression for T:
N8.36=T
78 • Picture the Problem The magnitude of the accelerations of Peter and the counterweight are the same. Choose a coordinate system in which up the incline is the positive x direction for the counterweight and downward is the positive x direction for Peter. The pulley changes the direction the tension in the rope acts. Let Peter’s mass be mP. Ignoring the mass of the rope, draw free-body diagrams for the counterweight and Peter, apply Newton’s 2nd law to each of them, and solve the resulting equations simultaneously. (a) Using a constant-acceleration equation, relate Peter’s displacement to her acceleration and descent time:
( ) 221
0 tatvx ∆+∆=∆ or, because v0 = 0,
( ) 221 tax ∆=∆
Solve for the common acceleration of Peter and the counterweight: ( )2
2txa
∆∆
=
Substitute numerical values and evaluate a:
( )( )
22 m/s32.1
s2.2m2.32
==a
Newton’s Laws
243
Draw the FBD for the counterweight:
Apply ∑ = xx maF to the counterweight:
T – mg sin50° = ma
Draw the FBD for Peter:
Apply ∑ = xx maF to Peter:
mPg – T = mPa
Add the two equations and solve for m:
( )°+
−=
50sinP
gaagmm
Substitute numerical values and evaluate m:
( )( )( )
kg0.48
sin50m/s9.81m/s1.32m/s1.32m/s9.81kg50
22
22
=
°+−
=m
(b) Substitute for m in the force equation for the counterweight and solve for T:
( )°+= 50singamT
(b) Substitute numerical values and evaluate T:
( )[ ( ) ] N424sin50m/s9.81m/s1.32kg48.0 22 =°+=T
79 •• Picture the Problem The magnitude of the accelerations of the two blocks are the same. Choose a coordinate system in which up the incline is the positive x direction for the 8-kg object and downward is the positive x direction for the 10-kg object. The peg changes the direction the tension in the rope acts. Draw free-body diagrams for each object, apply Newton’s 2nd law of motion to both of them, and solve the resulting equations simultaneously.
Chapter 4
244
(a) Draw the FBD for the 3-kg object:
Apply ∑ = xx maF to the 3-kg block:
T – m8g sin40° = m3a
Draw the FBD for the 10-kg object:
Apply ∑ = xx maF to the 10-kg block:
m10g sin50° − T = m10a
Add the two equations and solve for and evaluate a:
( )
2
108
810
m/s37.1
40sin50sin
=
+°−°
=mm
mmga
Substitute for a in the first of the two force equations and solve for T:
amgmT 88 40sin +°=
Substitute numerical values and evaluate T:
( ) ( )[]
N4.61
m/s37.140sinm/s81.9kg8
2
2
=
+
°=T
(b) Because the system is in equilibrium, set a = 0, express the force equations in terms of m1 and m2, add the two force equations, and solve for and evaluate the ratio m1/m2:
T – m1g sin40° = 0 m2g sin50°− T = 0 ∴ m2g sin50°– m1g sin40° = 0 and
19.140sin50sin
2
1 =°°
=mm
Newton’s Laws
245
80 •• Picture the Problem The pictorial representations shown to the right summarize the information given in this problem. While the mass of the rope is distributed over its length, the rope and the 6-kg block have a common acceleration. Choose a coordinate system in which the direction of the 100-N force is the positive x direction. Because the surface is horizontal and frictionless, the only force that influences our solution is the 100-N force. (a) Apply ∑ = xx maF to the objects shown for part (a):
100 N = (m1 + m2)a
Solve for a to obtain:
21
N100mm
a+
=
Substitute numerical values and evaluate a:
2m/s0.10kg10N100==a
(b) Let m represent the mass of a length x of the rope. Assuming that the mass of the rope is uniformly distributed along its length:
m5kg4
rope
2 ==Lm
xm
xm ⎟⎟⎠
⎞⎜⎜⎝
⎛=
m5kg4
and
Let T represent the tension in the rope at a distance x from the point at which it is attached to the 6-kg block. Apply ∑ = xx maF to the system shown for part (b) and solve for T:
T = (m1 + m)a
( )
( )x
x
N/m8N60
m/s10m5kg4kg6 2
+=
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+=
*81 •• Picture the Problem Choose a coordinate system in which upward is the positive y direction and draw the free-body diagram for the frame-plus- painter. Noting that
,TFrr
−= apply Newton’s 2nd law of motion.
(a) Letting mtot = mframe + mpainter, 2T – mtotg = mtota
and
Chapter 4
246
apply ∑ = yy maF to the frame-plus-painter and solve T:
( )2
tot gamT +=
Substitute numerical values and evaluate T:
( )( )
N3982
m/s81.9m/s8.0kg75 22
=
+=T
Because F = T: F = N398
(b) Apply ∑ = yy maF with a = 0 to obtain:
2T – mtotg = 0
Solve for T:
gmT tot21=
Substitute numerical values and evaluate T:
( )( ) N368m/s81.9kg75 221 ==T
82 ••• Picture the Problem Choose a coordinate system in which up the incline is the positive x direction and draw free-body diagrams for each block. Noting that ,1020 aa rr
−= apply Newton’s 2nd law of motion to each block and solve the resulting equations simultaneously. Draw a FBD for the 20-kg block:
Apply ∑ = xx maF to the block to obtain:
T – m20gsin20° = m20a20
Draw a FBD for the 10-kg block. Because all the surfaces, including the surfaces between the blocks, are frictionless, the force the 20-kg block exerts on the 10-kg block must be normal to their surfaces as shown to the right.
Apply ∑ = xx maF to the block to obtain:
T – m10gsin20° = m10a10
Newton’s Laws
247
Because the blocks are connected by a taut string:
a20 = −a10
Substitute for a20 and eliminate T between the two equations to obtain:
220
210
m/s12.1
and
m/s12.1
−=
=
a
a
Substitute for either of the accelerations in the force equations and solve for T:
N8.44=T
83 ••• Picture the Problem Choose a coordinate system in which the positive x direction is to the right and draw free-body diagrams for each block. Because of the pulley, the string exerts a force of 2T. Apply Newton’s 2nd law of motion to both blocks and solve the resulting equations simultaneously. (a) Noting the effect of the pulley, express the distance the 20-kg block moves in a time ∆t:
( ) cm00.5cm1021
521
20 ==∆=∆ xx
(b) Draw a FBD for the 20-kg block:
Apply ∑ = xx maF to the block to obtain:
2T = m20a20
Draw a FBD for the 5-kg block:
Apply ∑ = xx maF to the block to obtain:
m5g − T = m5a5
Using a constant-acceleration equation, relate the displacement of the 5-kg block to its acceleration
( )2521
5 tax ∆=∆
Chapter 4
248
and the time during which it is accelerated: Using a constant-acceleration equation, relate the displacement of the 20-kg block to its acceleration and the time during which it is accelerated:
( )22021
20 tax ∆=∆
Divide the first of these equations by the second to obtain:
( )( ) 20
52
2021
252
1
20
5
aa
tata
xx
=∆∆
=∆∆
Use the result of part (a) to obtain:
205 2aa =
Let a20 = a. Then a5 = 2a and the force equations become:
2T = m20a and m5g – T = m5(2a)
Eliminate T between the two equations to obtain:
2021
5
520 2 mm
gmaa+
==
Substitute numerical values and evaluate a20 and a5:
( )( )( ) ( )
2
21
2
20 m/s45.2kg20kg52
m/s81.9kg5=
+=a
and ( ) 22
5 m/s91.4m/s45.22 ==a
Substitute for either of the accelerations in either of the force equations and solve for T:
N5.24=T
Free-Body Diagrams: The Atwood’s Machine *84 •• Picture the Problem Assume that m1 > m2. Choose a coordinate system in which the positive y direction is downward for the block whose mass is m1 and upward for the block whose mass is m2 and draw free-body diagrams for each block. Apply Newton’s 2nd law of motion to both blocks and solve the resulting equations simultaneously. Draw a FBD for the block whose mass is m2:
Newton’s Laws
249
Apply ∑ = yy maF to this block:
T – m2g = m2a2
Draw a FBD for the block whose mass is m1:
Apply ∑ = yy maF to this block:
m1g – T = m1a1
Because the blocks are connected by a taut string, let a represent their common acceleration:
a = a1 = a2
Add the two force equations to eliminate T and solve for a:
m1g − m2g = m1a + m2a and
gmmmma
21
21
+−
=
Substitute for a in either of the force equations and solve for T:
21
212mm
gmmT+
=
85 •• Picture the Problem The acceleration can be found from the given displacement during the first second. The ratio of the two masses can then be found from the acceleration using the first of the two equations derived in Problem 89 relating the acceleration of the Atwood’s machine to its masses. Using a constant-acceleration equation, relate the displacement of the masses to their acceleration and solve for the acceleration:
( )221
0 tatvy ∆+=∆ or, because v0 = 0,
( )221 tay ∆=∆
Solve for and evaluate a:
( )( )( )
222 m/s600.0
s1m3.022
==∆∆
=tya
Solve for m1 in terms of 2m using the first of the two equations given in Problem 84:
222
2
21 13.1m/s9.21m/s10.41 mm
agagmm ==
−+
=
Find the second mass for m2 or m1 = 1.2 kg:
kg06.1orkg36.1mass2nd =m
Chapter 4
250
86 •• Picture the Problem Let Fnm be the force the block of mass m2 exerts on the pebble of mass m. Because m2 < m1, the block of mass m2 accelerates upward. Draw a free-body diagram for the pebble and apply Newton’s 2nd law and the acceleration equation given in Problem 84.
Apply ∑ = yy maF to the pebble:
Fnm – mg = ma
Solve for Fnm: ( )gamF +=nm
From Problem 84: g
mmmm
a21
21
+−
=
Substitute for a and simplify to obtain: g
mmmmgg
mmmmmF
21
1
21
21nm
2+
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
+−
=
87 •• Picture the Problem Note from the free-body diagrams for Problem 89 that the net force exerted by the accelerating blocks is 2T. Use this information, together with the expression for T given in Problem 84, to derive an expression for F = 2T. From Problem 84 we have:
21
212mm
gmmT+
=
The net force, F, exerted by the Atwood’s machine on the hanger is:
21
21
42
mmgmmTF
+==
If m1 = m2 = m, then: mg
mgmF 2
24 2
== … as expected.
If either m1 or m2 = 0, then: F = 0 … also as expected. 88 ••• Picture the Problem Use a constant-acceleration equation to relate the displacement of the descending (or rising) mass as a function of its acceleration and then use one of the results from Problem 84 to relate a to g. Differentiation of our expression for g will allow us to relate uncertainty in the time measurement to uncertainty in the measured value for g … and to the values of m2 that would yield an experimental value for g that is good to within 5%.
Newton’s Laws
251
(a) Use the result given in Problem 84 to express g in terms of a:
21
21
mmmmag
−+
= (1)
Using a constant-acceleration equation, express the displacement, L, as a function of t and solve for the acceleration:
( )221
0 tatvLy ∆+∆==∆ or, because v0 = 0 and ∆t = t,
22tLa = (2)
Substitute this expression for a:
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=21
212
2mmmm
tLg
(b) Evaluate dg/dt to obtain:
tg
mmmm
tL
t
mmmmLt
dtdg
222
4
21
212
21
213
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−+
⎥⎦⎤
⎢⎣⎡−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
−= −
Divide both sides of this expression by g and multiply both sides by dt: t
dtg
dg 2−=
(c) We have:
05.0±=g
dgand 025.0±=
tdt
Solve the second of these equations for t to obtain:
s40.025
s1025.0
===dtt
Substitute in equation (2) to obtain: ( )( )
22 m/s375.0
s4m32
==a
Solve equation (1) for m2 to obtain:
12 magagm
+−
=
Evaluate m2 with m1 = 1 kg:
( )
kg926.0
kg1m/s375.0m/s81.9m/s375.0m/s81.9
22
22
2
=+−
=m
Solve equation (1) for m1 to obtain:
agagmm
−+
= 21
Substitute numerical values to obtain:
( )
kg08.1m/s375.0m/s81.9m/s375.0m/s81.9kg926.0 22
22
1
=−+
=m
Chapter 4
252
Because the masses are interchangeable: kg1.08orkg0.9262 =m
*89 •• Picture the Problem We can reason to this conclusion as follows: In the two extreme cases when the mass on one side or the other is zero, the tension is zero as well, because the mass is in free-fall. By symmetry, the maximum tension must occur when the masses on each side are equal. An alternative approach that is shown below is to treat the problem as an extreme-value problem.
Express m2 in terms of M and m1:
m2 = M − m1
Substitute in the equation from Problem 84 and simplify to obtain:
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−=
−+−
=Mmmg
mMmmMgmT
21
111
11 22
Differentiate this expression with respect to m1 and set the derivative equal to zero for extreme values:
valuesextremefor0212 1
1
=⎟⎠⎞
⎜⎝⎛ −=
Mmg
dmdT
Solve for m1 to obtain:
Mm 21
1 =
Show that m1 = M/2 is a maximum value by evaluating the second derivative of T with respect to m1 at m1 = M/2:
121
2
oftlyindependen,04 mMg
dmTd
<−=
and we have shown that
. whenmaximum a is
21
21 MmmT
==
Remarks: An alternative solution is to use a graphing calculator to show that T as a function of m1 is concave downward and has its maximum value when m1 = m2 = M/2.
90 ••• Picture the Problem The free-body diagrams show the forces acting on the objects whose masses are m1 and m2. The application of Newton’s 2nd law to these forces and the accelerations the net forces are responsible for will lead us to an expression for the tension in the string as a function of m1 and m2. Examination of this expression as for m2 >> m1 will yield the predicted result.
(a) Apply ∑ = yy maF to the objects whose masses are m1 and m2 to obtain:
2222
1111
andamTgm
amgmT
=−
=−
Newton’s Laws
253
Assume that the role of the pulley is simply to change the direction the tension acts. Then T1 = T2 = T. Because the two objects have a common acceleration, let a = a1 = a2. Eliminate a between the two equations and solve for T to obtain:
gmm
mmT21
212+
=
Divide the numerator and denominator of this fraction by m2:
2
1
1
1
2
mmgmT
+=
Take the limit of this fraction as m2 → ∞ to obtain:
gmT 12=
(b) Imagine the situation when m2 >> m1:
Under these conditions, the object whose mass is m2 is essentially in free- fall, so the object whose mass is m1 is accelerating upward with an acceleration of magnitude g.
Under these conditions, the net force acting on the object whose mass is m1 is m1g and:
T – m1g = m1g ⇒ T = 2m1g. Note that this result agrees with that obtained using more analytical methods.
General Problems 91 • Picture the Problem Choose a coordinate system in which the force the tree exerts on the woodpecker’s head is in the negative-x direction and determine the acceleration of the woodpecker’s head from Newton’s 2nd law of motion. The depth of penetration, under the assumption of constant acceleration, can be determined using a constant- acceleration equation. Knowing the acceleration of the woodpecker’s head and the depth of penetration of the tree, we can calculate the time required to bring the head to rest. (a) Apply ∑ = xx maF to the woodpecker’s head to obtain:
2m/s100kg 0.060
N6−=
−=∑=
mFa x
x
(b) Using a constant-acceleration equation, relate the depth-of-penetration into the bark to the acceleration of the woodpecker’s head:
xavv ∆+= 220
2 or, because v = 0,
xav ∆+= 20 20
Solve for and evaluate ∆x: ( )( ) cm13.6
m/s1002m/s5.3
2 2
220 =
−−
=−
=∆avx
Chapter 4
254
(c) Use the definition of acceleration to express the time required for the woodpecker’s head to come to rest:
avvt 0−
=∆
or, because v = 0,
avvt 0−
=∆
Substitute numerical values and evaluate ∆t: ms0.35
m/s100m/s3.5
20 =
−−
=−
=∆avt
*92 •• Picture the Problem The free-body diagram shown to the right shows the forces acting on an object suspended from the ceiling of a car that is accelerating to the right. Choose the coordinate system shown and use Newton’s laws of motion and constant- acceleration equations in the determination of the influence of the forces on the behavior of the suspended object.
The second free-body diagram shows the forces acting on an object suspended from the ceiling of a car that is braking while it moves to the right.
(a) on.accelerati theof that oppositedirection in the be will
ntdisplaceme sobject' theinertia, of law sNewton' with accordanceIn
(b) Resolve the tension, T, into its components and apply ∑ = aF rr
m to the object:
ΣFx = Tsinθ = ma and ΣFy = Tcosθ − mg = 0
Take the ratio of these two equations to eliminate T and m:
θθ
θθ
tantan
orcossin
gaga
mgma
TT
=⇒=
=
(c) forward. swing ter willaccelerome the
moving, iscar thedirection theopposite ison accelerati theBecause
Newton’s Laws
255
Using a constant-acceleration equation, express the velocity of the car in terms of its acceleration and solve for the acceleration:
xavv ∆+= 220
2 or, because v = 0,
xav ∆+= 20 20
Solve for a:
xva∆−
=2
20
Substitute numerical values and evaluate a:
( )( )
22
m/s61.1m602
km/h50−=
−=a
Solve the equation derived in (b) for θ : ⎟⎟
⎠
⎞⎜⎜⎝
⎛= −
ga1tanθ
Substitute numerical values and evaluate θ : °=⎟⎟
⎠
⎞⎜⎜⎝
⎛= − 32.9
m/s9.81m/s1.61tan 2
21θ
93 •• Picture the Problem The free-body diagram shows the forces acting at the top of the mast. Choose the coordinate system shown and use Newton’s 2nd and 3rd laws of motion to analyze the forces acting on the deck of the sailboat.
Apply∑ = xx maF to the top of the mast:
TFsinθF − TBsinθB = 0
Find the angles that the forestay and backstay make with the vertical:
°=⎟⎟⎠
⎞⎜⎜⎝
⎛=
°=⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
−
1.28m12m4.6tan
and
7.16m12m3.6tan
1B
1F
θ
θ
Solve the x-direction equation for TB: ( )
N305
1.28sin7.16sinN500
sinsin
B
FFB
=
°°
==θθTT
Find the downward forces that TB and TF exert on the mast:
∑ =−−= 0coscos BBFFmast θθ TTFFy
Solve for Fmast to obtain: BBFFmast coscos θθ TTF +=
Chapter 4
256
Substitute numerical values and evaluate Fmast:
( ) ( ) N7481.28cosN3057.16cosN500mast =°+°=F
The force that the mast exerts on the deck is the sum of its weight and the downward forces exerted on it by the forestay and backstay:
kN55.1
N800N748decktheonmast
=
+=F
94 •• Picture the Problem Let m be the mass of the block and M be the mass of the chain. The free-body diagrams shown below display the forces acting at the locations identified in the problem. We can apply Newton’s 2nd law with ay = 0 to each of the segments of the chain to determine the tensions. (a)
(b)
(c)
(a) Apply ∑ = yy maF to the block and solve for Ta:
ya mamgT =− or, because ay = 0,
mgTa =
Substitute numerical values and evaluate Ta:
( )( ) N491m/s9.81kg50 2 ==aT
(b) Apply ∑ = yy maF to the block and half the chain and solve for Tb:
yb magMmT =⎟⎠⎞
⎜⎝⎛ +−
2
or, because ay = 0,
gMmTb ⎟⎠⎞
⎜⎝⎛ +=
2
Substitute numerical values and evaluate Tb:
( )( ) N589m/s9.81kg10kg50 2 =+=bT
(c) Apply ∑ = yy maF to the block and chain and solve for Tc:
( ) yc magMmT =+− or, because ay = 0,
( )gMmTc +=
Newton’s Laws
257
Substitute numerical values and evaluate Tc:
( )( )N687
m/s9.81kg20kg50 2
=
+=cT
*95 ••• Picture the Problem The free-body diagram shows the forces acting on the box as the man pushes it across a frictionless floor. Because the force is time-dependent, the acceleration will be, too. We can obtain the acceleration as a function of time from the application of Newton’s 2nd law and then find the velocity of the box as a function of time by integration. Finally, we can derive an expression for the displacement of the box as a function of time by integration of the velocity function.
(a) The velocity is related to the acceleration according to: ( )ta
dtdv
= (1)
Apply ∑ = xx maF to the box and solve for its acceleration:
maF = and
( ) ( )ttmFa 3
31 m/s
kg24N/s8
===
Because the box’s acceleration is a function of time, separate variables in equation (1) and integrate to find v as a function of time:
( ) ( ) ( )
( ) ( ) 2361
23
31
0
331
0
m/s2
m/s
''m/s''
tt
dttdttatvtt
==
== ∫∫
Evaluate v at t = 3 s: ( ) ( )( ) m/s50.1s3m/ss3 23
61 ==v
(b) Integrate v = dx/dt between 0 and 3 s to find the displacement of the box during this time:
( ) ( )
( ) m50.13'm/s
''m/s''
s3
0
33
61
s3
0
2361
s3
0
=⎥⎦
⎤⎢⎣
⎡=
==∆ ∫∫
t
dttdttvx
(c) The average velocity is given by:
m/s500.0s3m1.5
ave ==∆∆
=txv
(d) Use Newton’s 2nd law to express the average force exerted on the box by the man: t
vmmaF∆∆
== avav
Chapter 4
258
Substitute numerical values and evaluate Fav: ( ) N0.12
s3m/s0m/s1.5kg24av =
−=F
96 ••
Picture the Problem The application of Newton’s 2nd law to the glider and the hanging weight will lead to simultaneous equations in their common acceleration a and the tension T in the cord that connects them. Once we know the acceleration of this system, we can use a constant-acceleration equation to predict how long it takes the cart to travel 1 m from rest. Note that the magnitudes of T
rand 'T
rare equal.
(a) The free-body diagrams are shown to the right. m1 represents the mass of the cart and m2 the mass of the hanging weight.
(b) Apply ∑ = xx maF to the cart and the suspended mass:
222
111
andsin
amTgm
amgmT
=−
=− θ
Letting a represent the common accelerations of the two objects, eliminate T between the two equations and solve a:
gmm
mma21
12 sin+
−=
θ
Substitute numerical values and evaluate a:
( )
( )2
2
m/s71.1
m/s9.81kg0.270kg0.075sin30kg0.270kg0.075
−=
×
+°−
=a
i.e., the acceleration is down the incline.
Substitute for a in either of the force equations to obtain:
N863.0=T
(c) Using a constant-acceleration equation, relate the displacement of the cart down the incline to its initial speed and acceleration:
( )221
0 tatvx ∆+∆=∆ or, because v0 = 0,
( )221 tax ∆=∆
Solve for ∆t:
axt ∆
=∆2
Substitute numerical values and evaluate ∆t:
( ) s08.1m/s1.71m12
2 ==∆t
Newton’s Laws
259
97 •• Picture the Problem Note that, while the mass of the rope is distributed over its length, the rope and the block have a common acceleration. Because the surface is horizontal and smooth, the only force that influences our solution is F
r. The figure misrepresents
the situation in that each segment of the rope experiences a gravitational force; the combined effect of which is that the rope must sag. (a) Apply totnet / mFa
rr= to the rope-
block system to obtain:
21 mmFa+
=
(b) Apply ∑ = aF rrm to the rope,
substitute the acceleration of the system obtained in (a), and simplify to obtain: F
mmm
mmFmamF
21
2
2122net
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
==
(c) Apply ∑ = aF rr
m to the block, substitute the acceleration of the system obtained in (a), and simplify to obtain: F
mmm
mmFmamT
21
1
2111
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
==
(d) The rope sags and so F
rhas both
vertical and horizontal components; with its horizontal component being less than .F
r Consequently, a will be
somewhat smaller.
*98 •• Picture the Problem The free-body diagram shows the forces acting on the block. Choose the coordinate system shown on the diagram. Because the surface of the wedge is frictionless, the force it exerts on the block must be normal to its surface.
(a) Apply ∑ = yy maF to the block to obtain:
ymawF =−°30sinn or, because ay = 0 and w = mg,
030sinn =−° mgF or
Chapter 4
260
mgF =°30sinn (1)
Apply ∑ = xx maF to the block: xmaF =°30cosn (2)
Divide equation (2) by equation (1) to obtain:
°= 30cotgax
Solve for and evaluate ax:
( )2
2
m/s0.17
30cotm/s81.930cot
=
°=°= gax
(b) An acceleration of the wedge greater than g cot30° would require that the normal force exerted on the body by the wedge be greater than that given in part (a); i.e., Fn > mg/sin30°.
wedge. theup acceleratedblock woul the
anddirection in the forcenet a be would therecondition, Under this
y
99 •• Picture the Problem Because the system is initially in equilibrium, it follows that T0 = 5mg. When one washer is removed on the left side, the remaining washers will accelerate upward (and those on the right side downward) in response to the net force that results. The free-body diagrams show the forces under this unbalanced condition. Applying Newton’s 2nd law to each collection of washers will allow us to determine both the acceleration of the system and the mass of a single washer.
(a) Apply ∑ = yy maF to the rising masses:
( )ammgT 44 =− (1)
Apply ∑ = yy maF to the descending masses:
( )amTmg 55 =− (2)
Eliminate T between these equations to obtain:
ga 91=
Use this acceleration in equation (1) or equation (2) to obtain:
mgT940
=
Express the difference between T0 and T and solve for m: N3.0
94050 =−=− mgmgTT
and
Newton’s Laws
261
g0.55kg0550.0 ==m
(b) Proceed as in (a) to obtain: T – 3mg = 3ma
and 5mg – T = 5ma
Eliminate T and solve for a: ( ) 2241
41 m/s45.2m/s81.9 === ga
Eliminate a in either of the motion equations and solve for T to obtain: mgT
415
=
Substitute numerical values and evaluate T: ( )( )
N03.2
m/s81.9kg0550.04
15 2
=
=T
100 •• Picture the Problem The free-body diagram represents the Atwood’s machine with N washers moved from the left side to the right side. Application of Newton’s 2nd law to each collection of washers will result in two equations that can be solved simultaneously to relate N, a, and g. The acceleration can then be found from the given data. Apply ∑ = yy maF to the rising washers:
T – (5 – N)mg = (5 – N)ma
Apply ∑ = yy maF to the descending washers:
(5 +N)mg – T = (5 + N)ma
Add these equations to eliminate T:
( ) ( )( ) ( )maNmaN
mgNmgN++−=
−−+55
55
Simplify to obtain:
maNmg 102 =
Solve for N:
N = 5a/g
Using a constant-acceleration equation, relate the distance the washers fell to their time of fall:
( )221
0 tatvy ∆+∆=∆ or, because v0 = 0,
( )221 tay ∆=∆
Chapter 4
262
Solve for the acceleration:
( )22
tya
∆∆
=
Substitute numerical values and evaluate a:
( )( )
22 m/s89.5
s40.0m471.02
==a
Substitute in the expression for N:
3m/s81.9m/s89.55 2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛=N
101 •• Picture the Problem Draw the free-body diagram for the block of mass m and apply Newton’s 2nd law to obtain the acceleration of the system and then the tension in the rope connecting the two blocks.
(a) Letting T be the tension in the connecting string, apply
∑ = xx maF to the block of mass m:
T – F1 = ma
Apply ∑ = xx maF to both blocks to determine the acceleration of the system:
F2 – F1 = (m + 2m)a = (3m)a
Substitute and solve for a:
a = (F2 – F1)/3m
Substitute for a in the first equation and solve for T:
T = ( )1231 2FF +
(b) Substitute for F1 and F2 in the equation derived in part (a):
T = (2Ct +2Ct)/3 = 4Ct/3
Evaluate this expression for T = T0 and t = t0 and solve for t0: t0 =
CT
43 0
Newton’s Laws
263
*102 ••• Picture the Problem Because a constant-upward acceleration has the same effect as an increase in the acceleration due to gravity, we can use the result of Problem 89 (for the tension) with a replaced by a + g. The application of Newton’s 2nd law to the object whose mass is m2 will connect the acceleration of this body to tension from Problem 84. In Problem 84 it is given that, when the support pulley is not accelerating, the tension in the rope and the acceleration of the masses are related according to:
gmmmmT
21
212+
=
Replace a with a + g: ( )gammmmT ++
=21
212
Apply ∑ = yy maF to the object whose mass is m2 and solve for a2:
T – m2g = m2a2 and
2
22 m
gmTa −=
Substitute for T and simplify to obtain:
( )21
1212
2mm
amgmma+
+−=
The expression for a1 is the same as for a2 with all subscripts interchanged (note that a positive value for a1 represents acceleration upward):
( )21
2121
2mm
amgmma+
+−=
Chapter 4
264
265
Chapter 5 Applications of Newton’s Laws Conceptual Problems
1 • Determine the Concept Because the objects are speeding up (accelerating), there must be a net force acting on them. The forces acting on an object are the normal force exerted by the floor of the truck, the weight of the object, and the friction force; also exerted by the floor of the truck.
Of these forces, the only one that acts in the direction of the acceleration (chosen to be to the right in the free-body diagram) is the friction force. .accelerate object to
thecauses that force thebemust truck theoffloor theandobject
ebetween thfriction of force The
*2 •
Determine the Concept The forces acting on an object are the normal force exerted by the floor of the truck, the weight of the object, and the friction force; also exerted by the floor of the truck. Of these forces, the only one that acts in the direction of the acceleration (chosen to be to the right in the free-body diagram) is the friction force. Apply Newton’s 2nd law to the object to determine how the critical acceleration depends on its weight.
Taking the positive x direction to be to the right, apply ΣFx = max and solve for ax:
f = µsw = µsmg = max and ax = µsg
same. theare onsaccelerati critical the
and oft independen is Becausew,
max
Chapter 5
266
3 • Determine the Concept The forces acting on the block are the normal force nF
r
exerted by the incline, the weight of the block g
rm exerted by the earth, and the
static friction force sfr
exerted by an external agent. We can use the definition of µs and the conditions for equilibrium to determine the relationship between µs and θ.
Apply xx maF =∑ to the block: fs − mgsinθ = 0 (1)
Apply yy maF =∑ in the y
direction:
Fn − mgcosθ = 0 (2)
Divide equation (1) by equation (2) to obtain:
n
stanFf
=θ
Substitute for fs (≤ µsFn):
sn
nstan µµθ =≤FF
and correct. is )(d
*4 • Determine the Concept The block is in equilibrium under the influence of ,nF
r,mgr
and ;sfr
i.e.,
nFr
+ gr
m + sfr
= 0
We can apply Newton’s 2nd law in the x direction to determine the relationship between fs and mg.
Apply 0=∑ xF to the block: fs − mgsinθ = 0
Solve for fs: fs = mgsinθ and correct. is )(d
Applications of Newton’s Laws
267
5 •• Picture the Problem The forces acting on the car as it rounds a curve of radius R at maximum speed are shown on the free-body diagram to the right. The centripetal force is the static friction force exerted by the roadway on the tires. We can apply Newton’s 2nd law to the car to derive an expression for its maximum speed and then compare the speeds under the two friction conditions described.
Apply ∑ = aF rrm to the car: ∑ ==
RvmfFx
2max
maxs,
and
∑ =−= 0n mgFFy
From the y equation we have: Fn = mg
Express fs,max in terms of Fn in the x equation and solve for vmax:
gRv smax µ=
or
smax constant µ=v
Express 'vmax for s2
1s µµ =' :
maxmaxs
max %71707.2
constant vvv' ≈==µ
and correct. is )(b
*6 •• Picture the Problem The normal reaction force Fn provides the centripetal force and the force of static friction, µsFn, keeps the cycle from sliding down the wall. We can apply Newton’s 2nd law and the definition of fs,max to derive an expression for vmin.
Apply ∑ = aF rr
m to the motorcycle: ∑ ==RvmFFx
2
n
and
Chapter 5
268
∑ =−= 0s mgfFy
For the minimum speed: fs = fs,max = µsFn
Substitute for fs, eliminate Fn between the force equations, and solve for vmin:
smin µ
Rgv =
Assume that R = 6 m and µs = 0.8 and solve for vmin:
( )( )
km/h30.9m/s8.580.8
m/s9.81m6 2
min
==
=v
7 •• Determine the Concept As the spring is extended, the force exerted by the spring on the block increases. Once that force is greater than the maximum value of the force of static friction on the block, the block will begin to move. However, as it accelerates, it will shorten the length of the spring, decreasing the force that the spring exerts on the block. As this happens, the force of kinetic friction can then slow the block to a stop, which starts the cycle over again. One interesting application of this to the real world is the bowing of a violin string: The string under tension acts like the spring, while the bow acts as the block, so as the bow is dragged across the string, the string periodically sticks and frees itself from the bow. 8 • True. The velocity of an object moving in a circle is continually changing independently of whether the object’s speed is changing. The change in the velocity vector and the acceleration vector and the net force acting on the object all point toward the center of circle. This center-pointing force is called a centripetal force. 9 • Determine the Concept A particle traveling in a vertical circle experiences a downward gravitational force plus an additional force that constrains it to move along a circular path. Because the net force acting on the particle will vary with location along its trajectory, neither (b), (c), nor (d) can be correct. Because the velocity of a particle moving along a circular path is continually changing, (a) cannot be correct. correct. is )(e
*10 • Determine the Concept We can analyze these demonstrations by drawing force diagrams for each situation. In both diagrams, h denotes ″hand″, g denotes ″gravitational″, m denotes ″magnetic″, and n denotes ″normal″.
Applications of Newton’s Laws
269
(a) Demonstration 1:
Demonstration 2:
(b) Because the magnet doesn’t lift the iron in the first demonstration, the force exerted on the iron must be less than its (the iron’s) weight. This is still true when the two are falling, but the motion of the iron is not restrained by the table, and the motion of the magnet is not restrained by the hand. Looking at the second diagram, the net force pulling the magnet down is greater than its weight, implying that its acceleration is greater than g. The opposite is true for the iron: the magnetic force acts upwards, slowing it down, so its acceleration will be less than g. Because of this, the magnet will catch up to the iron piece as they fall. *11 ••• Picture the Problem The free-body diagrams show the forces acting on the two objects some time after block 2 is dropped. Note that, while ,21 TT
rr≠ T1 = T2.
The only force pulling block 2 to the left is the horizontal component of the tension. Because this force is smaller than the magnitude of the tension, the acceleration of block 1, which is identical to block 2, to the right (T1 = T2) will always be greater than the acceleration of block 2 to the left.
wall.the hits 2block beforepulley hit the will1block wall, the to2block of distance
initial theas same theispulley the to1block from distance initial theBecause
12 • True. The terminal speed of an object is given by ( ) ,1 n
t bmgv = where b depends on the
shape and area of the falling object as well as upon the properties of the medium in which the object is falling. 13 • Determine the Concept The terminal speed of a sky diver is given by ( ) ,1 n
t bmgv =
where b depends on the shape and area of the falling object as well as upon the properties of the medium in which the object is falling. The sky diver’s orientation as she falls
Chapter 5
270
determines the surface area she presents to the air molecules that must be pushed aside. correct. is )( d
14 •• Determine the Concept In your frame of reference (the accelerating reference frame of the car), the direction of the force must point toward the center of the circular path along which you are traveling; that is, in the direction of the centripetal force that keeps you moving in a circle. The friction between you and the seat you are sitting on supplies this force. The reason you seem to be "pushed" to the outside of the curve is that your body’s inertia "wants" , in accordance with Newton’s law of inertia, to keep it moving in a straight line–that is, tangent to the curve.
*15 • Determine the Concept The centripetal force that keeps the moon in its orbit around the earth is provided by the gravitational force the earth exerts on the moon. As described by Newton’s 3rd law, this force is equal in magnitude to the force the moon exerts on the earth. correct. is )(d
16 • Determine the Concept The only forces acting on the block are its weight and the force the surface exerts on it. Because the loop-the-loop surface is frictionless, the force it exerts on the block must be perpendicular to its surface. Point A: the weight is downward and the normal force is to the right.
Free-body diagram 3
Point B: the weight is downward, the normal force is upward, and the normal force is greater than the weight so that their difference is the centripetal force.
Free-body diagram 4
Point C: the weight is downward and the normal force is to the left.
Free-body diagram 5
Point D: both the weight and the normal forces are downward.
Free-body diagram 2
Applications of Newton’s Laws
271
17 •• Picture the Problem Assume that the drag force on an object is given by the Newtonian formula ,2
21
D vCAF ρ= where A is the projected surface area, v is the object’s speed, ρ is the density of air, and C a dimensionless coefficient. Express the net force acting on the falling object:
maFmgF =−= Dnet
Substitute for FD under terminal speed conditions and solve for the terminal speed:
02T2
1 =− vCAmg ρ or
ρCAmgv 2
T =
Thus, the terminal velocity depends on the ratio of the mass of the object to its surface area.
For a rock, which has a relatively small surface area compared to its mass, the terminal speed will be relatively high; for a lightweight, spread-out object like a feather, the opposite is true. Another issue is that the higher the terminal velocity is, the longer it takes for a falling object to reach terminal velocity. From this, the feather will reach its terminal velocity quickly, and fall at an almost constant speed very soon after being dropped; a rock, if not dropped from a great height, will have almost the same acceleration as if it were in free-fall for the duration of its fall, and thus be continually speeding up as it falls. An interesting point is that the average drag force acting on the rock will be larger than that acting on the feather precisely because the rock’s average speed is larger than the feather's, as the drag force increases as v2. This is another reminder that force is not the same thing as acceleration. Estimation and Approximation *18 • Picture the Problem The free-body diagram shows the forces on the Tercel as it slows from 60 to 55 mph. We can use Newton’s 2nd law to calculate the average force from the rate at which the car’s speed decreases and the rolling force from its definition. The drag force can be inferred from the average and rolling friction forces and the drag coefficient from the defining equation for the drag force. (a) Apply∑ = xx maF to the car to relate the average force acting on it to its average velocity:
tvmmaF
∆∆
== avav
Chapter 5
272
Substitute numerical values and evaluate Fav:
( ) N581s3.92
kmm1000
s3600h1
mikm1.609
hmi5
kg1020av =×××
=F
(b) Using its definition, express and evaluate the force of rolling friction:
( )( )( )N200
m/s9.81kg102002.0 2
rollingnrollingrolling
=
=
== mgFf µµ
Assuming that only two forces are acting on the car in the direction of its motion, express their relationship and solve for and evaluate the drag force:
rollingdragav FFF += and
N381N200N815
rollingavdrag
=−=
−= FFF
(c) Convert 57.5 mi/h to m/s:
m/s7.25km
m10s3600
h1mi
km1.609h
mi57.5h
mi5.57
3
=
××
×=
Using the definition of the drag force and its calculated value from (b) and the average speed of the car during this 5 mph interval, solve for C:
221
drag AvCF ρ= ⇒ 2drag2
AvF
Cρ
=
Substitute numerical values and evaluate C:
( )( )( )( )
499.0
m/s7.25m1.91kg/m1.21N3812
223
=
=C
19 • Picture the Problem We can use the dimensions of force and velocity to determine the dimensions of the constant b and the dimensions of ρ, r, and v to show that, for n = 2, Newton’s expression is consistent dimensionally with our result from part (b). In parts (d) and (e), we can apply Newton’s 2nd law under terminal velocity conditions to find the terminal velocity of the sky diver near the surface of the earth and at a height of 8 km. (a) Solve the drag force equation for b with n = 1:
vFb d=
Applications of Newton’s Laws
273
Substitute the dimensions of Fd and v and simplify to obtain: [ ]
TM
TL
TML
b ==2
and the units of b are kg/s
(b) Solve the drag force equation for b with n = 2:
2d
vFb =
Substitute the dimensions of Fd and v and simplify to obtain: [ ]
LM
TLTML
b =
⎟⎠⎞
⎜⎝⎛
= 2
2
and the units of b are kg/m
(c) Express the dimensions of Newton’s expression:
[ ] [ ] ( )
2
22
322
21
d
TML
TLL
LMvrF
=
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛== ρπ
From part (b) we have: [ ] [ ]
2
22
d
TML
TL
LMbvF
=
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛==
(d) Letting the downward direction be the positive y direction, apply
∑ = yy maF to the sky diver:
02221 =− tvrmg ρπ
Solve for and evaluate vt: ( )( )( )( )
m/s9.56
m0.3kg/m1.2m/s9.81kg5622
23
2
2
=
==πr
mgvt ρπ
(e) Evaluate vt at a height of 8 km: ( )( )
( )( )m/s9.86
m0.3kg/m514.0m/s9.81kg562
23
2
=
=π
vt
Chapter 5
274
20 •• Picture the Problem From Newton’s 2nd law, the equation describing the motion of falling raindrops and large hailstones is mg – Fd = ma where 222
21
d bvvrF == ρπ is the
drag force. Under terminal speed conditions (a = 0), the drag force is equal to the weight of the falling object. Take the radius of a raindrop rr to be 0.5 mm and the radius of a golf-ball sized hailstone rh to be 2 cm. Using 2
21 rb πρ= , evaluate br and bh: ( )( )
kg/m1071.4
m105.0kg/m2.17
23321
r
−
−
×=
×= πb
and ( )( )
kg/m1054.7
m102kg/m2.14
22321
h
−
−
×=
×= πb
Express the mass of a sphere in terms of its volume and density: 3
4 3ρπρ rVm ==
Using ρr = 103 kg/m3 and ρh = 920 kg/m3, evaluate mr and mh:
( ) ( )
kg1024.53
kg/m10m105.04
7
3333
r
−
−
×=
×=
πm
and ( ) ( )
kg1008.33
kg/m920m1024
2
332
h
−
−
×=
×=
πm
Express the relationship between vt and the weight of a falling object under terminal speed conditions and solve for vt:
bmgvmgbv =⇒= t
2t
Use numerical values to evaluate vt,r and vt,h:
( )( )
m/s30.3
kg/m104.71m/s9.81kg105.24
7
27
rt,
=
××
= −
−
v
and ( )( )
m/s0.20
kg/m1054.7m/s9.81kg1008.3
4
22
ht,
=
××
= −
−
v
Applications of Newton’s Laws
275
Friction *21 • Picture the Problem The block is in equilibrium under the influence of nF
r,
,mgr
and ;kfr
i.e.,
nFr
+ gr
m + kfr
= 0
We can apply Newton’s 2nd law to determine the relationship between fk, θ, and mg.
Using its definition, express the coefficient of kinetic friction:
n
kk F
f=µ (1)
Apply ∑ = xx maF to the block: fk − mgsinθ = max = 0 because ax = 0
Solve for fk: fk = mgsinθ
Apply ∑ = yy maF to the block: Fn − mgcosθ = may = 0 because ay = 0
Solve for Fn: Fn = mgcosθ
Substitute in equation (1) to obtain: θθθµ tan
cossin
k ==mgmg
and correct. is )(b
22 • Picture the Problem The block is in equilibrium under the influence of nF
r,
,mgr
,appFr
and ;kfr
i.e.,
nFr
+ gr
m + appFr
+ kfr
= 0
We can apply Newton’s 2nd law to determine fk.
Apply ∑ = xx maF to the block: Fapp − fk = max = 0 because ax = 0
Solve for fk: fk = Fapp = 20 N
and correct. is )(e
Chapter 5
276
*23 • Picture the Problem Whether the friction force is that due to static friction or kinetic friction depends on whether the applied tension is greater than the maximum static friction force. We can apply the definition of the maximum static friction to decide whether fs,max or T is greater.
Calculate the maximum static friction force:
fs,max = µsFn = µsw = (0.8)(20 N) = 16 N
(a) Because fs,max > T: f = fs = T = N0.15
(b) Because T > fs,max: f = fk = µkw = (0.6)(20 N) = N0.12
24 • Picture the Problem The block is in equilibrium under the influence of the forces ,T
r ,kfr
and ;gr
m i.e.,
Tr
+ kfr
+ gr
m = 0
We can apply Newton’s 2nd law to determine the relationship between T and fk.
Apply ∑ = xx maF to the block: T cosθ − fk = max = 0 because ax = 0
Solve for fk: fk = T cosθ and correct. is )(b
25 • Picture the Problem Whether the friction force is that due to static friction or kinetic friction depends on whether the applied tension is greater than the maximum static friction force.
Calculate the maximum static fs,max = µsFn = µsw
Applications of Newton’s Laws
277
friction force: = (0.6)(100 kg)(9.81 m/s2) = 589 N
Because fs,max > Fapp, the box does not move and :
Fapp = f s = N500
26 • Picture the Problem Because the box is moving with constant velocity, its acceleration is zero and it is in equilibrium under the influence of ,appF
r,nF
r ,wr
and
;fr
i.e.,
appFr
+ nFr
+ wr + fr
= 0
We can apply Newton’s 2nd law to determine the relationship between f and mg.
The definition of µk is:
n
kk F
f=µ
Apply ∑ = yy maF to the box: Fn – w = may = 0 because ay = 0
Solve for Fn: Fn = w = 600 N
Apply ∑ = xx maF to the box: ΣFx = Fapp – f = max = 0 because ax = 0
Solve for fk: Fapp = fk = 250 N
Substitute to obtain µk: µk = (250 N)/(600 N) = 417.0
27 • Picture the Problem Assume that the car is traveling to the right and let the positive x direction also be to the right. We can use Newton’s 2nd law of motion and the definition of µs to determine the maximum acceleration of the car. Once we know the car’s maximum acceleration, we can use a constant-acceleration equation to determine the least stopping distance.
Chapter 5
278
(a) Apply∑ = xx maF to the car: −fs,max = −µsFn = max (1)
Apply∑ = yy maF to the car and
solve for Fn:
Fn − w = may = 0 or, because ay = 0, Fn = mg (2)
Substitute (2) in (1) and solve for ax,max: 2
2smaxx,
m/s5.89 =
)m/s (0.6)(9.81 = =
−
ga µ
(b) Using a constant-acceleration equation, relate the stopping distance of the car to its initial velocity and its acceleration and solve for its displacement:
xavv ∆+= 220
2
or, because v = 0,
avx
2
20−
=∆
Substitute numerical values and evaluate ∆x:
( )( ) m4.76
m/s89.52m/s30
2
2
=−−
=∆x
*28 • Picture the Problem The free-body diagram shows the forces acting on the drive wheels, the ones we’re assuming support half the weight of the car. We can use the definition of acceleration and apply Newton’s 2nd law to the horizontal and vertical components of the forces to determine the minimum coefficient of friction between the road and the tires.
(a) slip.not do wheels theifgreater be will , Because ks fµµ >
(b) Apply∑ = xx maF to the car: fs = µsFn = max (1)
Apply∑ = yy maF to the car and
solve for Fn:
ymamgF =− 21
n
Because ay = 0, 02
1n =− mgF ⇒ mgF 2
1n =
Find the acceleration of the car: ( )( )
2m/s08.2s12
m/km1000km/h90
=
=∆∆
=tvax
Applications of Newton’s Laws
279
Solve equation (1) for µs: ga
mgma xx 2
21s ==µ
Substitute numerical values and evaluate ax:
( ) 424.0m/s9.81m/s2.082
2
2
s ==µ
29 • Picture the Problem The block is in equilibrium under the influence of the forces shown on the free-body diagram. We can use Newton’s 2nd law and the definition of µs to solve for fs and Fn.
(a) Apply∑ = yy maF to the block
and solve for fs:
ymamgf =−s
or, because ay = 0, 0s =− mgf
Solve for and evaluate fs:
( )( )N1.49
m/s81.9kg5 2s
=
== mgf
(b) Use the definition of µs to express Fn: s
maxs,n µ
fF =
Substitute numerical values and evaluate Fn:
N1230.4
N1.49n ==F
30 • Picture the Problem The free-body diagram shows the forces acting on the book. The normal force is the net force the student exerts in squeezing the book. Let the horizontal direction be the x direction and upward the y direction. Note that the normal force is the same on either side of the book because it is not accelerating in the horizontal direction. The book could be accelerating downward. We can apply Newton’s 2nd law to relate the minimum force required to hold the book in place to its mass and to the coefficients of static friction. In part (b), we can proceed similarly to relate the acceleration of the
Chapter 5
280
book to the coefficients of kinetic friction. (a) Apply ∑ = aF rr
m to the book:
0and
0
'min,2s,2
'min,11,s
min,1min,2
=−+=
=−=
∑
∑
mgFFF
FFF
y
x
µµ
Noting that ,min,2min,1
'' FF = solve the y equation for Fmin: s,21,
min µµ +=
mgF
Substitute numerical values and evaluate Fmin:
( )( ) N2080.160.32
m/s9.81kg10.2 2
min =+
=F
(b) Apply ∑ = yy maF with the book accelerating downward, to obtain:
mamgFFF ky =−+=∑ k,21, µµ
Solve for a to obtain: gF
ma kk −
+= 2,, µµ
Substitute numerical values and evaluate a: ( )
2
2
m/s27.4
m/s9.81N195kg10.20.090.2
−=
−+
=a
31 • Picture the Problem A free-body diagram showing the forces acting on the car is shown to the right. The friction force that the ground exerts on the tires is the force fs shown acting up the incline. We can use the definition of the coefficient of static friction and Newton’s 2nd law to relate the angle of the incline to the forces acting on the car.
Apply aF rr
m=∑ to the car: 0sins =−=∑ θmgfFx (1)
and 0cosn =−=∑ θmgFFy (2)
Solve equation (1) for fs and equation (2) for Fn:
θsins mgf =
and
Applications of Newton’s Laws
281
θcosn mgF =
Use the definition of µs to relate fs and Fn:
θθθµ tan
cossin
n
ss ===
mgmg
Ff
Solve for and evaluate θ : ( ) ( ) °=== −− 57.408.0tantan 1
s1 µθ
*32 • Picture the Problem The free-body diagrams for the two methods are shown to the right. Method 1 results in the box being pushed into the floor, increasing the normal force and the static friction force. Method 2 partially lifts the box,, reducing the normal force and the static friction force. We can apply Newton’s 2nd law to obtain expressions that relate the maximum static friction force to the applied force .F
r
(a) s.n , thereforeand, reducesit as preferable is 2 Method fF
(b) Apply ∑ = xx maF to the box: F cosθ − fs = Fcosθ − µsFn = 0
Method 1: Apply ∑ = yy maF to
the block and solve for Fn:
Fn – mg − Fsinθ = 0 ∴ Fn = mg + Fsinθ
Relate fs,max to Fn: fs,max = µsFn = µs(mg + Fsinθ) (1)
Method 2: Apply ∑ = yy maF to
the forces in the y direction and solve for Fn:
Fn – mg + Fsinθ = 0 and Fn = mg − Fsinθ
Relate fs,max to Fn: fs,max = µsFn = µs(mg − Fsinθ) (2)
Express the condition that must be satisfied to move the box by either method:
fs,max = Fcosθ (3)
Method 1: Substitute (1) in (3) and solve for F: θµθ
µsincos s
s1 −
=mgF (4)
Chapter 5
282
Method 2: Substitute (2) in (3) and solve for F: θµθ
µsincos s
s2 +
=mgF (5)
Evaluate equations (4) and (5) with θ = 30°:
( ) N520301 =°F
( ) N252302 =°F
Evaluate (4) and (5) with θ = 0°: ( ) ( ) N29400 s21 ==°=° mgFF µ
33 • Picture the Problem Draw a free-body diagram for each object. In the absence of friction, the 3-kg box will move to the right, and the 2-kg box will move down. The friction force is indicated by f
rwithout
subscript; it is sfr
for (a) and kfr
for (b). For values of µs less than the value found in part (a) required for equilibrium, the system will accelerate and the fall time for a given distance can be found using a constant-acceleration equation. (a) Apply ∑ = xx maF to the 3-kg
box:
T – fs = 0 because ax = 0 (1)
Apply∑ = yy maF to the 3-kg box,
solve for Fn,3, and substitute in (1):
Fn,3 – m3g = 0 because ay = 0 and T – µs m3g = 0 (2)
Apply∑ = xx maF to the 2-kg box:
m2g – T = 0 because ax = 0 (3)
Solve (2) and (3) simultaneously and solve for µs:
667.03
2s ==
mmµ
(b) The time of fall is related to the acceleration, which is constant:
( )221
0 tatvx ∆+∆=∆
or, because v0 =0, ( )2
21 tax ∆=∆
Solve for ∆t:
axt ∆
=∆2
(4)
Applications of Newton’s Laws
283
Apply ∑ = xx maF to each box: T – µk m3g = m3a (5) and m2g – T = m2a (6)
Add equations (5) and (6) and solve for a:
( )32
3k2
mmgmma
+−
=µ
Substitute numerical values and evaluate a:
( )[ ]( )
2
2
m/s16.2
kg3kg2m/s9.81kg30.3kg2
=
+−
=a
Substitute numerical values in equation (4) and evaluate ∆t:
( ) s36.1m/s2.16m22
2 ==∆t
34 •• Picture the Problem The application of Newton’s 2nd law to the block will allow us to express the coefficient of kinetic friction in terms of the acceleration of the block. We can then use a constant-acceleration equation to determine the block’s acceleration. The pictorial representation summarizes what we know about the motion.
A free-body diagram showing the forces acting on the block is shown to the right.
Apply∑ = xx maF to the block:
– fk = −µkFn = ma (1)
Apply∑ = yy maF to the block and
solve for Fn:
Fn – mg = 0 because ay = 0 and Fn = mg (2)
Chapter 5
284
Substitute (2) in (1) and solve for µk:
µk = −a/g (3)
Using a constant-acceleration equation, relate the initial and final velocities of the block to its displacement and acceleration:
220
21 xavv ∆+=
or, because v1 = 0, v0 = v, and ∆x = d, 20 2 adv +=
Solve for a to obtain: d
va2
2−=
Substitute for a in equation (3) to obtain: gd
v2
2
k =µ
*35 •• Picture the Problem We can find the speed of the system when it has moved a given distance by using a constant-acceleration equation. Under the influence of the forces shown in the free-body diagrams, the blocks will have a common acceleration a. The application of Newton’s 2nd law to each block, followed by the elimination of the tension T and the use of the definition of fk, will allow us to determine the acceleration of the system.
Using a constant-acceleration equation, relate the speed of the system to its acceleration and displacement; solve for its speed:
220
2 xavv ∆+=
and, because v0 = 0, xav ∆= 2 (1)
Apply aF rrmt =ne to the block whose
mass is m1:
ΣFx = T – fk – m1gsin30° = m1a (2) and ΣFy = Fn,1 – m1gcos30° = 0 (3)
Using fk = µkFn, substitute (3) in (2) to obtain:
T – µk m1g cos30° – m1gsin30° = m1a
Apply∑ = xx maF to the block
whose mass is m2:
m2g – T = m2a
Add the last two equations to eliminate T and solve for a to
( )21
11k2 30sin30cosmm
gmmma+
°−°−=
µ
Applications of Newton’s Laws
285
obtain: Substitute numerical values and evaluate a:
2m/s16.1=a
Substitute numerical values in equation (1) and evaluate v:
( )( ) m/s835.0m0.3m/s1.162 2 ==v
and correct. is )(a
36 •• Picture the Problem Under the influence of the forces shown in the free-body diagrams, the blocks are in static equilibrium. While fs can be either up or down the incline, the free-body diagram shows the situation in which motion is impending up the incline. The application of Newton’s 2nd law to each block, followed by the elimination of the tension T and the use of the definition of fs, will allow us to determine the range of values for m2.
(a) Apply aF rr
m=∑ to the block
whose mass is m1:
ΣFx = T ± fs,max – m1gsin30° = 0 (1) and ΣFy = Fn,1 – m1gcos30° = 0 (2)
Using fs,max = µsFn, substitute (2) in (1) to obtain:
amgmgmT s
11
1
30sin30cos
=°−°± µ
(3)
Apply∑ = xx maF to the block
whose mass is m2:
m2g – T = 0 (4)
Add equations (3) and (4) to eliminate T and solve for m2:
( )( ) ( )[ ]°+°±=
°+°±=30sin30cos4.0kg4
30sin30coss12 µmm (5)
Evaluate (5) denoting the value of m2 with the plus sign as m2,+ and the value of m2 with the minus sign as m2,- to determine the range of values of m2 for which the system is in static equilibrium:
kg39.3kg614.0
kg614.0mandkg39.3
2
2,-,2
≤≤∴
==+
m
m
Chapter 5
286
(b) With m2 = 1 kg, the impending motion is down the incline and the static friction force is up the incline. Apply∑ = xx maF to the block
whose mass is m1:
T + fs – m1gsin30° = 0 (6)
Apply∑ = xx maF to the block
whose mass is m2:
m2g – T = 0 (7)
Add equations (6) and (7) and solve for and evaluate fs:
fs = (m1sin30° – m2)g = [(4 kg)sin30° – 1 kg](9.81 m/s2) = N81.9
37 •• Picture the Problem Under the influence of the forces shown in the free-body diagrams, the blocks will have a common acceleration a. The application of Newton’s 2nd law to each block, followed by the elimination of the tension T and the use of the definition of fk, will allow us to determine the acceleration of the system. Finally, we can substitute for the tension in either of the motion equations to determine the acceleration of the masses.
Apply aF rr
m=∑ to the block
whose mass is m1:
ΣFx = T – fk – m1gsin30° = m1a (1) and ΣFy = Fn,1 – m1gcos30° = 0 (2)
Using fk = µkFn, substitute (2) in (1) to obtain:
amgmgmT
11
1k
30sin30cos
=°−°− µ
(3)
Apply∑ = xx maF to the block
whose mass is m2:
m2g – T = m2a (4)
Add equations (3) and (4) to eliminate T and solve for a to obtain:
( )21
11k2 30sin30cosmm
gmmma+
°−°−=
µ
Substituting numerical values and evaluating a yields:
2m/s36.2=a
Applications of Newton’s Laws
287
Substitute for a in equation (3) to obtain:
N3.37=T
*38 •• Picture the Problem The truck will stop in the shortest possible distance when its acceleration is a maximum. The maximum acceleration is, in turn, determined by the maximum value of the static friction force. The free-body diagram shows the forces acting on the box as the truck brakes to a stop. Assume that the truck is moving in the positive x direction and apply Newton’s 2nd law and the definition of fs,max to find the shortest stopping distance.
Using a constant-acceleration equation, relate the truck’s stopping distance to its acceleration and initial velocity; solve for the stopping distance: max
20
min
20
2
2
,0since or,2
avx
vxavv
−=∆
=∆+=
Apply aFrr
mt =ne to the block: ΣFx = – fs,max = mamax (1) and ΣFy = Fn – mg = 0 (2)
Using the definition of fs,max, solve equations (1) and (2) simultaneously for a:
fs,max ≡ µsFn and amax = −µsg = − (0.3)(9.81 m/s2) = −2.943 m/s2
Substitute numerical values and evaluate ∆xmin:
( ) ( ) ( )( ) m16.9
m/s2.9432sh/36001km/m1000km/h80
2
222
min =−
−=∆x
Chapter 5
288
39 •• Picture the Problem We can find the coefficient of friction by applying Newton’s 2nd law and determining the acceleration from the given values of displacement and initial velocity. We can find the displacement and speed of the block by using constant-acceleration equations. During its motion up the incline, the sum of the kinetic friction force and a component of the object’s weight will combine to bring the object to rest. When it is moving down the incline, the difference between the weight component and the friction force will be the net force.
(a) Draw a free-body diagram for the block as it travels up the incline:
Apply aF rr
m=∑ to the block: ΣFx = – fk – mgsin37°= ma (1) and ΣFy = Fn – mg cos37° = 0 (2)
Substitute fk = µkFn and Fn from (2) in (1) and solve for µk:
°−°−=
°−°−
=
37cos37tan
37cos37sin
k
ga
gagµ
(3)
Using a constant-acceleration equation, relate the final velocity of the block to its initial velocity, acceleration, and displacement:
xavv ∆+= 220
21
Solving for a yields: xvva
∆−
=2
20
21
Substitute numerical values and evaluate a:
( ) ( )( )
222
m/s6.10m82
m/s14m/s5.2−=
−=a
Applications of Newton’s Laws
289
Substitute for a in (3) to obtain: ( )
599.0
cos37m/s9.81m/s10.637tan 2
2
k
=
°−
−°−=µ
(b) Use the same constant-acceleration equation used above but with v1 = 0 to obtain:
xav ∆+= 20 20
Solve for ∆x to obtain: avx
2
20−
=∆
Substitute numerical values and evaluate ∆x:
( )( ) m25.9
m/s10.62m/s14
2
2
=−−
=∆x
(c) When the block slides down the incline, fk is in the positive x direction:
ΣFx = fk – mgsin37°= ma and ΣFy = Fn – mgcos37° = 0
Solve for a as in part (a): ( ) 2k m/s21.137sin37cos −=°−°= µga
Use the same constant-acceleration equation used in part (b) to obtain:
xavv ∆+= 220
2
Set v0 = 0 and solve for v: xav ∆= 2
Substitute numerical values and evaluate v:
( )( )m/s73.4
m25.9m/s21.12 2
=
−−=v
40 •• Picture the Problem We can find the stopping distances by applying Newton’s 2nd law to the automobile and then using a constant-acceleration equation. The friction force the road exerts on the tires and the component of the car’s weight along the incline combine to provide the net force that stops the car. The pictorial representation summarizes what we know about the motion of the car. We can use Newton’s 2nd law to determine the acceleration of the car and a constant-acceleration equation to obtain its stopping distance.
Chapter 5
290
(a) Using a constant-acceleration equation, relate the final speed of the car to its initial speed, acceleration, and displacement; solve for its displacement:
max
20
min
1
minmax20
21
2
,0because or,2
avx
vxavv
−=∆
=∆+=
Draw the free-body diagram for the car going up the incline:
Apply ∑ = aF rr
m to the car: ΣFx = −fs,max – mgsin15° = ma (1) and ΣFy = Fn – mgcos15° = 0 (2)
Substitute fs,max = µsFn and Fn from (2) in (1) and solve for a:
( )2
smax
m/s17.9
15sin15cos
−=
°+°−= µga
Substitute numerical values in the expression for ∆xmin to obtain:
( )( ) m1.49
m/s9.172m/s30
2
2
min =−−
=∆x
(b) Draw the free-body diagram for the car going down the incline:
Applications of Newton’s Laws
291
Apply ∑ = aF rrm to the car: ΣFx = fs,max – mgsin15° = ma
and ΣFy = Fn – mgcos15° = 0
Proceed as in (a) to obtain amax: ( ) 2smax m/s09.415sin15cos =°−°= µga
Again, proceed as in (a) to obtain the displacement of the car:
( )( ) m110
m/s09.42m/s30
2 2
2
max
20
min ==−
=∆avx
41 •• Picture the Problem The friction force the road exerts on the tires provides the net force that accelerates the car. The pictorial representation summarizes what we know about the motion of the car. We can use Newton’s 2nd law to determine the acceleration of the car and a constant-acceleration equation to calculate how long it takes it to reach 100 km/h.
(a) Because 40% of the car’s weight is on its two drive wheels and the accelerating friction forces act just on these wheels, the free-body diagram shows just the forces acting on the drive wheels.
Apply ∑ = aF rrm to the car: ΣFx = fs,max = ma (1)
and ΣFy = Fn – 0.4mg = 0 (2)
Use the definition of fs,max in equation (1) and eliminate Fn between the two equations to obtain:
( )( )2
2s
m/s75.2
m/s81.97.04.04.0
=
== ga µ
(b) Using a constant-acceleration equation, relate the initial and final
tavv ∆+= 01
Chapter 5
292
velocities of the car to its acceleration and the elapsed time; solve for the time:
or, because v0 = 0 and ∆t = t1,
avt 1
1 =
Substitute numerical values and evaluate t1:
( )( )( ) s1.10m/s2.75
m/km1000sh/36001km/h10021 ==t
*42 •• Picture the Problem To hold the box in place, the acceleration of the cart and box must be great enough so that the static friction force acting on the box will equal the weight of the box. We can use Newton’s 2nd law to determine the minimum acceleration required.
(a) Apply ∑ = aF rr
m to the box:
ΣFx = Fn = mamin (1) and ΣFy = fs,max – mg = 0 (2)
Substitute µFn for fs,max in equation (2), eliminate Fn between the two equations and solve for and evaluate amin:
µFn − mg = 0 , µ(mamin ) − mg = 0 and
22
smin m/s4.16
0.6m/s81.9
===µga
(b) Solve equation (2) for fs,max, and substitute numerical values and evaluate fs,max:
fs,max = mg = (2 kg)(9.81 m/s2) = N6.19
(c) If a is twice that required to hold the box in place, fs will still have its maximum value given by:
fs,max = N6.19
(d) . if fallnot box will the, is Because smins µµ gaag ≥
43 •• Picture the Problem Note that the blocks have a common acceleration and that the tension in the string acts on both blocks in accordance with Newton’s third law of motion. Let down the incline be the positive x direction. Draw the free-body diagrams for each block and apply Newton’s second law of motion and the definition of the kinetic friction force to each block to obtain simultaneous
Applications of Newton’s Laws
293
equations in ax and T. Draw the free-body diagram for the block whose mass is m1:
Apply ∑ = aF rrm to the upper block:
ΣFx = −fk,1 + T1 + m1gsinθ = m1ax (1) and ΣFy = Fn,1 – m1gcosθ = 0 (2)
The relationship between fk,1 and Fn,1 is:
fk,1 = µk,1Fn,1 (3)
Eliminate fk,1 and Fn,1 between (1), (2), and (3) to obtain:
−µk,1m1gcosθ + T1 + m1gsinθ = m1ax (4)
Draw the free-body diagram for the block whose mass is m2:
Apply ∑ = aF rrm to the block: ΣFx = − fk,2 – T2 + m2gsinθ = m2ax (5)
and ΣFy = Fn,2 – m2gcosθ = 0 (6)
The relationship between fk,2 and Fn,2 is:
fk,2 = µk,2Fn,2 (7)
Eliminate fk,2 and Fn,2 between (5), (6), and (7) to obtain:
−µk,2m2gcosθ – T2 + m2gsinθ = m2ax (8)
Chapter 5
294
Noting that T2 = T1 = T, add equations (4) and (8) to eliminate T and solve for ax:
gmm
mmax ⎥
⎦
⎤⎢⎣
⎡++
−= θµµ
θ cossin21
2k,21k,1
Substitute numerical values and evaluate ax to obtain:
2m/s965.0−=xa where the minus
sign tells us that the acceleration is directed up the incline.
(b) Eliminate ax between equations (4) and (8) and solve for T = T1 = T2 to obtain:
( )21
k,1k,221 cosmm
gmmT
+−
=θµµ
Substitute numerical values and evaluate T:
N184.0=T
*44 •• Picture the Problem The free-body diagram shows the forces acting on the two blocks as they slide down the incline. Down the incline has been chosen as the positive x direction. T is the force transmitted by the stick; it can be either tensile (T > 0) or compressive (T < 0). By applying Newton’s 2nd law to these blocks, we can obtain equations in T and a from which we can eliminate either by solving them simultaneously. Once we have expressed T, the role of the stick will become apparent. (a) Apply ∑ = aF rr
m to block 1:
∑
∑
=−=
=−+=
0cosand
sin
1n,1
1k,111
θ
θ
gmFF
amfgmTF
y
x
Apply ∑ = aF rr
m to block 2:
∑
∑
=−=
=−−=
0cosand
sin
2n,2
2k,222
θ
θ
gmFF
amfTgmF
y
x
Letting T1 = T2 = T, use the definition of the kinetic friction force to eliminate fk,1 and Fn,1 between the equations for block 1 and fk,2 and Fn,1 between the equations for block 2 to obtain:
θµθ cossin 1111 gmTgmam −+= (1) and
θµθ cossin 2222 gmTgmam −−= (2)
Applications of Newton’s Laws
295
Add equations (1) and (2) to eliminate T and solve for a: ⎟⎟
⎠
⎞⎜⎜⎝
⎛++
−= θµµ
θ cossin21
2211
mmmmga
(b) Rewrite equations (1) and (2) by dividing both sides of (1) by m1 and both sides of (2) by m2 to obtain.
θµθ cossin 11
gmTga −+= (3)
and
θµθ cossin 22
gmTga −−= (4)
Subtracting (4) from (3) and rearranging yields: ( ) θµµ cos21
21
21 gmm
mmT −⎟⎟⎠
⎞⎜⎜⎝
⎛−
=
( )block.either on force noexert must stick the, therefore this;change
tcan' embetween thstick a Inserting .cossin ofon accelerati same with theincline down the move blocks theand 0 , If 21
θµθµµ
−==
gT
45 •• Picture the Problem The pictorial representation shows the orientation of the two blocks on the inclined surface. Draw the free-body diagrams for each block and apply Newton’s 2nd law of motion and the definition of the static friction force to each block to obtain simultaneous equations in θc and T. (a) Draw the free-body diagram for the lower block:
Apply ∑ = aF rr
m to the block: ΣFx = m1gsinθc – fs,1 − T = 0 (1) and ΣFy = Fn,1 – m1gcosθc = 0 (2)
The relationship between fs,1 and Fn,1 is:
fs,1 = µs,1Fn,1 (3)
Chapter 5
296
Eliminate fs,1 and Fn,1 between (1), (2), and (3) to obtain:
m1gsinθc − µs,1m1gcosθc − T = 0 (4)
Draw the free-body diagram for the upper block:
Apply ∑ = aF rrm to the block: ΣFx =T + m2gsinθc – fs,2 = 0 (5)
and ΣFy = Fn,2 – m2gcosθc = 0 (6)
The relationship between fs,2 and Fn,2 is:
fs,2 = µs,2Fn,2 (7)
Eliminate fs,2 and Fn,2 between (5), (6), and (7) to obtain:
T + m2gsinθc – µs,2m2gcosθc = 0 (8)
Add equations (4) and (8) to eliminate T and solve for θc:
( )( ) ( )( )
°=
⎥⎦
⎤⎢⎣
⎡++
=
⎥⎦
⎤⎢⎣
⎡++
=
−
−
0.25
kg0.2kg0.1kg0.10.6kg0.20.4tan
tan
1
21
2s,21s,11c mm
mm µµθ
(b) Because θc is greater than the angle of repose (tan−1(µs,1) = tan−1(0.4) = 21.8°) for the lower block, it would slide if T = 0. Solve equation (4) for T:
( )Cs,1C1 cossin θµθ −= gmT
Substitute numerical values and evaluate T:
( )( ) ( )[ ] N118.0cos250.4sin25m/s9.81kg0.2 2 =°−°=T
Applications of Newton’s Laws
297
46 •• Picture the Problem The pictorial representation shows the orientation of the two blocks with a common acceleration on the inclined surface. Draw the free-body diagrams for each block and apply Newton’s 2nd law and the definition of the kinetic friction force to each block to obtain simultaneous equations in a and T.
(a) Draw the free-body diagram for the lower block:
Apply ∑ = aF rr
m to the lower
block:
ΣFx = m1gsin20° − fk,1 – T = m1a (1) and ΣFy = Fn,1 − m1gcos20° = 0 (2)
Express the relationship between fk,1 and Fn,1:
fk,1 = µk,1Fn,1 (3)
Eliminate fk,1 and Fn,1 between (1), (2), and (3) to obtain:
amTgmgm
1
1k,11 20cos20sin=−
°−° µ (4)
Draw the free-body diagram for the upper block:
Apply ∑ = aF rr
m to the upper
block:
ΣFx = T + m2gsin20° − fk,2 = m2a (5) and ΣFy = Fn,2 – m2gcos20° = 0 (6)
Express the relationship between fk,2 and Fn,2 :
fk,2 = µk,2Fn,2 (7)
Chapter 5
298
Eliminate fk,2 and Fn,2 between (5), (2), and (7) to obtain:
amgmgmT
2
2k,22 20cos20sin=
°−°+ µ (8)
Add equations (4) and (8) to eliminate T and solve for a:
⎟⎟⎠
⎞⎜⎜⎝
⎛°
++
−°= 20cos20sin21
2211
mmmmga µµ
Substitute the given values and evaluate a:
2m/s944.0=a
(b) Substitute for a in either equation (4) or equation (8) to obtain:
N426.0−=T ; i.e., the rod is under
compression. *47 •• Picture the Problem The vertical component of F
rreduces the normal force;
hence, the static friction force between the surface and the block. The horizontal component is responsible for any tendency to move and equals the static friction force until it exceeds its maximum value. We can apply Newton’s 2nd law to the box, under equilibrium conditions, to relate F to θ. (a) The static-frictional force opposes the motion of the object, and the maximum value of the static-frictional force is proportional to the normal force FN. The normal force is equal to the weight minus the vertical component FV of the force F. Keeping the magnitude F constant while increasing θ from zero results in a decrease in FV and thus a corresponding decrease in the maximum static-frictional force fmax. The object will begin to move if the horizontal component FH of the force F exceeds fmax. An increase in θ results in a decrease in FH. As θ increases from 0, the decrease in FN is larger than the decrease in FH, so the object is more and more likely to slip. However, as θ approaches 90°, FH approaches zero and no movement will be initiated. If F is large enough and if θ increases from 0, then at some value of θ the block will start to move. (b) Apply ∑ = aF rr
m to the block:
ΣFx =Fcosθ – fs = 0 (1) and ΣFy = Fn + Fsinθ – mg = 0 (2)
Assuming that fs = fs,max, eliminate fs and Fn between equations (1) and (2) and solve for F:
θµθµ
sincos s
s
+=
mgF
Applications of Newton’s Laws
299
Use this function with mg = 240 N to generate the table shown below:
θ (deg) 0 10 20 30 40 50 60 F (N) 240 220 210 206 208 218 235
The following graph of F(θ) was plotted using a spreadsheet program.
205
210
215
220
225
230
235
240
0 10 20 30 40 50 60
theta (degrees)
F (N
)
From the graph, we can see that the minimum value for F occurs when θ ≈ 32°. Remarks: An alternative to manually plotting F as a function of θ or using a spreadsheet program is to use a graphing calculator to enter and graph the function. 48 ••• Picture the Problem The free-body diagram shows the forces acting on the block. We can apply Newton’s 2nd law, under equilibrium conditions, to relate F to θ and then set its derivative with respect to θ equal to zero to find the value of θ that minimizes F.
(a) Apply ∑ = aF rr
m to the block: ΣFx =Fcosθ – fs = 0 (1) and ΣFy = Fn + Fsinθ – mg = 0 (2)
Chapter 5
300
Assuming that fs = fs,max, eliminate fs and Fn between equations (1) and (2) and solve for F:
θµθµ
sincos s
s
+=
mgF (3)
To find θmin, differentiate F with respect to θ and set the derivative equal to zero for extrema of the function:
( ) ( )( )
( )( )
( )( )
extremafor 0sincos
cossinsincos
sincos
sincos
sincos
2s
ss
2s
ss
2s
ss
=+
+−=
+
+−
+
+=
θµθθµθµ
θµθ
θµθθ
µ
θµθ
µθ
θµθ
θmg
ddmgmg
dd
ddF
Solve for θmin to obtain:
s1
min tan µθ −=
(b) Use the reference triangle shown below to substitute for cosθ and sinθ in equation (3):
mg
mg
mgF
2s
s
2s
2s
s
2s
ss2
s
smin
1
11
111
µ
µ
µ
µµ
µ
µµµ
µ
+=
+
+=
++
+
=
(c)
decreased. becan angle theso decrease, willfriction oft coefficien themoving isblock theonce Therefore, .tan
bygiven anglean at applied be should movingblock thekeep apply to should one force minimum that theshows above one the toidentical analysisAn
friction. static oft coefficien than theless isfriction kinetic oft coefficien The
k1
min µθ −=
49 •• Picture the Problem The vertical component of F
rincreases the normal force and the
static friction force between the surface and the block. The horizontal component is responsible for any tendency to move and equals the static friction force until it exceeds its maximum value. We can apply Newton’s 2nd law to the box, under equilibrium conditions, to relate F to θ.
Applications of Newton’s Laws
301
(a) As θ increases from zero, F increases the normal force exerted by the surface and the static friction force. As the horizontal component of F decreases with increasing θ, one would expect F to continue to increase.
(b) Apply ∑ = aF rr
m to the block:
ΣFx =Fcosθ – fs = 0 (1) and ΣFy = Fn – Fsinθ – mg = 0 (2)
Assuming that fs = fs,max, eliminate fs and Fn between equations (1) and (2) and solve for F:
θµθµ
sincos s
s
−=
mgF (3)
Use this function with mg = 240 N to generate the table shown below.
θ (deg) 0 10 20 30 40 50 60 F (N) 240 273 327 424 631 1310 very
large The graph of F as a function of θ, plotted using a spreadsheet program, confirms our prediction that F continues to increase with θ.
0
200
400
600
800
1000
1200
1400
0 10 20 30 40 50
theta (degrees)
F (N
)
(a) From the graph we see that: °= 0minθ
Chapter 5
302
(b) Evaluate equation (3) for θ = 0° to obtain:
mgmgF ss
s
0sin0cosµ
µµ
=°−°
=
(c) .0at angle thekeep shouldYou °
Remarks: An alternative to the use of a spreadsheet program is to use a graphing calculator to enter and graph the function. 50 •• Picture the Problem The forces acting on each of these masses are shown in the free-body diagrams below. m1 represents the mass of the 20-kg mass and m2 that of the 100-kg mass. As described by Newton’s 3rd law, the normal reaction force Fn,1 and the friction force fk,1 (= fk,2) act on both masses but in opposite directions. Newton’s 2nd law and the definition of kinetic friction forces can be used to determine the various forces and the acceleration called for in this problem. (a) Draw a free-body diagram showing the forces acting on the 20-kg mass:
Apply ∑ = aF rrm to this mass: ΣFx = fk,1 = m1a1 (1)
and ΣFy = Fn,1 – m1g = 0 (2)
Solve equation (1) for fk,1: fk,1 = m1a1 = (20 kg)(4 m/s2) = N0.80
(b) Draw a free-body diagram showing the forces acting on the 100-kg mass:
Apply ∑ = xx maF to the 100-kg
object and evaluate Fnet:
( )( ) N600m/s6kg100 2
22net
==
= amF
Applications of Newton’s Laws
303
Express F in terms of Fnet and fk,2: F = Fnet + fk,2 = 600 N + 80 N = N680
(c) When the 20-kg mass falls off, the 680-N force acts just on the 100-kg mass and its acceleration is given by Newton’s 2nd law:
2net m/s80.6kg100N680
===m
Fa
51 •• Picture the Problem The forces acting on each of these blocks are shown in the free-body diagrams to the right. m1 represents the mass of the 60-kg block and m2 that of the 100-kg block. As described by Newton’s 3rd law, the normal reaction force Fn,1 and the friction force fk,1 (= fk,2) act on both objects but in opposite directions. Newton’s 2nd law and the definition of kinetic friction forces can be used to determine the coefficient of kinetic friction and acceleration of the 100-kg block.
(a) Apply ∑ = aF rr
m to the 60-kg
block:
ΣFx = F − fk,1 = m1a1 (1) and ΣFy = Fn,1 – m1g = 0 (2)
Apply ∑ = xx maF to the 100-kg
block:
fk,2 = m2a2 (3)
Using equation (2), express the relationship between the kinetic friction forces 1,kf
rand 2,kf
r:
fk,1 = fk,2 = fk = µ kFn,1 = µ km1g (4)
Substitute equation (4) into equation (1) and solve for µ k: gm
amF
1
11k
−=µ
Substitute numerical values and evaluate µ k:
( )( )( )( ) 238.0
m/s9.81kg60m/s3kg60N320
2
2
k =−
=µ
(b) Substitute equation (4) into equation (3) and solve for a2:
2
1k2 m
gma µ=
Chapter 5
304
Substitute numerical values and evaluate a2:
( )( )( )
2
2
2
m/s40.1
kg100m/s9.81kg600.238
=
=a
*52 •• Picture the Problem The accelerations of the truck can be found by applying Newton’s 2nd law of motion. The free-body diagram for the truck climbing the incline with maximum acceleration is shown to the right.
(a) Apply ∑ = aF rr
m to the truck
when it is climbing the incline:
ΣFx = fs,max – mgsin12° = ma (1) and ΣFy = Fn – mgcos12° = 0 (2)
Solve equation (2) for Fn and use the definition of fs,max to obtain:
fs,max = µsmgcos12° (3)
Substitute equation (3) into equation (1) and solve for a:
( )°−°= 12sin12cossµga
Substitute numerical values and evaluate a:
( ) ( )[ ]2
2
m/s12.6
12sin12cos85.0m/s81.9
=
°−°=a
(b) When the truck is descending the incline with maximum acceleration, the static friction force points down the incline; i.e., its direction is reversed on the FBD. Apply
∑ = xx maF to the truck under
these conditions:
– fs,max – mgsin12° = ma (4)
Substitute equation (3) into equation (4) and solve for a:
( )°+°−= 12sin12cossµga
Substitute numerical values and evaluate a:
( ) ( )[ ]2
2
m/s2.10
12sin12cos85.0m/s81.9
−=
°+°−=a
Applications of Newton’s Laws
305
53 •• Picture the Problem The forces acting on each of the blocks are shown in the free-body diagrams to the right. m1 represents the mass of the 2-kg block and m2 that of the 4-kg block. As described by Newton’s 3rd law, the normal reaction force Fn,1 and the friction force fs,1 (= fs,2) act on both objects but in opposite directions. Newton’s 2nd law and the definition of the maximum static friction force can be used to determine the maximum force acting on the 4-kg block for which the 2-kg block does not slide.
(a) Apply ∑ = aF rr
m to the 2-kg
block:
ΣFx = fs,1,max = m1amax (1) and ΣFy = Fn,1 – m1g = 0 (2)
Apply ∑ = aF rrm to the 4-kg block: ΣFx = F – fs,2,max = m2amax (3)
and ΣFy = Fn,2 – Fn,1 - m2g = 0 (4)
Using equation (2), express the relationship between the static friction forces max,1,sf
rand :max,2,sf
r
fs,1,max = fs,2,max = µs m1g (5)
Substitute (5) in (1) and solve for amax:
amax = µsg = (0.3)g = 2.94 m/s2
Solve equation (3) for F = Fmax: gmamF 1smax2max µ+=
Substitute numerical values and evaluate Fmax:
( )( ) ( )( )( )
N7.17
m/s9.81
kg20.3m/s2.94kg42
2max
=
×
+=F
(b) Use Newton’s 2nd law to express the acceleration of the blocks moving as a unit:
21 mmFa+
=
Substitute numerical values and evaluate a:
( ) 221
m/s47.1kg4kg2N7.17
=+
=a
Chapter 5
306
Because the friction forces are an action-reaction pair, the friction force acting on each block is given by:
fs = m1a = (2 kg)(1.47 m/s2) = N94.2
(c) If F = 2Fmax, then m1 slips on m2 and the friction force (now kinetic) is given by:
f = fk = µkm1g
Use∑ = xx maF to relate the
acceleration of the 2-kg block to the net force acting on it and solve for a1:
fk = µkm1g = m1a1 and a1 = µkg = (0.2)g = 2m/s1.96
Use∑ = xx maF to relate the
acceleration of the 4-kg block to the net force acting on it:
F − µkm1g = m2a2
Solve for a2:
2
1k2 m
gmFa µ−=
Substitute numerical values and evaluate a2:
( ) ( )( )( )
2
2
2
m/s87.7
kg4m/s9.81kg20.2N17.72
=
−=a
54 •• Picture the Problem Let the positive x direction be the direction of motion of these blocks. The forces acting on each of the blocks are shown, for the static friction case, on the free-body diagrams to the right. As described by Newton’s 3rd law, the normal reaction force Fn,1 and the friction force fs,1 (= fs,2) act on both objects but in opposite directions. Newton’s 2nd law and the definition of the maximum static friction force can be used to determine the maximum acceleration of the block whose mass is m1.
(a) Apply ∑ = aF rr
m to the 2-kg
block:
ΣFx = fs,1,max = m1amax (1) and
Applications of Newton’s Laws
307
ΣFy = Fn,1 – m1g = 0 (2)
Apply ∑ = aF rrm to the 4-kg
block:
ΣFx = T – fs,2,max = m2amax (3) and ΣFy = Fn,2 – Fn,1 – m2g = 0 (4)
Using equation (2), express the relationship between the static friction forces max,1,sf
rand max,2,sf
r:
fs,1,max = fs,2,max = µs m1g (5)
Substitute (5) in (1) and solve for amax:
amax = µsg = (0.6)g = 2m/s89.5
(b) Use ∑ = xx maF to express the
acceleration of the blocks moving as a unit:
T = (m1 + m2) amax (6)
Apply ∑ = xx maF to the object
whose mass is m3:
m3g – T = m3 amax (7)
Add equations (6) and (7) to eliminate T and then solve for and evaluate m3:
( ) ( )( )
kg5.22
6.01kg5kg106.0
1 s
21s3
=
−+
=−
+=
µµ mmm
(c) If m3 = 30 kg, then m1 will slide on m2 and the friction force (now kinetic) is given by:
f = fk = µkm1g
Use ∑ = xx maF to relate the
acceleration of the 30-kg block to the net force acting on it:
m3g – T = m3a3 (8)
Noting that a2 = a3 and that the friction force on the body whose mass is m2 is due to kinetic friction, add equations (3) and (8) and solve for and evaluate the common acceleration:
( )
( ) ( )( )[ ]
2
232
1k332
m/s87.6
kg30kg10kg50.4kg30m/s9.81
=
+−
=
+−
==mm
mmgaa µ
With block 1 sliding on block 2, the fk = µkm1g = m1a1 (1′)
Chapter 5
308
friction force acting on each is kinetic and equations (1) and (3) become:
T – fk = T – µkm1g = m2a2 (3′)
Solve equation (1′) for and evaluate a1:
( )( )2
2k1
m/s92.3
m/s81.94.0
=
== ga µ
Solve equation (3′) for T:
gmamT 1k22 µ+=
Substitute numerical values and evaluate T:
( )( ) ( )( )( ) N3.88m/s9.81kg50.4m/s6.87kg10 22 =+=T
55 • Picture the Problem Let the direction of motion be the positive x direction. The free-body diagrams show the forces acting on both the block (M) and the counterweight (m). While ,21 TT
rr≠ T1 = T2.
By applying Newton’s 2nd law to these blocks, we can obtain equations in T and a from which we can eliminate the tension. Once we know the acceleration of the block, we can use constant-acceleration equations to determine how far it moves in coming to a momentary stop.
(a) Apply ∑ = aF rr
m to the block on the incline:
∑
∑
=−=
=−−=
0cosand
sin
n
k1
θ
θ
MgFF
MafMgTF
y
x
Apply ∑ = aF rr
m to the counterweight:
maTmgFx =−=∑ 2 (1)
Letting T1 = T2 = T and using the definition of the kinetic friction force, eliminate fk and Fn between the equations for the block on the incline to obtain:
MaMgMgT =−− θµθ cossin k (2)
Eliminate T from equations (1) and (2) by adding them and solve for a:
( ) gMm
Mma k
++−
=θµθ cossin
Applications of Newton’s Laws
309
Substitute numerical values and evaluate a:
( ) ( )( ) 22 m/s163.0m/s81.9kg1600kg550
10cos15.010sinkg1600kg550=
+°+°−
=a
(b) Using a constant-acceleration equation, relate the speed of the block at the instant the rope breaks to its acceleration and displacement as it slides to a stop. Solve for its displacement:
xavv ∆+= 22i
2f
or, because vf = 0,
avx
2
2i−
=∆ (3)
The block had been accelerating up the incline for 3 s before the rope broke, so it has an initial speed of :
(0.163 m/s2)(3 s) = 0.489 m/s
From equation (2) we can see that, when the rope breaks (T = 0) and:
( )( ) ( )[ ]
2
2
m/s15.310cos15.010sinm/s81.9
cossin
−=
°+°−=
+−= θµθ kga
where the minus sign indicates that the block is being accelerated down the incline, although it is still sliding up the incline.
Substitute in equation (3) and evaluate ∆x:
( )( ) m0380.0
m/s15.32m/s489.0
2
2
=−
−=∆x
(c) When the block is sliding down the incline, the kinetic friction force will be up the incline. Express the block’s acceleration:
( )( ) ( )[ ]
2
2
m/s254.0
10cos15.010sinm/s81.9
cossin
−=
°−°−=
−−= θµθ kga
56 ••• Picture the Problem If the 10-kg block is not to slide on the bracket, the maximum value for F
rmust be equal to the maximum
value of fs and will produce the maximum acceleration of this block and the bracket. We can apply Newton’s 2nd law and the definition of fs,max to first calculate the maximum acceleration and then the maximum value of F. (a) and (b) Apply ∑ = aF rr
m to the
10-kg block when it is experiencing its maximum acceleration:
ΣFx = fs,max – F = m2a2,max (1) and ΣFy = Fn,2 – m2g = 0 (2)
Chapter 5
310
Express the static friction force acting on the 10-kg block:
fs,max = µsFn,2 (3)
Eliminate fs,max and Fn,2 from equations (1), (2) and (3) to obtain:
µsm2g – F = m2a2,max (4)
Apply ∑ = xx maF to the bracket
to obtain:
2F – µsm2g = m1a1,max (5)
Because a1,max = a2,max, denote this acceleration by amax. Eliminate F from equations (4) and (5) and solve for amax:
21
2smax 2mm
gma+
=µ
Substitute numerical values and evaluate amax:
( )( )( )( )
2
2
max
m/s57.1
kg102kg5m/s9.81kg100.4
=
+=a
Solve equation (4) for F = Fmax: ( )maxs2max22s agmamgmF −=−= µµ
Substitute numerical values and evaluate F:
( ) ( )( )[ ]N5.23
m/s1.57m/s9.810.4kg10 22
=
−=F
*57 •• Picture the Problem The free-body diagram shows the forces acting on the block as it is moving up the incline. By applying Newton’s 2nd law, we can obtain expressions for the accelerations of the block up and down the incline. Adding and subtracting these equations, together with the data found in the notebook, will lead to values for gV and µk.
Apply aF rr
mi i =∑ to the block when
it is moving up the incline:
0cosand
sin
Vn
upVk
=−=
=−−=
∑
∑
θ
θ
mgFF
mamgfF
y
x
Using the definition of fk, eliminate Fn between the two equations to obtain:
θθµ sincos VVkup gga −−= (1)
Applications of Newton’s Laws
311
When the block is moving down the incline, fk is in the positive x direction, and its acceleration is:
θθµ sincos VVkdown gga −= (2)
Add equations (1) and (2) to obtain: θsin2 Vdownup gaa −=+ (3)
Solve equation (3) for gV:
θsin2downup
V −
+=
aag (4)
Determine θ from the figure:
°=⎥⎦
⎤⎢⎣
⎡= − 8.10
glapp3.82glapp0.73tan 1θ
Substitute the data from the notebook in equation (4) to obtain:
222
V pglapp/plip41.88.10sin2
pglapp/plip1.42pglapp/plip73.1−=
°−+
=g
Subtract equation (1) from equation (2) to obtain:
θµ cos2 Vkupdown gaa =−
Solve for µk:
θµ
cos2 V
updownk g
aa −=
Substitute numerical values and evaluate µk:
( ) 191.08.10cospglapp/plip41.82
pglapp/plip1.73pglapp/plip1.422
22
k =°−
−−=µ
*58 •• Picture the Problem The free-body diagram shows the block sliding down the incline under the influence of a friction force, its weight, and the normal force exerted on it by the inclined surface. We can find the range of values for m for the two situations described in the problem statement by applying Newton’s 2nd law of motion to, first, the conditions under which the block will not move or slide if pushed, and secondly, if pushed, the block will move up the incline. (a) Assume that the block is sliding down the incline with a constant velocity and with no hanging weight (m = 0) and apply aF rr
m=∑ to ∑
∑
=−=
=+−=
0cosand
0sin
n
k
θ
θ
MgFF
MgfF
y
x
Chapter 5
312
the block: Using fk = µkFn, eliminate Fn between the two equations and solve for the net force acting on the block:
θθµ sincosknet MgMgF +−=
If the block is moving, this net force must be nonnegative and:
( ) 0sincosk ≥+− Mgθθµ
This condition requires that: 325.018tantank =°=≤ θµ
Because µk = 0.2, this condition is satisfied and:
0min =m
To find the maximum value, note that the maximum possible value for the tension in the rope is mg. For the block to move down the incline, the component of the block’s weight parallel to the incline minus the frictional force must be greater than or equal to the tension in the rope:
Mgsinθ – µkMgcosθ ≥ mg
Solve for mmax: ( )θµθ cossin kmax −≤ Mm
Substitute numerical values and evaluate mmax:
( ) ( )[ ]kg9.11
18cos2.018sinkg100max
=°−°≤m
The range of values for m is: kg9.110 ≤≤ m
(b) If the block is being dragged up the incline, the frictional force will point down the incline, and:
Mg sinθ + µkMg cosθ < mg
Solve for and evaluate mmin: mmin > M (sinθ + µk cosθ) = (100 kg)[sin18° + (0.2)cos18°] = kg9.49
If the block is not to move unless pushed:
Mg sinθ + µs Mg cosθ > mg
Solve for and evaluate mmax: mmax < M (sinθ + µs cosθ) = (100 kg)[sin18° + (0.4)cos18°] = kg9.68
The range of values for m is: kg9.68kg9.49 ≤≤ m
Applications of Newton’s Laws
313
59 ••• Picture the Problem The free-body diagram shows the forces acting on the 0.5 kg block when the acceleration is a minimum. Note the choice of coordinate system is consistent with the direction of Fr
. Apply Newton’s 2nd law to the block and solve the resulting equations for amin and amax. (a) Apply ∑ = aF rr
m to the 0.5-kg
block:
ΣFx = Fnsinθ – fscosθ = ma (1) and ΣFy = Fncosθ + fssinθ – mg = 0 (2)
Under minimum acceleration, fs = fs,max. Express the relationship between fs,max and Fn:
fs,max = µsFn (3)
Substitute fs,max for fs in equation (2) and solve for Fn: θµθ sincos s
n +=
mgF
Substitute for Fn in equation (1) and solve for a = amin: θµθ
θµθsincoscossin
s
smin +
−= ga
Substitute numerical values and evaluate amin:
( ) ( )( )
2
2min
m/s627.0sin350.8cos35cos350.8sin35m/s9.81
−=
°+°°−°
=a
Treat the block and incline as a single object to determine Fmin:
Fmin = mtotamin = (2.5 kg)( –0.627 m/s2) = N57.1−
To find the maximum acceleration, reverse the direction of sf
rand apply
∑ = aF rrm to the block:
ΣFx = Fnsinθ + fscosθ = ma (4) and ΣFy = Fncosθ – fssinθ – mg = 0 (5)
Proceed as above to obtain: θµθθµθ
sincoscossin
s
smax −
+= ga
Substitute numerical values and evaluate amax:
( ) ( )( )
2
2max
m/s5.33sin350.8cos35cos350.8sin35m/s9.81
=
°−°°+°
=a
Chapter 5
314
Treat the block and incline as a single object to determine Fmax:
Fmax = mtotamax = (2.5 kg)(33.5 m/s2) = N8.83
(b) Repeat (a) with µs = 0.4 to obtain: N5.37 andN75.5 maxmin == FF
60 • Picture the Problem The kinetic friction force fk is the product of the coefficient of sliding friction µk and the normal force Fn the surface exerts on the sliding object. By applying Newton’s 2nd law in the vertical direction, we can see that, on a horizontal surface, the normal force is the weight of the sliding object. Note that the acceleration of the block is opposite its direction of motion. (a) Relate the force of kinetic friction to µk and the normal force acting on the sliding wooden object:
( ) mgv
Ff 224nkk103.2111.0
−×+== µ
Substitute v = 10 m/s and evaluate fk:
( )( )( )( ) N103
m/s10103.21
m/s9.81kg10011.0224
2
k =×+
=−
f
(b) Substitute v = 20 m/s and evaluate fk:
( )( )( )( )
N5.90
m/s20103.21
m/s9.81kg10011.0224
2
k
=
×+=
−f
61 •• Picture the Problem The pictorial representation shows the block sliding from left to right and coming to rest when it has traveled a distance ∆x. Note that the direction of the motion is opposite that of the block’s acceleration. The acceleration and stopping distance of the blocks can be found from constant-acceleration equations. Let the direction of motion of the sliding blocks be the positive x direction. Because the surface is horizontal, the normal force acting on the sliding block is the block’s weight.
Applications of Newton’s Laws
315
(a) Using a constant-acceleration equation, relate the block’s stopping distance to its initial speed and acceleration; solve for the stopping distance:
xavv ∆+= 220
2 or, because v = 0,
avx
2
20−
=∆ (1)
Apply ∑ = xx maF to the sliding block, introduce Konecny’s empirical expression, and solve for the block’s acceleration:
( )mmg
mF
mf
mF
a x
91.0
0.91nknet,
4.0
4.0
−=
−=−
==
Evaluate a with m = 10 kg: ( ) ( )( )[ ]
2
0.912
m/s60.2
kg10m/s9.81kg100.4
−=
−=a
Substitute in equation (1) and evaluate the stopping distance when v0 = 10 m/s:
( )( ) m2.19
m/s2.602m/s10
2
2
=−−
=∆x
(b) Proceed as in (a) with m = 100 kg to obtain:
( ) ( )( )[ ]
2
0.912
m/s11.2
kg100m/s9.81kg1000.4
−=
−=a
Find the stopping distance as in (a): ( )
( ) m7.23m/s2.112
m/s102
2
=−−
=∆x
*62 ••• Picture the Problem The kinetic friction force fk is the product of the coefficient of sliding friction µk and the normal force Fn the surface exerts on the sliding object. By applying Newton’s 2nd law in the vertical direction, we can see that, on a horizontal surface, the normal force is the weight of the sliding object. We can apply Newton’s 2nd law in the horizontal (x) direction to relate the block’s acceleration to the net force acting on it. In the spreadsheet program, we’ll find the acceleration of the block from this net force (which is velocity dependent), calculate the increase in the block’s speed from its acceleration and the elapsed time and add this increase to its speed at end of the previous time interval, determine how far it has moved in this time interval, and add this distance to its previous position to find its current position. We’ll also calculate the position of the block x2, under the assumption that µk = 0.11, using a constant-acceleration equation.
Chapter 5
316
The spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows:
Cell Formula/Content Algebraic Form C9 C8+$B$6 tt ∆+ D9 D8+F9*$B$6 tav ∆+ E9 $B$5−($B$3)*($B$2)*$B$5/
(1+$B$4*D9^2)^2 ( )224k
1034.21 vmgF
−×+−
µ
F9 E10/$B$5 mF /net G9 G9+D10*$B$6 tvx ∆+ K9 0.5*5.922*I10^2 2
21 at
L9 J10-K10 2xx −
A B C D E F G H I J 1 g= 9.81 m/s^2 2 Coeff1= 0.11 3 Coeff2= 2.30E-
04
4 Mass= 10 kg 5 Applied
Force= 70 N
6 Time step=
0.05 s t x x2 x−x2
7 8 9
t v Net
force a x mu=variable mu=constant 10 0.00 0.00 0.00 0.00 0.00 0.00 0.0011 0.05 0.30 59.22 5.92 0.01 0.05 0.01 0.01 0.0112 0.10 0.59 59.22 5.92 0.04 0.10 0.04 0.03 0.0113 0.15 0.89 59.22 5.92 0.09 0.15 0.09 0.07 0.0214 0.20 1.18 59.22 5.92 0.15 0.20 0.15 0.12 0.0315 0.25 1.48 59.23 5.92 0.22 0.25 0.22 0.19 0.04
205 9.75 61.06 66.84 6.68 292.37 9.75 292.37 281.48 10.89206 9.80 61.40 66.88 6.69 295.44 9.80 295.44 284.37 11.07
Applications of Newton’s Laws
317
207 9.85 61.73 66.91 6.69 298.53 9.85 298.53 287.28 11.25208 9.90 62.07 66.94 6.69 301.63 9.90 301.63 290.21 11.42209 9.95 62.40 66.97 6.70 304.75 9.95 304.75 293.15 11.61210 10.00 62.74 67.00 6.70 307.89 10.00 307.89 296.10 11.79 The displacement of the block as a function of time, for a constant coefficient of friction (µk = 0.11) is shown as a solid line on the graph and for a variable coefficient of friction, is shown as a dotted line. Because the coefficient of friction decreases with increasing particle speed, the particle travels slightly farther when the coefficient of friction is variable.
0
50
100
150
200
250
300
0 2 4 6 8 10
t (s)
x (m
)
mu = variablemu = constant
The velocity of the block, with variable coefficient of kinetic friction, is shown below.
0
10
20
30
40
50
60
70
0 2 4 6 8 10
t (s)
v (m
/s)
Chapter 5
318
63 •• Picture the Problem The free-body diagram shows the forces acting on the block as it moves to the right. The kinetic friction force will slow the block and, eventually, bring it to rest. We can relate the coefficient of kinetic friction to the stopping time and distance by applying Newton’s 2nd law and then using constant-acceleration equations. (a) Apply ∑ = aF rr
m to the block of wood:
0and
n
k
=−=
=−=
∑
∑
mgFF
mafF
y
x
Using the definition of fk, eliminate Fn between the two equations to obtain:
ga kµ−= (1)
Use a constant-acceleration equation to relate the acceleration of the block to its displacement and its stopping time:
( )221
0 tatvx ∆+∆=∆ (2)
Relate the initial speed of the block, v0, to its displacement and stopping distance:
0. since 2
021
0av
=∆=
∆+
=∆=∆
vtv
tvvtvx (3)
Use this result to eliminate v0 in equation (2):
( )221 tax ∆−=∆ (4)
Substitute equation (1) in equation (4) and solve for µk: ( )2
2tgx
k ∆∆
=µ
Substitute for ∆x = 1.37 m and ∆t = 0.97 s to obtain:
( )( )( )
297.0s0.97m/s9.81
m1.37222 ==kµ
(b) Use equation (3) to find v0: ( ) m/s82.2
s0.97m1.3722
0 ==∆∆
=txv
Applications of Newton’s Laws
319
*64 •• Picture the Problem The free-body diagram shows the forces acting on the block as it slides down an incline. We can apply Newton’s 2nd law to these forces to obtain the acceleration of the block and then manipulate this expression algebraically to show that a graph of a/cosθ versus tanθ will be linear with a slope equal to the acceleration due to gravity and an intercept whose absolute value is the coefficient of kinetic friction.
(a) Apply ∑ = aF rr
m to the block as it slides down the incline:
∑
∑
=−=
=−=
0cosand
sin
n
k
θ
θ
mgFF
mafmgF
y
x
Substitute µkFn for fk and eliminate Fn between the two equations to obtain:
( )θµθ cossin k−= ga
Divide both sides of this equation by cosθ to obtain: ktan
cosµθ
θgga
−=
Note that this equation is of the form y = mx + b:
Thus, if we graph a/cosθ versus tanθ, we should get a straight line with slope g and y-intercept −gµk.
(b) A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows:
Cell Formula/Content Algebraic FormC7 θ D7 a E7 TAN(C7*PI()/180)
⎟⎠⎞
⎜⎝⎛ ×
180tan πθ
F7 D7/COS(C7*PI()/180)
⎟⎠⎞
⎜⎝⎛ ×
180cos πθ
a
C D E F
6 theta a tan(theta) a/cos(theta)7 25 1.691 0.466 1.866 8 27 2.104 0.510 2.362 9 29 2.406 0.554 2.751
Chapter 5
320
10 31 2.888 0.601 3.370 11 33 3.175 0.649 3.786 12 35 3.489 0.700 4.259 13 37 3.781 0.754 4.735 14 39 4.149 0.810 5.338 15 41 4.326 0.869 5.732 16 43 4.718 0.933 6.451 17 45 5.106 1.000 7.220
A graph of a/cosθ versus tanθ is shown below. From the curve fit (Excel’s Trendline
was used), g = 9.77 m/s2 and .268.0m/s9.77m/s2.62
2
2
k ==µ
The percentage error in g from the commonly accepted value of 9.81 m/s2 is
%408.0m/s81.9
m/s77.9m/s81.9100 2
22
=⎟⎟⎠
⎞⎜⎜⎝
⎛ −
y = 9.7681x - 2.6154R2 = 0.9981
0
1
2
3
4
5
6
7
8
0.4 0.5 0.6 0.7 0.8 0.9 1.0
tan(theta)
a/c
os(th
eta)
Applications of Newton’s Laws
321
Motion Along a Curved Path 65 • Picture the Problem The free-body diagram showing the forces acting on the stone is superimposed on a sketch of the stone rotating in a horizontal circle. The only forces acting on the stone are the tension in the string and the gravitational force. The centripetal force required to maintain the circular motion is a component of the tension. We’ll solve the problem for the general case in which the angle with the horizontal is θ by applying Newton’s 2nd law of motion to the forces acting on the stone. Apply ∑ = aF rr
m to the stone: ΣFx = Tcosθ = mac = mv2/r (1) and ΣFy= Tsinθ – mg = 0 (2)
Use the right triangle in the diagram to relate r, L, and θ :
r = Lcosθ (3)
Eliminate T and r between equations (1), (2) and (3) and solve for v2:
θθ coscot2 gLv = (4)
Express the velocity of the stone in terms of its period:
rev1
2t
rv π= (5)
Eliminate v between equations (4) and (5) and solve for θ : ⎟
⎟⎠
⎞⎜⎜⎝
⎛= −
Lgt
2
2rev11
4sin
πθ
Substitute numerical values and evaluate θ :
( )( )( ) °=⎥
⎦
⎤⎢⎣
⎡= − 8.25
m0.854πs1.22m/s9.81sin 2
221θ
and correct. is )(c
Chapter 5
322
66 • Picture the Problem The free-body diagram showing the forces acting on the stone is superimposed on a sketch of the stone rotating in a horizontal circle. The only forces acting on the stone are the tension in the string and the gravitational force. The centripetal force required to maintain the circular motion is a component of the tension. We’ll solve the problem for the general case in which the angle with the horizontal is θ by applying Newton’s 2nd law of motion to the forces acting on the stone. Apply ∑ = aF rr
m to the stone: ΣFx = Tcosθ = mac = mv2/r (1) and ΣFy= Tsinθ – mg = 0 (2)
Use the right triangle in the diagram to relate r, L, and θ:
r = Lcosθ (3)
Eliminate T and r between equations (1), (2), and (3) and solve for v:
θθ coscotgLv =
Substitute numerical values and evaluate v:
( )( )m/s50.4
20cos20cotm0.8m/s9.81 2
=
°°=v
67 • Picture the Problem The free-body diagram showing the forces acting on the stone is superimposed on a sketch of the stone rotating in a horizontal circle. The only forces acting on the stone are the tension in the string and the gravitational force. The centripetal force required to maintain the circular motion is a component of the tension. We’ll solve the problem for the general case in which the angle with the vertical is θ by applying Newton’s 2nd law of motion to the forces acting on the stone. (a) Apply ∑ = aF rr
m to the stone: ΣFx = Tsinθ = mac = mv2/r (1) and
Applications of Newton’s Laws
323
ΣFy= Tcosθ – mg = 0 (2)
Eliminate T between equations (1) and (2) and solve for v:
θtanrgv =
Substitute numerical values and evaluate v:
( )( )m/s1.41
tan30m/s9.81m0.35 2
=
°=v
(b) Solve equation (2) for T:
θcosmgT =
Substitute numerical values and evaluate T:
( )( ) N50.8cos30
m/s9.81kg0.75 2
=°
=T
*68 •• Picture the Problem The sketch shows the forces acting on the pilot when her plane is at the lowest point of its dive. nF
ris the
force the airplane seat exerts on her. We’ll apply Newton’s 2nd law for circular motion to determine Fn and the radius of the circular path followed by the airplane.
(a) Apply yy maF =∑ to the pilot:
Fn − mg = mac
Solve for and evaluate Fn: Fn = mg + mac = m(g + ac) = m(g + 8.5g) = 9.5mg = (9.5) (50 kg) (9.81 m/s2) = kN66.4
(b) Relate her acceleration to her velocity and the radius of the circular arc and solve for the radius:
rvac
2
= ⇒ c
2
avr =
Substitute numerical values and evaluate r :
( )( )( )[ ]( ) m110
m/s9.818.5m/km1000sh/36001km/h345
2
2
==r
Chapter 5
324
69 •• Picture the Problem The diagram shows the forces acting on the pilot when her plane is at the lowest point of its dive.
nFr
is the force the airplane seat exerts on her. We’ll use the definitions of centripetal acceleration and centripetal force and apply Newton’s 2nd law to calculate these quantities and the normal force acting on her.
(a) Her acceleration is centripetal and given by:
upward ,2
c rva =
Substitute numerical values and evaluate ac:
( )( )( )[ ]
upward,m/s33.8
m300/km10sh/36001km/h180
2
23
c
=
=a
(b) The net force acting on her at the bottom of the circle is the force responsible for her centripetal acceleration:
( )( )upward N,541
m/s33.8kg65 2cnet
=
== maF
(c) Apply ∑ = yy maF to the pilot:
Fn – mg = mac
Solve for Fn: Fn = mg + mac = m(g + ac)
Substitute numerical values and evaluate Fn:
Fn = (65 kg)(9.81 m/s2 + 8.33 m/s2) = upwardkN,18.1
70 •• Picture the Problem The free-body diagrams for the two objects are shown to the right. The hole in the table changes the direction the tension in the string (which provides the centripetal force required to keep the object moving in a circular path) acts. The application of Newton’s 2nd law and the definition of centripetal force will lead us to an expression for r as a function of m1, m2, and the time T for one revolution.
Applications of Newton’s Laws
325
Apply ∑ = xx maF to both objects
and use the definition of centripetal acceleration to obtain:
m2g – F2 = 0 and F1 = m1ac = m1v2/r
Because F1 = F2 we can eliminate both of them between these equations to obtain:
02
12 =−rvmgm
Express the speed v of the object in terms of the distance it travels each revolution and the time T for one revolution:
Trv π2
=
Substitute to obtain: 04
2
22
12 =−rT
rmgm π
or
042
2
12 =−T
rmgm π
Solve for r:
12
22
4 mgTmr
π=
*71 •• Picture the Problem The free-body diagrams show the forces acting on each block. We can use Newton’s 2nd law to relate these forces to each other and to the masses and accelerations of the blocks.
Apply ∑ = xx maF to the block
whose mass is m1: 1
21
121 LvmTT =−
Apply ∑ = xx maF to the block
whose mass is m2: 21
22
22 LLvmT+
=
Relate the speeds of each block to their common period and their distance from the center of the
( )T
LLvTLv 21
21
12and2 +
==ππ
Chapter 5
326
circle: Solve the first force equation for T2, substitute for v2, and simplify to obtain:
( )[ ]2
21222
⎟⎠⎞
⎜⎝⎛+=
TLLmT π
Substitute for T2 and v1 in the first force equation to obtain: ( )[ ]
2
1121212
⎟⎠⎞
⎜⎝⎛++=
TLmLLmT π
*72 •• Picture the Problem The path of the particle and its position at 1-s intervals are shown. The displacement vectors are also shown. The velocity vectors for the average velocities in the first and second intervals are along 01rr and ,12r
rrespectively,
and are shown in the lower diagram. ∆ vr points toward the center of the circle.
Use the diagram to the right to find ∆r:
∆r = 2rsin22.5°= 2(4 cm) sin22.5° = 3.06 cm
Find the average velocity of the particle along the chords:
vav = ∆r/∆t = (3.06 cm)/(1 s) = 3.06 cm/s
Using the lower diagram and the fact that the angle between
21 and vvrr
is 45°, express ∆v in
terms of v1 (= v2):
∆v = 2v1sin22.5°
Evaluate ∆v using vav as v1: ∆v = 2(3.06 cm/s)sin22.5° = 2.34 cm/s
Now we can determine a = ∆v/∆t: 2cm/s34.2s1cm/s2.34
==a
Applications of Newton’s Laws
327
Find the speed v (= v1 = v2 …) of the particle along its circular path:
( ) cm/s14.3s8cm42π2
===T
rv π
Calculate the radial acceleration of the particle:
( ) 222
c cm/s46.2cm4cm/s3.14
===rva
Compare ac and a by taking their ratio:
05.1cm/s34.2cm/s46.2
2
2c ==
aa
or aa 05.1c =
73 •• Picture the Problem The diagram to the right has the free-body diagram for the child superimposed on a pictorial representation of her motion. The force her father exerts is F
rand the angle it makes
with respect to the direction we’ve chosen as the positive y direction is θ. We can infer her speed from the given information concerning the radius of her path and the period of her motion. Applying Newton’s 2nd law as it describes circular motion will allow us to find both the direction and magnitude of F
r.
Apply ∑ = aF rr
m to the child: ΣFx = Fsinθ = mv2/r and ΣFy = Fcosθ − mg = 0
Eliminate F between these equations and solve for θ : ⎥
⎦
⎤⎢⎣
⎡= −
rgv2
1tanθ
Express v in terms of the radius and period of the child’s motion: T
rv π2=
Substitute for v in the expression for θ to obtain: ⎥
⎦
⎤⎢⎣
⎡= −
2
21 4tan
gTrπθ
Chapter 5
328
Substitute numerical values and evaluate θ :
( )( )( )
°=⎥⎦
⎤⎢⎣
⎡= − 3.53
s1.5m/s9.81m0.754tan 22
21 πθ
Solve the y equation for F:
θcosmgF =
Substitute numerical values and evaluate F:
( )( ) N410cos53.3
m/s9.81kg25 2
=°
=F
74 •• Picture the Problem The diagram to the right has the free-body diagram for the bob of the conical pendulum superimposed on a pictorial representation of its motion. The tension in the string is F
rand the angle it
makes with respect to the direction we’ve chosen as the positive x direction isθ. We can findθ from the y equation and the information provided about the tension. Then, by using the definition of the speed of the bob in its orbit and applying Newton’s 2nd law as it describes circular motion, we can find the period T of the motion.
Apply ∑ = aF rr
m to the pendulum
bob:
ΣFx = Fcosθ = mv2/r and ΣFy = Fsinθ − mg = 0
Using the given information that F = 6mg, solve the y equation for θ:
°=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎠⎞
⎜⎝⎛= −− 59.9
6sinsin 11
mgmg
Fmgθ
With F = 6mg, solve the x equation for v:
θcos6rgv =
Relate the period T of the motion to the speed of the bob and the radius of the circle in which it moves:
θππcos6
22rg
rv
rT ==
From the diagram, one can see that:
r =Lcosθ
Applications of Newton’s Laws
329
Substitute for r in the expression for the period to obtain: g
LT6
2π=
Substitute numerical values and evaluate T: ( ) s579.0
m/s9.816m0.52 2 == πT
75 •• Picture the Problem The static friction force fs is responsible for keeping the coin from sliding on the turntable. Using Newton’s 2nd law of motion, the definition of the period of the coin’s motion, and the definition of the maximum static friction force, we can find the magnitude of the friction force and the value of the coefficient of static friction for the two surfaces.
(a) Apply ∑ = aF rr
m to the coin: ∑ ==rvmfF sx
2
and
∑ =−= 0n mgFFy
If T is the period of the coin’s motion, its speed is given by: T
rv π2=
Substitute for v in the force equation and simplify to obtain: 2
24T
mrfsπ
=
Substitute numerical values and evaluate fs:
( )( )( )
N395.0s1
m0.1kg(0.14π2
2
==sf
(b) Determine Fn from the y equation:
Fn = mg
If the coin is about to slide at r = 16 cm, fs = fs,max. Solve for µs in terms of fs,max and Fn:
2
22
2
n
max,s
44
gTr
mgT
mr
Ffs π
π
µ ===
Chapter 5
330
Substitute numerical values and evaluate µs:
( )( )( )
644.0s1m/s9.81
m0.164π22
2
s ==µ
76 •• Picture the Problem The forces acting on the tetherball are shown superimposed on a pictorial representation of the motion. The horizontal component of T
ris the
centripetal force. Applying Newton’s 2nd law of motion and solving the resulting equations will yield both the tension in the cord and the speed of the ball. (a) Apply ∑ = aF rr
m to the tetherball: ∑ =°=rvmTFx
2
20sin
and
∑ =−°= 020cos mgTFy
Solve the y equation for T:
°=
20cosmgT
Substitute numerical values and evaluate T:
( )( ) N61.220cos
m/s9.81kg0.25 2
=°
=T
(b) Eliminate T between the force equations and solve for v:
°= 20tanrgv
Note from the diagram that: r = Lsin20°
Substitute for r in the expression for v to obtain:
°°= 20tan20singLv
Substitute numerical values and evaluate v:
( )( )m/s21.1
20tan20sinm1.2m/s9.81 2
=
°°=v
Applications of Newton’s Laws
331
*77 •• Picture the Problem The diagram includes a pictorial representation of the earth in its orbit about the sun and a force diagram showing the force on an object at the equator that is due to the earth’s rotation, ,RF
r and the force on the object
due to the orbital motion of the earth about the sun, .oF
r Because these are centripetal
forces, we can calculate the accelerations they require from the speeds and radii associated with the two circular motions.
Express the radial acceleration due to the rotation of the earth: R
va2R
R =
Express the speed of the object on the equator in terms of the radius of the earth R and the period of the earth’s rotation TR:
RR
2T
Rv π=
Substitute for vR in the expression for aR to obtain: 2
R
2
R4
TRa π
=
Substitute numerical values and evaluate aR:
( )( )
( )
g
a
3
22
2
2
R
1044.3
m/s1037.3
h1s3600h24
m/km1000km63704
−
−
×=
×=
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛=
π
Express the radial acceleration due to the orbital motion of the earth: r
va2o
o =
Express the speed of the object on the equator in terms of the earth-sun distance r and the period of the earth’s motion about the sun To:
oo
2T
rv π=
Chapter 5
332
Substitute for vo in the expression for ao to obtain: 2
o
2
o4T
ra π=
Substitute numerical values and evaluate ac:
( )( )
g
a
423
2
112
o
1007.6m/s1095.5
h1s3600
d1h24d365
m10.514
−− ×=×=
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
×=
π
78 • Picture the Problem The most significant force acting on the earth is the gravitational force exerted by the sun. More distant or less massive objects exert forces on the earth as well, but we can calculate the net force by considering the radial acceleration of the earth in its orbit. Similarly, we can calculate the net force acting on the moon by considering its radial acceleration in its orbit about the earth. (a) Apply ∑ = rr maF to the earth: r
vmF2
earthon =
Express the orbital speed of the earth in terms of the time it takes to make one trip around the sun (i.e., its period) and its average distance from the sun:
Trv π2
=
Substitute for v to obtain: 2
2
earthon4
TmrF π
=
Substitute numerical values and evaluate Fon earth:
( )( ) N1055.3
hs3600
dh24d24.365
m10496.1kg1098.54 222
11242
earthon ×=
⎟⎠⎞
⎜⎝⎛ ××
××=
πF
(b) Proceed as in (a) to obtain:
( )( ) N1000.2
hs3600
dh24d32.27
m10844.3kg1035.74 202
8222
moonon ×=
⎟⎠⎞
⎜⎝⎛ ××
××=
πF
Applications of Newton’s Laws
333
79 •• Picture the Problem The semicircular wire of radius 10 cm limits the motion of the bead in the same manner as would a 10-cm string attached to the bead and fixed at the center of the semicircle. The horizontal component of the normal force the wire exerts on the bead is the centripetal force. The application of Newton’s 2nd law, the definition of the speed of the bead in its orbit, and the relationship of the frequency of a circular motion to its period will yield the angle at which the bead will remain stationary relative to the rotating wire.
Apply∑ = aF rr
m to the bead: ∑ ==rvmFFx
2
n sinθ
and
∑ =−= 0cosn mgFFy θ
Eliminate Fn from the force equations to obtain: rg
v2
tan =θ
The frequency of the motion is the reciprocal of its period T. Express the speed of the bead as a function of the radius of its path and its period:
Trv π2
=
Using the diagram, relate r to L and θ :
θsinLr =
Substitute for r and v in the expression for tanθ and solve for θ : ⎥
⎦
⎤⎢⎣
⎡= −
LgT
2
21
4cos
πθ
Substitute numerical values and evaluate θ :
( )( )( ) °=⎥
⎦
⎤⎢⎣
⎡= − 6.51
m0.14πs0.5m/s9.81cos 2
221θ
Chapter 5
334
80 ••• Picture the Problem Note that the acceleration of the bead has two components, the radial component perpendicular to ,vr and a tangential component due to friction that is opposite to .vr The application of Newton’s 2nd law will result in a differential equation with separable variables. Its integration will lead to an expression for the speed of the bead as a function of time.
Apply ∑ = aF rr
m to the bead in the
radial and tangential directions: ∑ ==
rvmFFr
2
n
and
∑ ==−=dtdvmmafF tt k
Express fk in terms of µk and Fn:
fk = µkFn
Substitute for Fn and fk in the tangential equation to obtain the differential equation:
2k vrdt
dv µ−=
Separate the variables to obtain: dtrv
dv k2
µ−=
Express the integral of this equation with the limits of integration being from v0 to v on the left-hand side and from 0 to t on the right-hand side:
∫∫ −=tv
v
dt'r
dv'v' 0
k2
0
1 µ
Evaluate these integrals to obtain: t
rvv⎟⎠⎞
⎜⎝⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−− k
0
11 µ
Solve this equation for v:
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞
⎜⎝⎛+
=t
rv
vv0k
0
1
1µ
Applications of Newton’s Laws
335
81 ••• Picture the Problem Note that the acceleration of the bead has two components−the radial component perpendicular to ,vr and a tangential component due to friction that is opposite to .vr The application of Newton’s 2nd law will result in a differential equation with separable variables. Its integration will lead to an expression for the speed of the bead as a function of time.
(a) In Problem 81 it was shown that:
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞
⎜⎝⎛+
=t
rv
vv0k
0
1
1µ
Express the centripetal acceleration of the bead:
2
0
20
2
c
1
1
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞
⎜⎝⎛+
==t
rvr
vrva
kµ
(b) Apply Newton’s 2nd law to the bead: ∑ ==
rvmFFr
2
n
and
∑ ==−=dtdvmmafF tt k
Eliminate Fn and fk to rewrite the radial force equation and solve for at:
ck
2
k arvat µµ −=−=
(c) Express the resultant acceleration in terms of its radial and tangential components:
( )2k
22k
22
1 µ
µ
+=
+−=+=
c
ccct
a
aaaaa
Chapter 5
336
Concepts of Centripetal Force *82 • Picture the Problem The diagram depicts a seat at its highest and lowest points. Let ″t″ denote the top of the loop and ″b″ the bottom of the loop. Applying Newton’s 2nd law to the seat at the top of the loop will establish the value of mv2/r; this can then be used at the bottom of the loop to determine Fn,b. Apply ∑ = rr maF to the seat at the
top of the loop:
mg +Fn,t = 2mg = mar = mv2/r
Apply ∑ = rr maF to the seat at the
bottom of the loop:
Fn,b – mg = mv2/r
Solve for Fn,b and substitute for mv2/r to obtain:
Fn,b = 3mg and correct. is )(d
83 • Picture the Problem The speed of the roller coaster is imbedded in the expression for its radial acceleration. The radial acceleration is determined by the net radial force acting on the passenger. We can use Newton’s 2nd law to relate the net force on the passenger to the speed of the roller coaster. Apply ∑ = radialradial maF to the
passenger:
mg + 0.4mg = mv2/r
Solve for v: grv 4.1=
Substitute numerical values and evaluate v:
( )( )m/s8.12
m12.0m/s9.811.4 2
=
=v
Applications of Newton’s Laws
337
84 • Picture the Problem The force F the passenger exerts on the armrest of the car door is the radial force required to maintain the passenger’s speed around the curve and is related to that speed through Newton’s 2nd law of motion. Apply ∑ = xx maF to the forces
acting on the passenger: rvmF
2
=
Solve this equation for v:
mrFv =
Substitute numerical values and evaluate v:
( )( ) m/s9.15kg70
N220m80==v
and correct. is )(a
*85 ••• Picture the Problem The forces acting on the bicycle are shown in the force diagram. The static friction force is the centripetal force exerted by the surface on the bicycle that allows it to move in a circular path.
sn fFrr
+ makes an angle θ with the vertical
direction. The application of Newton’s 2nd law will allow us to relate this angle to the speed of the bicycle and the coefficient of static friction.
(a) Apply ∑ = aF rr
m to the bicycle: r
mvfFx
2
s ==∑
and 0n =−=∑ mgFFy
Relate Fn and fs to θ :
rgv
mgr
mv
Ff 2
2
n
stan ===θ
Chapter 5
338
Solve for v: θtanrgv =
Substitute numerical values and evaluate v:
( )( )m/s7.25
tan15m/s9.81m20 2
=
°=v
(b) Relate fs to µs and Fn: mgff s2
1maxs,2
1s µ==
Solve for µs and substitute for fs to obtain:
rgv
mgf 2
ss
22==µ
Substitute numerical values and evaluate µs
( )( )( ) 536.0
m/s9.81m20m/s7.252
2
2
s ==µ
86 •• Picture the Problem The diagram shows the forces acting on the plane as it flies in a horizontal circle of radius R. We can apply Newton’s 2nd law to the plane and eliminate the lift force in order to obtain an expression for R as a function of v and θ.
Apply aF rr
m=∑ to the plane: RvmFFx
2
lift sin ==∑ θ
and 0coslift =−=∑ mgFFy θ
Eliminate Flift between these equations to obtain:
Rgv2
tan =θ
Solve for R: θtan
2
gvR =
Substitute numerical values and evaluate R: ( ) km2.16
tan40m/s9.81s3600
h1h
km480
2
2
=°
⎟⎟⎠
⎞⎜⎜⎝
⎛×
=R
Applications of Newton’s Laws
339
87 • Picture the Problem Under the conditions described in the problem statement, the only forces acting on the car are the normal force exerted by the road and the gravitational force exerted by the earth. The horizontal component of the normal force is the centripetal force. The application of Newton’s 2nd law will allow us to express θ in terms of v, r, and g.
Apply ∑ = aF rr
m to the car:
∑
∑
=−=
==
0cosand
sin
n
2
n
mgFF
rvmFF
y
x
θ
θ
Eliminate Fn from the force equations to obtain: rg
v2
tan =θ
Solve for θ :
⎥⎦
⎤⎢⎣
⎡= −
rgv2
1tanθ
Substitute numerical values and evaluate θ:
( )( )( )[ ]( )( ) °=
⎭⎬⎫
⎩⎨⎧
= − 7.21m/s9.81m160
m/km1000s3600h1km/h90tan 2
21θ
*88 •• Picture the Problem Both the normal force and the static friction force contribute to the centripetal force in the situation described in this problem. We can apply Newton’s 2nd law to relate fs and Fn and then solve these equations simultaneously to determine each of these quantities.
Chapter 5
340
(a) Apply ∑ = aF rrm to the car:
∑
∑
=−−=
=+=
0sincosand
cossin
n
2
n
mgfFF
rvmfFF
sy
sx
θθ
θθ
Multiply the x equation by sinθ and the y equation by cosθ to obtain:
θθθθ sinsincossin2
2ns r
vmFf =+
and 0coscossincos s
2n =−− θθθθ mgfF
Add these equations to eliminate fs: θθ sincos
2
n rvmmgF =−
Solve for Fn:
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=
θθ
θθ
sincos
sincos
2
2
n
rvgm
rvmmgF
Substitute numerical values and evaluate Fn:
( ) ( ) ( ) ( ) ( )
kN8.25
sin10m150
sh/36001m/km1000km/h85cos10m/s9.81kg800222
2n
=
⎥⎦
⎤⎢⎣
⎡°+°=F
(b) Solve the y equation for fs:
θθ
sincos mgFf n
s−
=
Substitute numerical values and evaluate fs:
( ) ( )( ) kN59.1sin10
m/s9.81kg800cos10kN8.25 2
=°
−°=sf
(c) Express µs,min in terms of fs and Fn: n
smins, F
f=µ
Substitute numerical values and evaluate µs,min:
193.0kN8.25kN1.59
mins, ==µ
Applications of Newton’s Laws
341
89 •• Picture the Problem Both the normal force and the static friction force contribute to the centripetal force in the situation described in this problem. We can apply Newton’s 2nd law to relate fs and Fn and then solve these equations simultaneously to determine each of these quantities.
(a) Apply ∑ = aF rr
m to the car:
∑∑
=−−=
=+=
0sincos
cossin
n
2
n
mgfFFrvmfFF
sy
sx
θθ
θθ
Multiply the x equation by sinθ and the y equation by cosθ :
θθθθ sinsincossin2
2ns r
vmFf =+
0coscossincos s2
n =−− θθθθ mgfF
Add these equations to eliminate fs: θθ sincos
2
n rvmmgF =−
Solve for Fn:
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=
θθ
θθ
sincos
sincos
2
2
n
rvgm
rvmmgF
Substitute numerical values and evaluate Fn:
( ) ( ) ( ) ( ) ( )
kN832.7
sin10m150
sh/36001m/km1000km/h38cos10m/s9.81kg800222
2n
=
⎥⎦
⎤⎢⎣
⎡°+°=F
(b) Solve the y equation for fs:
θθ
θθ
sincot
sincos
mgF
mgFf
n
ns
−=
−=
Substitute numerical values and evaluate fs:
Chapter 5
342
( ) ( )( ) N777sin10
m/s9.81kg80010cotkN832.72
−=°
−°=sf
The negative sign tells us that fs points upward along the inclined plane rather than as shown in the force diagram. *90 ••• Picture the Problem The free-body diagram to the left is for the car at rest. The static friction force up the incline balances the downward component of the car’s weight and prevents it from sliding. In the free-body diagram to the right, the static friction force points in the opposite direction as the tendency of the moving car is to slide toward the outside of the curve.
Apply ∑ = aF rr
m to the car that is
at rest:
∑ =−+= 0sincos sny mgfFF θθ (1)
and
∑ =−= 0cossin snx θθ fFF (2)
Substitute fs = fs,max = µsFn in equation (2) and solve for and evaluate the maximum allowable value of θ:
( ) ( ) °=== −− 57.408.0tantan 1s
1 µθ
Apply∑ = aF rrm to the car that is
moving with speed v:
∑ =−−= 0sincos sny mgfFF θθ (3)
∑ =+=rv
mfFF2
snx cossin θθ (4)
Substitute fs = µsFn in equations (3) and (4) and simplify to obtain:
( ) mgF =− θµθ sincos sn (5)
( )rvmcF
2
sn sinos =+ θθµ (6)
Substitute numerical values into (5) 0.9904Fn = mg
Applications of Newton’s Laws
343
and (6) to obtain: and
rvmF
2
n1595.0 =
Eliminate Fn and solve for r:
gvr
1610.0
2
=
Substitute numerical values and evaluate r:
( )( )
m176
m/s9.810.1610m/km 1000sh/36001km/h60
2
2
=
××=r
91 ••• Picture the Problem The free-body diagram to the left is for the car rounding the curve at the minimum (not sliding down the incline) speed. The static friction force up the incline balances the downward component of the car’s weight and prevents it from sliding. In the free-body diagram to the right, the static friction force points in the opposite direction as the tendency of the car moving with the maximum safe speed is to slide toward the outside of the curve. Application of Newton’s 2nd law and the simultaneous solution of the force equations will yield vmin and vmax.
Apply ∑ = aF rr
m to a car traveling
around the curve when the coefficient of static friction is zero:
rvmFF
2min
nx sin∑ == θ
and 0cosny∑ =−= mgFF θ
Divide the first of these equations by the second to obtain:
rgv2
tan =θ or ⎟⎟⎠
⎞⎜⎜⎝
⎛= −
rgv2
1tanθ
Substitute numerical values and evaluate the banking angle:
Chapter 5
344
( ) ( ) ( )( )( ) °=⎥
⎦
⎤⎢⎣
⎡= − 8.22
m/s81.9m30sh/36001m/km1000km/h40tan 2
2221θ
Apply∑ = aF rr
m to the car
traveling around the curve at minimum speed:
rvmfFF
2min
snx cossin∑ =−= θθ
and 0sincos sny∑ =−+= mgfFF θθ
Substitute fs = fs,max = µsFn in the force equations and simplify to obtain:
( )r
vmF2min
sn sincos =− θθµ
and ( ) mgF =+ θµθ sincos sn
Evaluate these equations for θ = 22.8° and µs = 0.3:
0.1102Fn= rvm
2min
and 1.038Fn = mg
Eliminate Fn between these two equations and solve for vmin:
rgv 106.0min =
Substitute numerical values and evaluate vmin:
( )( )km/h20.1m/s59.5
m/s9.81m30106.0 2min
==
=v
Apply∑ = aF rr
m to the car
traveling around the curve at maximum speed:
rvmfFF
2max
snx cossin∑ =+= θθ
and 0sincos sny∑ =−−= mgfFF θθ
Substitute fs = fs,max = µsFn in the force equations and simplify to obtain:
( )r
vmF2max
sn sincos =+ θθµ
and ( ) mgF =− θµθ sincos sn
Evaluate these equations for θ = 22.8° and µs = 0.3:
0.6641Fn= rvm
2max
and 0.8056Fn = mg
Applications of Newton’s Laws
345
Eliminate Fn between these two equations and solve for vmax:
rgv 8243.0max =
Substitute numerical values and evaluate vmax:
( )( )( )km/h1.65m/s5.61
m/s81.9m308243.0 2max
==
=v
Drag Forces 92 • Picture the Problem We can apply Newton’s 2nd law to the particle to obtain its equation of motion. Applying terminal speed conditions will yield an expression for b that we can evaluate using the given numerical values. Apply yy maF =∑ to the particle: ymabvmg =−
When the particle reaches its terminal speed v = vt and ay = 0:
0t =− bvmg
Solve for b to obtain:
t
v
mgb =
Substitute numerical values and evaluate b:
( )( )
kg/s1027.3
m/s103m/s81.9kg10
9
4
213
−
−
−
×=
×=b
93 • Picture the Problem We can apply Newton’s 2nd law to the Ping-Pong ball to obtain its equation of motion. Applying terminal speed conditions will yield an expression for b that we can evaluate using the given numerical values. Apply yy maF =∑ to the Ping-
Pong ball:
ymabvmg =− 2
When the Ping-Pong ball reaches its terminal speed v = vt and ay = 0:
02t =− bvmg
Solve for b to obtain: 2t
vmgb =
Chapter 5
346
Substitute numerical values and evaluate b:
( )( )( )
kg/m1079.2
m/s9m/s9.81kg102.3
4
2
23
−
−
×=
×=b
*94 • Picture the Problem Let the upward direction be the positive y direction and apply Newton’s 2nd law to the sky diver. (a) Apply yy maF =∑ to the sky
diver:
ymamgF =−d or, because ay = 0,
mgF =d (1)
Substitute numerical values and evaluate Fd:
( )( ) N589m/s81.9kg60 2d ==F
(b) Substitute Fd = b 2
tv in equation
(1) to obtain:
mgbv =2t
Solve for b: 2t
d2t v
Fvmgb ==
Substitute numerical values and evaluate b: ( )
kg/m942.0m/s25
N5892 ==b
95 •• Picture the Problem The free-body diagram shows the forces acting on the car as it descends the grade with its terminal velocity. The application of Newton’s 2nd law with a = 0 and Fd equal to the given function will allow us to solve for the terminal velocity of the car.
Apply ∑ = xx maF to the car:
xmaFmg =− dsinθ
or, because v = vt and ax = 0, 0sin d =− Fmg θ
Substitute for Fd to obtain: ( ) 0m/sN1.2N100sin 2
t22 =⋅−− vmg θ
Applications of Newton’s Laws
347
Solve for vt: 22t m/sN1.2N100sin
⋅−
=θmgv
Substitute numerical values and evaluate vt:
( )( )
km/h88.2m/s5.24
m/sN1.2N1006sinm/s81.9kg800
22
2
t
==
⋅−°
=v
96 ••• Picture the Problem Let the upward direction be the positive y direction and apply Newton’s 2nd law to the particle to obtain an equation from which we can find the particle’s terminal speed. (a) Apply yy maF =∑ to a
pollution particle:
ymarvmg =− πη6
or, because ay = 0, 06 t =− rvmg πη
Solve for vt to obtain: r
mgvπη6t =
Express the mass of a sphere in terms of its volume: ⎟⎟
⎠
⎞⎜⎜⎝
⎛==
34 3rVm πρρ
Substitute for m to obtain:
ηρ
92 2
tgrv =
Substitute numerical values and evaluate vt:
( ) ( )( )( )
cm/s42.2
s/mN101.89m/s9.81kg/m2000m102
25
2325
t
=
⋅×= −
−
v
(b) Use distance equals average speed times the fall time to find the time to fall 100 m at 2.42 cm/s:
h15.1s1013.4cm/s2.42cm10 3
4
=×==t
*97 ••• Picture the Problem The motion of the centrifuge will cause the pollution particles to migrate to the end of the test tube. We can apply Newton’s 2nd law and Stokes’ law to derive an expression for the terminal speed of the sedimentation particles. We can then use this terminal speed to calculate the sedimentation time. We’ll use the 12 cm distance
Chapter 5
348
from the center of the centrifuge as the average radius of the pollution particles as they settle in the test tube. Let R represent the radius of a particle and r the radius of the particle’s circular path in the centrifuge. Express the sedimentation time in terms of the sedimentation speed vt: t
sediment vxt ∆
=∆
Apply ∑ = radialradial maF to a
pollution particle:
ct6 maRv =πη
Express the mass of the particle in terms of its radius R and density ρ:
ρπρ 334 RVm ==
Express the acceleration of the pollution particles due to the motion of the centrifuge in terms of their orbital radius r and period T:
2
2
2
2
c4
2
Tr
rT
r
rva π
π
=⎟⎠⎞
⎜⎝⎛
==
Substitute for m and ac and simplify to obtain: 2
33
2
23
34
t 31646
TrR
TrRRv ρππρππη =⎟⎟
⎠
⎞⎜⎜⎝
⎛=
Solve for vt:
2
22
t 98
TrRv
ηρπ
=
Find the period T of the motion from the number of revolutions the centrifuge makes in 1 second:
s/rev1075.0s/min60min/rev1025.1
min/rev1025.1min/rev800
1
3-
3
3
×=
××=
×==
−
−T
Substitute numerical values and evaluate vt:
( )( )( )( )( )
m/s08.2s1075s/mN101.89m10m12.0kg/m20008
2325
2532
t
=×⋅×
=−−
−πv
Find the time it takes the particles to move 8 cm as they settle in the test tube:
ms38.5
cm/s208cm8
sediment
=
=∆
=∆vxt
Applications of Newton’s Laws
349
In Problem 96 it was shown that the rate of fall of the particles in air is 2.42 cm/s. Find the time required to fall 8 cm in air under the influence of gravity:
s31.3
cm/s42.2cm8
air
=
=∆
=∆vxt
Find the ratio of the two times: ∆tair/∆tsediment ≈ 100
Euler’s Method 98 •• Picture the Problem The free-body diagram shows the forces acting on the baseball sometime after it has been thrown downward but before it has reached its terminal speed. In order to use Euler’s method, we’ll need to determine how the acceleration of the ball varies with its speed. We can do this by applying Newton’s 2nd law to the ball and using its terminal speed to express the constant in the acceleration equation in terms of the ball’s terminal speed. We can then use
tavv nnn ∆+=+1 to find the speed of the ball at any given time.
Apply Newton’s 2nd law to the ball to obtain: dt
dvmbvmg =− 2
Solve for dv/dt to obtain:
2vmbg
dtdv
−=
When the ball reaches its terminal speed:
2t0 v
mbg −= ⇒ 2
tvg
mb
=
Substitute to obtain: ⎟⎟
⎠
⎞⎜⎜⎝
⎛−= 2
t
2
1vvg
dtdv
Express the position of the ball to obtain:
tvvxx nnnn ∆
++= +
+ 21
1
Letting an be the acceleration of the ball at time tn, express its speed when t = tn + 1:
tavv nnn ∆+=+1 where
⎟⎟⎠
⎞⎜⎜⎝
⎛−= 2
t
2
1vvga n
n
Chapter 5
350
and ∆t is an arbitrarily small interval of time.
A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows:
Cell Formula/Content Algebraic Form A10 B9+$B$1 t + ∆t B10 B9+0.5*(C9+C10)*$B$1
tvvxx nnnn ∆
++= +
+ 21
1
C10 C9+D9*$B$1 vn+1 = vn+ an∆t D10 $B$4*(1−C10^2/$B$5^2)
⎟⎟⎠
⎞⎜⎜⎝
⎛−= 2
t
2
1vvga n
n
A B C D 1 ∆t= 0.5 s 2 x0= 0 m 3 v0= 9.722 m/s 4 a0= 9.81 m/s^2 5 vt= 41.67 m/s 6 7 t x v a 8 (s) (m) (m/s) (m/s^2) 9 0.0 0 9.7 9.28
10 0.5 6 14.4 8.64 11 1.0 14 18.7 7.84 12 1.5 25 22.6 6.92
28 9.5 317 41.3 0.17 29 10.0 337 41.4 0.13 30 10.5 358 41.5 0.10
38 14.5 524 41.6 0.01 39 15.0 545 41.7 0.01 40 15.5 566 41.7 0.01 41 16.0 587 41.7 0.01 42 16.5 608 41.7 0.00
From the table we can see that the speed of the ball after 10 s is approximately
m/s.4.41 We can estimate the uncertainty in this result by halving ∆t and
recalculating the speed of the ball at t = 10 s. Doing so yields v(10 s) ≈ 41.3 m/s, a difference of about %.02.0
The graph shows the velocity of the ball thrown straight down as a function of time.
Applications of Newton’s Laws
351
Ball Throw n Straight Dow n
05
1015202530354045
0 5 10 15 20
t (s)
v (m
/s)
Reset ∆t to 0.5 s and set v0 = 0. Ninety-nine percent of 41.67 m/s is approximately 41.3 m/s. Note that the ball will reach this speed in about s5.10 and that the distance it
travels in this time is about m.322 The following graph shows the distance traveled by
the ball dropped from rest as a function of time.
Ball Dropped From Rest
0
50
100
150
200
250
300
350
400
0 2 4 6 8 10 12
t (s)
x (m
)
*99 •• Picture the Problem The free-body diagram shows the forces acting on the baseball after it has left your hand. In order to use Euler’s method, we’ll need to determine how the acceleration of the ball varies with its speed. We can do this by applying Newton’s 2nd law to the baseball. We can then use tavv nnn ∆+=+1 and
tvxx nnn ∆+=+1 to find the speed and
Chapter 5
352
position of the ball. Apply ∑ = yy maF to the baseball:
dtdvmmgvbv =−−
where vv = for the upward part of the
flight of the ball and vv −= for the downward part of the flight.
Solve for dv/dt: vv
mbg
dtdv
−−=
Under terminal speed conditions ( tvv −= ):
2t0 v
mbg +−=
and
2tv
gmb
=
Substitute to obtain:
⎟⎟⎠
⎞⎜⎜⎝
⎛+−=−−= 2
t2t
1vvv
gvvvgg
dtdv
Letting an be the acceleration of the ball at time tn, express its position and speed when t = tn + 1:
( ) tvvyy nnnn ∆++= −+ 121
1 and
tavv nnn ∆+=+1 where
⎟⎟⎠
⎞⎜⎜⎝
⎛+−= 2
t
1vvv
ga nnn
and ∆t is an arbitrarily small interval of time.
A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows:
Cell Formula/Content Algebraic Form D11 D10+$B$6 t + ∆t E10 41.7 v0 E11 E10−$B$4*
(1+E10*ABS(E10)/($B$5^2))*$B$6tavv nnn ∆+=+1
F10 0 y0 F11 F10+0.5*(E10+E11)*$B$6 ( ) tvvyy nnnn ∆++= −+ 12
11
G10 0 y0 G11 $E$10*D11−0.5*$B$4*D11^2 2
21
0 gttv −
Applications of Newton’s Laws
353
A B C D E F G
4 g= 9.81 m/s^2 5 vt= 41.7 m/s 6 ∆t= 0.1 s 7 8 9 t v y y no drag 10 0.0 41.70 0.00 0.00 11 0.1 39.74 4.07 4.12 12 0.2 37.87 7.95 8.14
40 3.0 3.01 60.13 81.00 41 3.1 2.03 60.39 82.18 42 3.2 1.05 60.54 83.26 43 3.3 0.07 60.60 84.25 44 3.4 −0.91 60.55 85.14 45 3.5 −1.89 60.41 85.93 46 3.6 −2.87 60.17 86.62
78 6.8 −28.34 6.26 56.98 79 6.9 −28.86 3.41 54.44 80 7.0 −29.37 0.49 51.80 81 7.1 −29.87 −2.47 49.06
From the table we can see that, after 3.5 s, the ball reaches a height of about m.4.60 It
reaches its peak a little earlier−at about s,3.3 and its height at t = 3.3 s is m.6.60
The ball hits the ground at about t = s7 −so it spends a little longer coming down than
going up. The solid curve on the following graph shows y(t) when there is no drag on the baseball and the dotted curve shows y(t) under the conditions modeled in this problem.
Chapter 5
354
0
10
20
30
40
50
60
70
80
90
0 1 2 3 4 5 6 7
t (s)
y (m
)
x with dragx with no drag
100 •• Picture the Problem The pictorial representation shows the block in its initial position against the compressed spring, later as the spring accelerates it to the right, and finally when it has reached its maximum speed at xf = 0. In order to use Euler’s method, we’ll need to determine how the acceleration of the block varies with its position. We can do this by applying Newton’s 2nd law to the box. We can then use tavv nnn ∆+=+1 and
tvxx nnn ∆+=+1 to find the speed and position of the block.
Apply ∑ = xx maF to the block: ( ) nn maxk =−m3.0
Solve for an: ( )nn xmka −= m3.0
Express the position and speed of the block when t = tn + 1:
tvxx nnn ∆+=+1 and
tavv nnn ∆+=+1 where
( )nn xmka −= m3.0
and ∆t is an arbitrarily small interval of time.
A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows:
Applications of Newton’s Laws
355
Cell Formula/Content Algebraic FormA10 A9+$B$1 t + ∆t B10 B9+C10*$B$1 tvx nn ∆+ C10 C9+D9*$B$1 tav nn ∆+ D10 ($B$4/$B$5)*(0.3−B10) ( )nx
mk
−3.0
A B C D 1 ∆t= 0.005 s 2 x0= 0 m 3 v0= 0 m/s 4 k = 50 N/m 5 m = 0.8 kg 6 7 t x v a 8 (s) (m) (m/s) (m/s^2) 9 0.000 0.00 0.00 18.75
10 0.005 0.00 0.09 18.72 11 0.010 0.00 0.19 18.69 12 0.015 0.00 0.28 18.63
45 0.180 0.25 2.41 2.85 46 0.185 0.27 2.42 2.10 47 0.190 0.28 2.43 1.34 48 0.195 0.29 2.44 0.58 49 0.200 0.30 2.44 −0.19
From the table we can see that it took about s200.0 for the spring to push the block 30
cm and that it was traveling about m/s44.2 at that time. We can estimate the
uncertainty in this result by halving ∆t and recalculating the speed of the ball at t = 10 s. Doing so yields v(0.200 s) ≈ 2.41 m/s, a difference of about %.2.1
Chapter 5
356
General Problems 101 • Picture the Problem The forces that act on the block as it slides down the incline are shown on the free-body diagram to the right. The acceleration of the block can be determined from the distance-and-time information given in the problem. The application of Newton’s 2nd law to the block will lead to an expression for the coefficient of kinetic friction as a function of the block’s acceleration and the angle of the incline.
Apply ∑ = aF rr
m to the block: ΣFx = mgsinθ − fk = ma and ΣFy = Fn − mg = 0
Set fk = µkFn, Fn between the two equations, and solve for µk: θ
θµcos
sink g
ag −=
Using a constant-acceleration equation, relate the distance the block slides to its sliding time:
( ) 0where 02
21
0 =∆+∆=∆ vtatvx
Solve for a: ( )22
txa
∆∆
=
Substitute numerical values and evaluate a:
( )( )
22 m/s1775.0
s5.2m2.42
==a
Find µk for a = 0.1775 m/s2 and θ = 28°:
( )( )
511.0
cos28m/s9.81m/s0.1775sin28m/s9.81
2
22
k
=
°−°
=µ
Applications of Newton’s Laws
357
102 • Picture the Problem The free-body diagram shows the forces acting on the model airplane. The speed of the plane can be calculated from the data concerning the radius of its path and the time it takes to make one revolution. The application of Newton’s 2nd law will give us the tension F in the string.
(a) Express the speed of the airplane in terms of the circumference of the circle in which it is flying and its period:
Trv π2
=
Substitute numerical values and evaluate v:
( ) m/s10.7s
1.24
m5.72π==v
(b) Apply ∑ = xx maF to the model
airplane:
rvmF
2
=
Substitute numerical values and evaluate F: ( ) ( ) N8.03
m5.7m/s10.7kg0.4
2
==F
*103 •• Picture the Problem The free-body diagram shows the forces acting on the box. If the student is pushing with a force of 200 N and the box is on the verge of moving, the static friction force must be at its maximum value. In part (b), the motion is impending up the incline; therefore the direction of fs,max is down the incline. (a) Apply ∑ = aF rr
m to the box: ∑ =−+= 0sins θmgFfFx
and
∑ =−= 0cosn θmgFFy
Chapter 5
358
Substitute fs = fs,max = µsFn, eliminate Fn between the two equations, and solve for µs:
θθµ
costans mg
F−=
Substitute numerical values and evaluate µs: ( )
289.0
cos30N800N20030tans
=
°−°=µ
(b) Find fs,max from the x-direction force equation:
Fmgf −= θsinmaxs,
Substitute numerical values and evaluate fs,max:
( )N200
N200sin30N800maxs,
=
−°=f
If the block is on the verge of sliding up the incline, fs,max
must act down the incline. The x-direction force equation becomes:
0sinmaxs, =−+− θmgFf
Solve the x-direction force equation for F:
maxs,sin fmgF += θ
Substitute numerical values and evaluate F:
( ) N600N200sin30N800 =+°=F
104 • Picture the Problem The path of the particle is a circle if r is a constant. Once we have shown that it is, we can calculate its value from its components. The direction of the particle’s motion can be determined by examining two positions of the particle at times that are close to each other. (a) and (b) Express the magnitude of rr
in terms of its components:
22yx rrr +=
Evaluate r with rx = −10 m cos ωt and ry = 10 m sinωt:
( )[ ] ( )[ ]( )m0.10
msincos100
sinm10cosm1022
22
=
+=
+−=
tt
ttr
ωω
ωω
Applications of Newton’s Laws
359
(c) Evaluate rx and ry at t = 0 s: ( )( ) 00sinm10
m100cosm10=°=
−=°−=
y
x
rr
Evaluate rx and ry at t = ∆t, where ∆t is small:
( ) ( )
( )positiveiswhere
sinm10m10
0cosm10cosm10
yy
tr
tr
y
x
∆∆=
∆=−=
°−≈∆−=
ω
ω
and clockwise ismotion the
(d) Differentiate rr with respect to time to obtain :vr
[ ] ( )[ ] ji
rvˆcos10ˆ)sin10(
/
mtmt
dtd
ωωωω +=
=rr
Use the components of vr to find its speed:
( )[ ] ( )[ ]( ) ( )( )
m/s0.20
s2m10m10
mcos10msin101
22
22
=
==
+=
+=
−ω
ωωωω tt
vvv yx
(e) Relate the period of the particle’s motion to the radius of its path and its speed:
( ) sm/s20
m1022 πππ===
vrT
105 •• Picture the Problem The free-body diagram shows the forces acting on the crate of books. The kinetic friction force opposes the motion of the crate up the incline. Because the crate is moving at constant speed in a straight line, its acceleration is zero. We can determine F by applying Newton’s 2nd law to the crate, substituting for fk, eliminating the normal force, and solving for the required force.
Apply∑ = aF rr
m to the crate, with
both ax and ay equal to zero, to the crate:
∑ =−−= 0sincos θθ mgfFF kx
and
∑ =−−= 0cossin θθ mgFFF ny
Chapter 5
360
Substitute µsFn for fk and eliminate Fn to obtain:
( )θµθ
θµθsincos
cossin
k
k
−+
=mgF
Substitute numerical values and evaluate F:
( )( ) ( )( )( ) kN49.1
sin300.5cos30cos300.5sin30m/s9.81kg100 2
=°−°
°+°=F
106 •• Picture the Problem The free-body diagram shows the forces acting on the object as it slides down the inclined plane. We can calculate its speed at the bottom of the incline from its acceleration and displacement and find its acceleration from Newton’s 2nd law. Using a constant-acceleration equation, relate the initial and final velocities of the object to its acceleration and displacement: solve for the final velocity:
xavv ∆+= 220
2
Because v0 = 0, xav ∆= 2 (1)
Apply ∑ = aF rrm to the sliding
object:
∑ =+−= mamgfFx θsink
and
∑ =−= 0cosn θmgFFy
Solve the y equation for Fn and using fk = µkFn, eliminate both Fn and fk from the x equation and solve for a:
( )θµθ cossin k−= ga (2)
Substitute equation (2) in equation (1) and solve for v:
( ) xgv ∆−= θµθ cossin2 k
Substitute numerical values and evaluate v:
( ) ( )( )( ) m/s7.16m7230cos35.030sinm/s9.812 2 =°−°=v and correct. is )(d
Applications of Newton’s Laws
361
*107 •• Picture the Problem The free-body diagram shows the forces acting on the brick as it slides down the inclined plane. We’ll apply Newton’s 2nd law to the brick when it is sliding down the incline with constant speed to derive an expression for µk in terms of θ0. We’ll apply Newton’s 2nd law a second time for θ = θ1 and solve the equations simultaneously to obtain an expression for a as a function of θ0 and θ1.
Apply ∑ = aF rr
m to the brick
when it is sliding with constant speed:
∑ =+−= 0sin 0k θmgfFx
and
∑ =−= 0cos 0n θmgFFy
Solve the y equation for Fn and using fk = µkFn, eliminate both Fn and fk from the x equation and solve for µk:
0k tanθµ =
Apply ∑ = aF rrm to the brick when
θ = θ1:
∑ =+−= mamgfFx 1k sinθ
and
∑ =−= 0cos 1n θmgFFy
Solve the y equation for Fn, use fk = µkFn to eliminate both Fn and fk from the x equation, and use the expression for µk obtained above to obtain:
( )101 costansin θθθ −= ga
108 •• Picture the Problem The fact that the object is in static equilibrium under the influence of the three forces means that .0321 =++ FFF
rrr Drawing the corresponding force
triangle will allow us to relate the forces to the angles between them through the law of sines and the law of cosines.
Chapter 5
362
(a) Using the fact that the object is in static equilibrium, redraw the force diagram connecting the forces head-to-tail:
Apply the law of sines to the triangle: ( ) ( ) ( )12
3
13
2
23
1
sinsinsin θπθπθπ −=
−=
−FFF
Use the trigonometric identity sin(π − α) = sinα to obtain:
12
3
13
2
23
1
sinsinsin θθθFFF
==
(b) Apply the law of cosines to the triangle:
( )23322
32
22
1 cos2 θπ −−+= FFFFF
Use the trigonometric identity cos(π − α) = −cosα to obtain: 2332
23
22
21 cos2 θFFFFF ++=
109 •• Picture the Problem We can calculate the acceleration of the passenger from his/her speed that, in turn, is a function of the period of the motion. To determine the longest period of the motion, we focus our attention on the situation at the very top of the ride when the seat belt is exerting no force on the rider. We can use Newton’s 2nd law to relate the period of the motion to the acceleration and speed of the rider.
(a) Because the motion is at constant speed, the acceleration is entirely radial and is given by:
rva
2
c =
Express the speed of the motion of the ride as a function of the radius of the circle and the period of its motion:
Trv π2
=
Applications of Newton’s Laws
363
Substitute in the expression for ac to obtain: 2
2
c4T
ra π=
Substitute numerical values and evaluate ac:
( )( )
22
2
c m/s3.49s2
m54==
πa
(b) Apply ∑ = aF rr
m to the
passenger when he/she is at the top of the circular path and solve for ac:
∑ == cr mamgF
and ac = g
Relate the acceleration of the motion to its radius and speed and solve for v:
grvr
vg =⇒=2
Express the period of the motion as a function of the radius of the circle and the speed of the passenger and solve for Tm:
gr
vrT ππ 22
m ==
Substitute numerical values and evaluate Tm: s49.4
m/s9.81m52 2m == πT
Remarks: The rider is ″weightless″ under the conditions described in part (b). *110 •• Picture the Problem The pictorial representation to the right shows the cart and its load on the inclined plane. The load will not slip provided its maximum acceleration is not exceeded. We can find that maximum acceleration by applying Newton’s 2nd law to the load. We can then apply Newton’s 2nd law to the cart-plus-load system to determine the tension in the rope when the system is experiencing its maximum acceleration.
Chapter 5
364
Draw the free-body diagram for the cart and its load:
Apply ∑ = xx maF to the cart plus
its load:
( ) ( ) max2121 sin ammgmmT +=+− θ (1)
Draw the free-body diagram for the load of mass m2 on top of the cart:
Apply ∑ = aF rrm to the load on
top of the cart:
∑ =−= max22maxs, sin amgmfFx θ
and 0cos22,n =−=∑ θgmFFy
Using fs,max = µsFn,2, eliminate Fn,2 between the two equations and solve for the maximum acceleration of the load:
( )θθµ sincossmax −= ga (2)
Substitute equation (2) in equation (1) and solve for T :
( ) θµ coss21 gmmT +=
111 •• Picture the Problem The free-body diagram for the sled while it is held stationary by the static friction force is shown to the right. We can solve this problem by repeatedly applying Newton’s 2nd law under the conditions specified in each part of the problem.
Applications of Newton’s Laws
365
(a) Apply∑ = yy maF to the sled: 0cos1n,1 =− θgmF
Solve for Fn,1:
θcos1n,1 gmF =
Substitute numerical values and evaluate Fn,1:
( ) N19315cosN200n,1 =°=F
(b) Apply∑ = xx maF to the sled: 0sin1s =− θgmf
Solve for fs:
θsin1s gmf =
Substitute numerical values and evaluate fs:
( ) N8.5115sinN200s =°=f
(c) Draw the free-body diagram for the sled when the child is pulling on the rope:
Apply∑ = aF rr
m to the sled to
determine whether it moves: maxs,1
net
sin30cos fgmFFFx
−−°=
=∑θ
and
∑ =−°+= 0cos30sin 1n,1 θgmFFFy
Solve the y-direction equation for Fn,1:
θcos30sin 1n,1 gmFF +°−=
Substitute numerical values and evaluate Fn,1:
( ) ( )N143
15cosN20030sinN100n,1
=
°+°−=F
Express fs,max: fs,max = µsFn,1 = (0.5)(143 N)
= 71.5 N
Use the x-direction force equation to evaluate Fnet:
Fnet = (100 N)cos30° − (200 N)sin15° − 71.5 N = −36.7 N
Chapter 5
366
Because the net force is negative, the sled does not move:
edundetermin is kf
(d) Because the sled does not move: edundetermin is kµ
(e) Draw the FBD for the child:
Express the net force Fc exerted on the child by the incline:
2maxs,
2n2c fFF += (1)
Noting that the child is stationary, apply∑ = aF rr
m to the child:
0
15sin30cos 2maxs,
=
°−°−=∑ gmFfFx
and 030sin15sin2n2 =°−°−=∑ FgmFFy
Solve the x equation for fs,max and the y equation for Fn2:
°+°= 15sin30cos 2maxs, gmFf
and °+°= 30sin15sin2n2 FgmF
Substitute numerical values and evaluate Fx and Fn2:
( ) ( )N459
15sinN10030cosN500maxs,
=
°+°=f
and ( ) ( )
N27630sinN50015sinN100n2
=°+°=F
Substitute numerical values in equation (1) and evaluate F:
( ) ( ) N536N459N276 22c =+=F
112 • Picture the Problem Let v represent the speed of rotation of the station, and r the distance from the center of the station. Because the O’Neill colony is, presumably, in deep space, the only acceleration one would experience in it would be that due to its rotation.
Applications of Newton’s Laws
367
(a) Express the acceleration of anyone who is standing inside the station:
a = v2/r
This acceleration is directed toward the axis of rotation. If someone inside the station drops an apple, the apple will not have any forces acting on it once released, but will move along a straight line at constant speed. However, from the point of view of our observer inside the station, if he views himself as unmoving, the apple is perceived to have an acceleration of mv2/r directed away from the axis of rotation (a "centrifugal" force). (b) Each deck must rotate the central axis with the same period T. Relate the speed of a person on a particular deck to his/her distance r from the center:
Trv π2
=
Express the "acceleration of gravity" perceived by someone a distance r from the center:
2
22 4T
rr
v π=
decreases. as decreases gravity" todueon accelerati" thei.e.,
r
(c) Relate the desired acceleration to the radius of Babylon 5 and its period:
2
24T
ra π=
Solve for T:
arT
24π=
Substitute numerical values and evaluate T:
min 0.735 s1.44m/s8.9
mikm1.609mi3.04
2
2
==
⎟⎠⎞
⎜⎝⎛ ×
=π
T
Take the reciprocal of this time to find the number of revolutions per minute Babylon 5 has to make in order to provide this ″earth-like″ acceleration:
min/rev36.11 =−T
Chapter 5
368
113 •• Picture the Problem The free-body diagram shows the forces acting on the child as she slides down the incline. We’ll first use Newton’s 2nd law to derive an expression for µk in terms of her acceleration and then use Newton’s 2nd law to find her acceleration when riding the frictionless cart. Using a constant-acceleration equation, we’ll relate these two accelerations to her descent times and solve for her acceleration when sliding. Finally, we can use this acceleration in the expression for µk.
Apply ∑ = aF rr
m to the child as
she slides down the incline:
∑ =−°= 1k30sin mafmgFx
and
∑ =°−= 030cosn mgFFy
Using fk = µkFn, eliminate fk and Fn between the two equations and solve for µk:
°−°=
30cos30tan 1
k ga
µ (1)
Apply∑ = xx maF to the child as
she rides the frictionless cart down the incline and solve for her acceleration a2:
230sin mamg =°
and
22
m/s91.4
30sin
=
°= ga
Letting s represent the distance she slides down the incline, use a constant-acceleration equation to relate her sliding times to her accelerations and distance traveled down the slide :
0where 02112
110 =+= vtatvs
and 0where 0
2222
120 =+= vtatvs
Equate these expressions, substitute t2 = 2
1 t1 and solve for a1:
241
241
1 m/s23.130sin =°== gaa
Applications of Newton’s Laws
369
Evaluate equation (1) with a1 = 1.23 m/s2: ( )
433.0
30cosm/s81.9m/s23.130tan 2
2
k
=
°−°=µ
*114 •• Picture the Problem The path of the particle is a circle if r is a constant. Once we have shown that it is, we can calculate its value from its components and determine the particle’s velocity and acceleration by differentiation. The direction of the net force acting on the particle can be determined from the direction of its acceleration.
(a) Express the magnitude of rr
in terms of its components:
22yx rrr +=
Evaluate r with rx = Rsinωt and ry = Rcosωt:
[ ] [ ]( ) m0.4cossin
cossin222
22
==+=
+=
RttR
tRtRr
ωω
ωω
origin. at the centered circle a is particle theofpath the∴
(b) Differentiate rr with respect to time to obtain :vr
[ ][ ]
( )[ ]( )[ ] j
i
j
irv
ˆm/s2sin8
ˆm/s2cos8
ˆsin
ˆcos/
t
t
tR
tRdtd
ππ
ππ
ωω
ωω
−=
−+
==rr
Express the ratio :y
x
vv
tt
tvv
y
x ωωπ
ωπ cotsin8
cos8−=
−=
Express the ratio :xy
− ttRtR
xy ω
ωω cot
sincos
−=−=−
xy
vv
y
x −=∴
(c) Differentiate vr with respect to time to obtain :ar
( )[ ]( )[ ]j
i
va
ˆcosm/s16
ˆsinm/s16
/
22
22
t
t
dtd
ωπ
ωπ
−+
−=
=rr
Chapter 5
370
Factor −4π2/s2 from ar to obtain: ( ) ( ) ( )[ ]( )r
jiar
r
22
22
s/4
ˆcos4ˆsin4s/4
π
ωωπ
−=
+−= tt
Because ar is in the opposite direction from ,rr it is directed toward the center of the
circle in which the particle is traveling.
Find the ratio r
v 2
: ( ) arv
=== 2222
m/s16m4m/s8 ππ
(d) Apply∑ = aF rr
m to the particle:
( )( )N8.12
m/s16kg8.02
22net
π
π
=
== maF
Because the direction of netF
ris the
same as that of ar
: circle. theofcenter the towardis netF
r
115 •• Picture the Problem The free-body diagram showing the forces acting on a rider being held in place by the maximum static friction force is shown to the right. The application of Newton’s 2nd law and the definition of the maximum static friction force will be used to determine the period T of the motion. The reciprocal of the period will give us the minimum number of revolutions required per unit time to hold the riders in place.
Apply ∑ = aF rr
m to the riders
while they are held in place by friction:
∑ ==r
vmFFx
2
n
and
∑ =−= 0maxs, mgfFy
Using nsmaxs, Ff µ= and ,2T
rv π=
eliminate Fn between the force equations and solve for the period of the motion:
grT s2 µπ=
Applications of Newton’s Laws
371
Substitute numerical values and evaluate T:
( )( )
min0.00423s54.2m/s9.81
m40.42 2
==
= πT
The number of revolutions per minute is the reciprocal of the period in minutes:
rev/min6.23
116 •• Picture the Problem The free-body diagrams to the right show the forces acting on the blocks whose masses are m1 and m2. The application of Newton’s 2nd law and the use of a constant-acceleration equation will allow us to find a relationship between the coefficient of kinetic friction and m1. The repetition of this procedure with the additional object on top of the object whose mass is m1 will lead us to a second equation that, when solved simultaneously with the former equation, leads to a quadratic equation in m1. Finally, its solution will allow us to substitute in an expression for µk and determine its value.
Using a constant-acceleration equation, relate the displacement of the system in its first configuration as a function of its acceleration and fall time:
( )212
10 tatvx ∆+∆=∆
or, because v0 = 0, ( )2
121 tax ∆=∆
Solve for a1: ( )212
txa
∆∆
=
Substitute numerical values and evaluate a1:
( )( )
221 m/s46.4
s82.0m5.12
==a
Apply ∑ = xx maF to the object
whose mass is m2 and solve for T1:
1212 amTgm =−
and ( )agmT −= 21
Chapter 5
372
Substitute numerical values and evaluate T1:
( )( )N375.13
m/s46.4m/s81.9kg5.2 221
=−=T
Apply ∑ = aF rr
m to the object
whose mass is m1:
∑ =−= 11k1 amfTFx
and
∑ =−= 01n,1 gmFFy
Using fk = µkFn, eliminate Fn between the two equations to obtain:
111k1 amgmT =− µ (1)
Find the acceleration a2 for the second run: ( )
( )( )
2222 m/s775.1
s3.1m5.122
==∆∆
=txa
Evaluate T2: ( )
( )( )N1.20
m/s775.1m/s81.9kg5.2 2222
=−=
−= agmT
Apply ∑ = xx maF to the 1.2-kg
object in place:
( )( ) 21
1k2
kg2.1kg2.1
amgmT
+=+− µ
(2)
Solve equation (1) for µk:
gmamT
1
111k
−=µ (3)
Substitute for µk in equation (2) and simplify to obtain the quadratic equation in m1:
005.16947.9685.2 121 =−+ mm
Solve the quadratic equation to obtain:
( ) kg22.1kg07.385.1 11 =⇒±−= mm
Substitute numerical values in equation (3) and evaluate µk:
( )( )( )( )
643.0
m/s81.9kg1.22m/s66.4kg 1.22N375.13
2
2
k
=
−=µ
Applications of Newton’s Laws
373
*117 ••• Picture the Problem The diagram shows a point on the surface of the earth at latitude θ. The distance R to the axis of rotation is given by R = rcosθ. We can use the definition of centripetal acceleration to express the centripetal acceleration of a point on the surface of the earth due to the rotation of the earth. (a) Referring to the figure, express ac for a point on the surface of the earth at latitude θ :
θcos where2
c rRRva ==
Express the speed of the point due to the rotation of the earth: .revolution onefor timetheiswhere
2
TT
Rv π=
Substitute for v in the expression for ac and simplify to obtain: 2
2
ccos4
Tra θπ
=
Substitute numerical values and evaluate ac:
( )( )( )[ ]( )
axis. searth' the toward,coscm/s37.3
s/h3600h24coskm63704
2
2
2
c
θ
θπ
=
=a
(b)
. of than thatlessslightly is of magnitude that theshow to used becan diagramaddition A vector rotation. searth' the todueon accelerati
theandgravity todueon accelerati theeight toapparent w therelateswhich , gives grearrangin and by equation isthrough th
gMultiplyin earth). theofrotation the todue surface theofon accelerati (the frame inertial the torelativeearth theof surface local theofon accelerati
theis where, toequal is stone on the force nalgravitatio The earth. theof surface local the torelative )resistanceair g(neglectin
stone falling theofon accelerati theis where, toequal is stone theof weight effective The earth.on location aat hand a from dropped stoneA
iner st,surf st,
iner surf,iner st,surf st,
iner st,inerst,
surf st,surf st,
aa
aaa
aa
aa
rr
rrr
rr
rr
mm
mmmm
m
m
−=
Chapter 5
374
(c) At the equator, the gravitational acceleration and the radial acceleration are both directed toward the center of the earth. Therefore:
( )2
22ceff
cm/s4.981
cos0cm/s3.37cm/s978
=
°+=
+= agg
At latitude θ the gravitational acceleration points toward the center of the earth whereas the centripetal acceleration points toward the axis of rotation. Use the law of cosines to relate geff, g, and ac:
θcos2 c2c
22eff gaagg −+=
Substitute for θ, geff, and ac and simplify to obtain the quadratic equation:
( ) 0/scm962350cm/s75.4 4222 =−− gg
Solve for the physically meaningful (i.e., positive) root to obtain:
2cm/s983=g
*118 ••• Picture the Problem The diagram shows the block in its initial position, an intermediate position, and as it is separating from the sphere. Because the sphere is frictionless, the only forces acting on the block are the normal and gravitational forces. We’ll apply Newton’s 2nd law and set Fn equal to zero to determine the angle θc at which the block leaves the surface.
Taking the inward direction to be positive, apply rr maF =∑ to the block:
RvmFmg
2
ncos =−θ
Apply the separation condition to obtain: R
vmmg2
ccos =θ
Solve for cosθc:
gRv2
ccos =θ (1)
Apply tt maF =∑ to the block: tsin mamg =θ
or
Applications of Newton’s Laws
375
θsint gdtdva ==
Note that a is not constant and, hence, we cannot use constant-acceleration equations.
Multiply the left-hand side of the equation by one in the form of dθ/dθ and rearrange to obtain:
θθθ sing
dd
dtdv
=
and
θθ
θ singddv
dtd
=
Relate the arc distance s the block travels to the angle θ and the radius R of the sphere:
Rs
=θ and Rv
dtds
Rdtd
==1θ
where v is the block’s instantaneous speed.
Substitute to obtain: θ
θsing
ddv
Rv
=
Separate the variables and integrate from v′ = 0 to v and θ = 0 to θc:
∫∫ =c
00
sinθ
θθdgRv'dv'v
or ( )c
2 cos12 θ−= gRv
Substitute in equation (1) to obtain: ( )
( )c
cc
cos12
cos12cos
θ
θθ
−=
−=
gRgR
Solve for and evaluate θc: °=⎟
⎠⎞
⎜⎝⎛= − 2.48
32cos 1
cθ
Chapter 5
376
371
Chapter 6 Work and Energy Conceptual Problems
*1 • Determine the Concept A force does work on an object when its point of application moves through some distance and there is a component of the force along the line of motion. (a) False. The net force acting on an object is the vector sum of all the forces acting on the object and is responsible for displacing the object. Any or all of the forces contributing to the net force may do work. (b) True. The object could be at rest in one reference frame and moving in another. If we consider only the frame in which the object is at rest, then, because it must undergo a displacement in order for work to be done on it, we would conclude that the statement is true. (c) True. A force that is always perpendicular to the velocity of a particle changes neither it’s kinetic nor potential energy and, hence, does no work on the particle.
2 • Determine the Concept If we ignore the work that you do in initiating the horizontal motion of the box and the work that you do in bringing it to rest when you reach the second table, then neither the kinetic nor the potential energy of the system changed as you moved the box across the room. Neither did any forces acting on the box produce displacements. Hence, we must conclude that the minimum work you did on the box is zero.
3 • False. While it is true that the person’s kinetic energy is not changing due to the fact that she is moving at a constant speed, her gravitational potential energy is continuously changing and so we must conclude that the force exerted by the seat on which she is sitting is doing work on her. *4 • Determine the Concept The kinetic energy of any object is proportional to the square of its speed. Because ,2
21 mvK = replacing v by 2v yields
( ) ( ) .442 2212
21 KmvvmK' === Thus doubling the speed of a car quadruples its kinetic
energy.
Chapter 6
372
5 • Determine the Concept No. The work done on any object by any force F
r is defined as
rF rrddW ⋅= . The direction of netF
r is toward the center of the circle in which the object
is traveling and rrd is tangent to the circle. No work is done by the net force because
netFr
and rrd are perpendicular so the dot product is zero.
6 • Determine the Concept The kinetic energy of any object is proportional to the square of its speed and is always positive. Because ,2
21 mvK = replacing v by 3v yields
( ) ( ) .993 2212
21 KmvvmK' === Hence tripling the speed of an object increases its
kinetic energy by a factor of 9 and correct. is )(d
*7 • Determine the Concept The work required to stretch or compress a spring a distance x is given by 2
21 kxW = where k is the spring’s stiffness constant. Because W ∝ x2, doubling
the distance the spring is stretched will require four times as much work. 8 • Determine the Concept No. We know that if a net force is acting on a particle, the particle must be accelerated. If the net force does no work on the particle, then we must conclude that the kinetic energy of the particle is constant and that the net force is acting perpendicular to the direction of the motion and will cause a departure from straight-line motion. 9 • Determine the Concept We can use the definition of power as the scalar product of force and velocity to express the dimension of power. Power is defined as: P ≡ F
r⋅ vr
Express the dimension of force: [M][L/T 2]
Express the dimension of velocity: [L/T]
Express the dimension of power in terms of those of force and velocity:
[M][L/T 2][L/T] = [M][L]2/[T]3
and correct. is )(d
Work and Energy
373
10 • Determine the Concept The change in gravitational potential energy, over elevation changes that are small enough so that the gravitational field can be considered constant, is mg∆h, where ∆h is the elevation change. Because ∆h is the same for both Sal and Joe, their gains in gravitational potential energy are the same. correct. is )(c
11 • (a) False. The definition of work is not limited to displacements caused by conservative forces. (b) False. Consider the work done by the gravitational force on an object in freefall. (c) True. This is the definition of work done by a conservative force. *12 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x; i.e., dxdUFx −= .
(a) Examine the slopes of the curve at each of the lettered points, remembering that Fx is the negative of the slope of the potential energy graph, to complete the table:
Point dU/dx Fx
A + − B 0 0 C − + D 0 0 E + − F 0 0
(b) Find the point where the slope is steepest:
greatest. is point At xFC
(c) If d2U/dx2 < 0, then the curve is concave downward and the equilibrium is unstable.
unstable. is mequilibriu thepoint At B
If d2U/dx2 > 0, then the curve is concave upward and the equilibrium is stable.
stable. is mequilibriu thepoint At D
Remarks: At point F, d2U/dx2 = 0 and the equilibrium is neither stable nor unstable; it is said to be neutral.
Chapter 6
374
13 • (a) False. Any force acting on an object may do work depending on whether the force produces a displacement … or is displaced as a consequence of the object’s motion. (b) False. Consider an element of area under a force-versus-time graph. Its units are N⋅s whereas the units of work are N⋅m. 14 • Determine the Concept Work ( )sF
rrddW ⋅= is done when a force F
r produces a
displacement .srd Because ( )ds,FFdsd θθ coscos =≡⋅ sFrr
W will be negative if the
value of θ is such that Fcosθ is negative. correct. is )(d
Estimation and Approximation *15 •• Picture the Problem The diagram depicts the situation when the tightrope walker is at the center of rope. M represents her mass and the vertical components of tensions
1Tr
and ,2Tr
equal in magnitude, support her weight. We can apply a condition for static equilibrium in the vertical direction to relate the tension in the rope to the angle θ and use trigonometry to find s as a function of θ.
(a) Use trigonometry to relate the sag s in the rope to its length L and θ :
Ls
21
tan =θ and θtan2Ls =
Apply 0=∑ yF to the tightrope walker when she is at the center of the rope to obtain:
0sin2 =− MgT θ where T is the
magnitude of 1Tr
and 2Tr
.
Solve for θ to obtain: ⎟
⎠⎞
⎜⎝⎛= −
TMg2
sin 1θ
Substitute numerical values and evaluate θ :
( )( )( ) °=⎥
⎦
⎤⎢⎣
⎡= − 81.2
N50002m/s9.81kg05sin
21θ
Work and Energy
375
Substitute to obtain: m0.24581.2tan
2m10
=°=s
(b) Express the change in the tightrope walker’s gravitational potential energy as the rope sags:
yMgUUU ∆=−=∆ endcenterat
Substitute numerical values and evaluate ∆U:
( )( )( )J120
m0.245m/s9.81kg50 2
−=
−=∆U
16 • Picture the Problem You can estimate your change in potential energy due to this change in elevation from the definition of ∆U. You’ll also need to estimate the height of one story of the Empire State building. We’ll assume your mass is 70 kg and the height of one story to be 3.5 m. This approximation gives us a height of 1170 ft (357 m), a height that agrees to within 7% with the actual height of 1250 ft from the ground floor to the observation deck. We’ll also assume that it takes 3 min to ride non-stop to the top floor in one of the high-speed elevators. (a) Express the change in your gravitational potential energy as you ride the elevator to the 102nd floor:
hmgU ∆=∆
Substitute numerical values and evaluate ∆U:
( )( )( )kJ452
m357m/s9.81kg70 2
=
=∆U
(b) Ignoring the acceleration intervals at the beginning and the end of your ride, express the work done on you by the elevator in terms of the change in your gravitational potential energy:
UFhW ∆==
Solve for and evaluate F: N686
m357kJ452
==∆
=hUF
(c) Assuming a 3 minute ride to the top, express and evaluate the average power delivered to the elevator:
( )( )kW36.1
s/min60min3kJ452
=
=∆
∆=
tUP
Chapter 6
376
17 • Picture the Problem We can find the kinetic energy K of the spacecraft from its definition and compare its energy to the annual consumption in the U.S. W by examining the ratio K/W. Using its definition, express and evaluate the kinetic energy of the spacecraft:
( )( )J104.50
m/s103kg1000018
27212
21
×=
×== mvK
Express this amount of energy as a percentage of the annual consumption in the United States:
%1J105
J104.5020
18
≈×
×≈
EK
*18 •• Picture the Problem We can find the orbital speed of the Shuttle from the radius of its orbit and its period and its kinetic energy from .2
21 mvK = We’ll ignore the variation in
the acceleration due to gravity to estimate the change in the potential energy of the orbiter between its value at the surface of the earth and its orbital value.
(a) Express the kinetic energy of the orbiter:
221 mvK =
Relate the orbital speed of the orbiter to its radius r and period T:
Trv π2
=
Substitute and simplify to obtain: 2
222
21 22
Tmr
TrmK ππ
=⎟⎠⎞
⎜⎝⎛=
Substitute numerical values and evaluate K:
( ) ( )( )[ ]( )( )[ ]
TJ43.2min/s60min90
km/mi1.609mi3960mi200kg10822
242
=+×
=πK
(b) Assuming the acceleration due to gravity to be constant over the 200 mi and equal to its value at the surface of the earth (actually, it is closer to 9 m/s2 at an elevation of 200 mi), express the change in gravitational potential energy of the orbiter, relative to the surface of the earth, as the Shuttle goes into orbit:
mghU =∆
Work and Energy
377
Substitute numerical values and evaluate ∆U:
( )( )( )( )
TJ253.0
km/mi1.609mi200m/s9.81kg108 24
=
××=∆U
(c)
energy. potentialin change actual thefind toshuttle theof weight under then deformatio itsin earth of surface theofenergy potentialin change the
ionconsiderat into take toneed would Weit.on upward exertsearth theforce normal by thegravity of force eagainst th supported isit earth, theof surface on the resting is shuttle When thehere.consider gravity to
of force just the than more is therebecause equal bet shouldn' they No,
19 • Picture the Problem Let’s assume that the width of the driveway is 18 ft. We’ll also assume that you lift each shovel full of snow to a height of 1 m, carry it to the edge of the driveway, and drop it. We’ll ignore the fact that you must slightly accelerate each shovel full as you pick it up and as you carry it to the edge of the driveway. While the density of snow depends on the extent to which it has been compacted, one liter of freshly fallen snow is approximately equivalent to 100 mL of water. Express the work you do in lifting the snow a distance h:
ghVmghUW snowsnowρ==∆= where ρ is the density of the snow.
Using its definition, express the densities of water and snow: snow
snowsnow V
m=ρ and
water
waterwater V
m=ρ
Divide the first of these equations by the second to obtain: snow
water
water
snow
VV
=ρρ
or snow
waterwatersnow V
Vρρ =
Substitute and evaluate the ρsnow: ( ) 333snow kg/m100
LmL100kg/m10 ==ρ
Calculate the volume of snow covering the driveway:
( )( )
3
33
33
snow
m2.21L
m10ft
L32.28ft750
ft1210ft18ft50
=
××=
⎟⎠⎞
⎜⎝⎛=
−
V
Substitute numerical values in the expression for W to obtain an estimate (a lower bound) for the work you would do on the snow in removing it:
( )( )( )( )kJ8.20
m1m/s81.9m2.21kg/m100 233
=
=W
Chapter 6
378
Work and Kinetic Energy *20 • Picture the Problem We can use 2
21 mv to find the kinetic energy of the bullet.
(a) Use the definition of K:
( )( )kJ8.10
m/s101.2kg0.015 2321
221
=
×=
= mvK
(b) Because K ∝ v2: kJ70.24
1 == KK'
(c) Because K ∝ v2: kJ2.434 == KK'
21 • Picture the Problem We can use 2
21 mv to find the kinetic energy of the baseball and the
jogger. (a) Use the definition of K: ( )( )
J147
m/s45kg0.145 2212
21
=
== mvK
(b) Convert the jogger’s pace of 9 min/mi into a speed:
m/s98.2mi1
m1609s60
min1min9mi1
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=v
Use the definition of K: ( )( )
J266
m/s98.2kg60 2212
21
=
== mvK
22 • Picture the Problem The work done in raising an object a given distance is the product of the force producing the displacement and the displacement of the object. Because the weight of an object is the gravitational force acting on it and this force acts downward, the work done by gravity is the negative of the weight of the object multiplied by its displacement. The change in kinetic energy of an object is equal to the work done by the net force acting on it. (a) Use the definition of W: yF
rr∆⋅=W = F∆y
Work and Energy
379
= (80 N)(3 m) = J240
(b) Use the definition of W: yF rr
∆⋅=W = −mg∆y, because Fr
and yr
∆ are in opposite directions.
∴ W = − (6 kg)(9.81 m/s2)(3 m) = J177−
(c) According to the work-kinetic energy theorem:
K = W + Wg = 240 J + (−177 J) = J0.63
23 • Picture the Problem The constant force of 80 N is the net force acting on the box and the work it does is equal to the change in the kinetic energy of the box. Using the work-kinetic energy theorem, relate the work done by the constant force to the change in the kinetic energy of the box:
( )2i
2f2
1if vvmKKW −=−=
Substitute numerical values and evaluate W:
( ) ( ) ( )[ ]kJ6.10
m/s20m/s68kg5 2221
=
−=W
*24 •• Picture the Problem We can use the definition of kinetic energy to find the mass of your friend. Using the definition of kinetic energy and letting ″1″ denote your mass and speed and ″2″ your girlfriend’s, express the equality of your kinetic energies and solve for your girlfriend’s mass as a function of both your masses and speeds:
2222
12112
1 vmvm =
and 2
2
112 ⎟⎟
⎠
⎞⎜⎜⎝
⎛=
vvmm (1)
Express the condition on your speed that enables you to run at the same speed as your girlfriend:
v2 = 1.25v1 (2)
Chapter 6
380
Substitute equation (2) in equation (1) to obtain: ( )
kg4.54
25.11kg85
22
2
112
=
⎟⎠⎞
⎜⎝⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
vvmm
Work Done by a Variable Force 25 •• Picture the Problem The pictorial representation shows the particle as it moves along the positive x axis. The particle’s kinetic energy increases because work is done on it. We can calculate the work done on it from the graph of Fx vs. x and relate its kinetic energy when it is at x = 4 m to its kinetic energy when it was at the origin and the work done on it by using the work-kinetic energy theorem.
(a) Calculate the kinetic energy of the particle when it is at x = 0:
( )J00.6
)m/s(2kg3 2212
21
0
=
== mvK
(b) Because the force and displacement are parallel, the work done is the area under the curve. Use the formula for the area of a triangle to calculate the area under the F as a function of x graph:
( )( )( )( )
J0.12
N6m4altitudebase
2121
40
=
==→W
(c) Express the kinetic energy of the particle at x = 4 m in terms of its speed and mass and solve for its speed:
mK
v 44
2= (1)
Using the work-kinetic energy theorem, relate the work done on the particle to its change in kinetic energy and solve for the particle’s kinetic energy at x = 4 m:
W0→4 = K4 – K0
J18.0J12.0J00.64004
=+=+= →WKK
Work and Energy
381
Substitute numerical values in equation (1) and evaluate v4:
( ) m/s46.3kg3
J18.024 ==v
*26 •• Picture the Problem The work done by this force as it displaces the particle is the area under the curve of F as a function of x. Note that the constant C has units of N/m3. Because F varies with position non-linearly, express the work it does as an integral and evaluate the integral between the limits x = 1.5 m and x = 3 m:
( )
( ) [ ]( ) ( ) ( )[ ]
J19
m5.1m34
N/m
'N/m
''N/m
443
m3m5.1
4413
m3
m5.1
33
C
C
xC
dxxCW
=
−=
=
= ∫
27 •• Picture the Problem The work done on the dog by the leash as it stretches is the area under the curve of F as a function of x. We can find this area (the work Lou does holding the leash) by integrating the force function. Because F varies with position non-linearly, express the work it does as an integral and evaluate the integral between the limits x = 0 and x = x1:
( )
[ ]313
1212
1
03
312
21
0
2
1
1
''
'''
axkx
axkx
dxaxkxW
x
x
−−=
−−=
−−= ∫
28 •• Picture the Problem The work done on an object can be determined by finding the area bounded by its graph of Fx as a function of x and the x axis. We can find the kinetic energy and the speed of the particle at any point by using the work-kinetic energy theorem. (a) Express W, the area under the curve, in terms of the area of one square, Asquare, and the number of squares n:
W = n Asquare
Determine the work equivalent of one square:
W = (0.5 N)(0.25 m) = 0.125 J
Chapter 6
382
Estimate the number of squares under the curve between x = 0 and x = 2 m:
n ≈ 22
Substitute to determine W: W = 22(0.125 J) = J75.2
(b) Relate the kinetic energy of the object at x = 2 m, K2, to its initial kinetic energy, K0, and the work that was done on it between x = 0 and x = 2 m:
( )( )J4.11
J2.75m/s2.40kg3 221
2002
=
+=
+= →WKK
(c) Calculate the speed of the object at x = 2 m from its kinetic energy at the same location:
( ) m/s2.76kg3
J11.422 2 ===mKv
(d) Estimate the number of squares under the curve between x = 0 and x = 4 m:
n ≈ 26
Substitute to determine W: ( ) J25.3J125.026 ==W
(e) Relate the kinetic energy of the object at x = 4 m, K4, to its initial kinetic energy, K0, and the work that was done on it between x = 0 and x = 4 m:
( )( )J9.11
J25.3m/s2.40kg3 221
4004
=
+=
+= →WKK
Calculate the speed of the object at x = 4 m from its kinetic energy at the same location:
( ) m/s82.2kg3
J.91122 4 ===mKv
*29 •• Picture the Problem We can express the mass of the water in Margaret’s bucket as the difference between its initial mass and the product of the rate at which it loses water and her position during her climb. Because Margaret must do work against gravity in lifting and carrying the bucket, the work she does is the integral of the product of the gravitational field and the mass of the bucket as a function of its position. (a) Express the mass of the bucket and the water in it as a function of
( ) ryym −= kg40
Work and Energy
383
its initial mass, the rate at which it is losing water, and Margaret’s position, y, during her climb:
Find the rate, ymr
∆∆
= , at which
Margaret’s bucket loses water:
kg/m1m20kg20
==∆∆
=ymr
Substitute to obtain:
( ) yryymmkg1kg40kg40 −=−=
(b) Integrate the force Margaret exerts on the bucket, m(y)g, between the limits of y = 0 and y = 20 m:
( ) ( ) ( )[ ] kJ89.5'kg/m1'kg40m/s81.9''mkg1kg40 m20
02
212
m20
0
=−=⎟⎠⎞
⎜⎝⎛ −= ∫ yydyygW
Remarks: We could also find the work Margaret did on the bucket, at least approximately, by plotting a graph of m(y)g and finding the area under this curve between y = 0 and y = 20 m. Work, Energy, and Simple Machines 30 • Picture the Problem The free-body diagram shows the forces that act on the block as it slides down the frictionless incline. We can find the work done by these forces as the block slides 2 m by finding their components in the direction of, or opposite to, the motion. When we have determined the work done on the block, we can use the work-kinetic energy theorem or a constant-acceleration equation to calculate its kinetic energy and its speed at any given location.
Chapter 6
384
(a) incline. thelarly toperpendicu exerts
incline that theforce normal theand downward acts that force nalgravitatio a areblock on the acting forces that thesee wediagram,body -free theFrom
Identify the component of mg that acts down the incline and calculate the work done by it:
Fx = mg sin 60°
Express the work done by this force: °∆=∆= 60sinx xmgxFW
Substitute numerical values and evaluate W:
( )( ) ( )J102
60sinm2m/s9.81kg6 2
=
°=W
Remarks: Fn and mgcos60°, being perpendicular to the motion, do no work on the block
(b) The total work done on the block is the work done by the net force: ( )( )( )
J102
60sinm2m/s9.81kg6
60sin2
net
=
°=
°∆=∆= xmgxFW
(c) Express the change in the kinetic energy of the block in terms of the distance, ∆x, it has moved down the incline:
∆K = Kf – Ki = W = (mgsin60°)∆x or, because Ki = 0, Kf = W = (mgsin60°)∆x
Relate the speed of the block when it has moved a distance ∆x down the incline to its kinetic energy at that location:
°∆=
°∆==
60sin2
60sin22
xgmxmg
mKv
Determine this speed when ∆x = 1.5 m:
( )( )m/s05.5
60sinm1.5m/s9.812 2
=
°=v
(d) As in part (c), express the change in the kinetic energy of the block in terms of the distance, ∆x, it has moved down the incline and
∆K = Kf – Ki = W = (mg sin 60°)∆x and
Work and Energy
385
solve for Kf:
Kf = (mg sin 60°)∆x + Ki
Substitute for the kinetic energy terms and solve for vf to obtain:
2f 60sin2 ivxgv +∆°=
Substitute numerical values and evaluate vf:
( )( ) ( ) m/s43.5m/s2sin60m1.5m/s9.812 22f =+°=v
31 • Picture the Problem The free-body diagram shows the forces acting on the object as in moves along its circular path on a frictionless horizontal surface. We can use Newton’s 2nd law to obtain an expression for the tension in the string and the definition of work to determine the amount of work done by each force during one revolution.
(a) Apply ∑ = rr maF to the 2-kg
object and solve for the tension: ( ) ( )
N17.4
m3m/s2.5kg2
22
=
==rvmT
(b) From the FBD we can see that the forces acting on the object are:
ng and ,, FFTrrr
any work. do themof none object, theofmotion of
direction thelarly toperpendicuact forces theseof all Because
Chapter 6
386
*32 • Picture the Problem The free-body diagram, with F
rrepresenting the force
required to move the block at constant speed, shows the forces acting on the block. We can apply Newton’s 2nd law to the block to relate F to its weight w and then use the definition of the mechanical advantage of an inclined plane. In the second part of the problem we’ll use the definition of work. (a) Express the mechanical advantage M of the inclined plane: F
wM =
Apply ∑ = xx maF to the block: 0sin =− θwF because ax = 0.
Solve for F and substitute to obtain: θθ sin
1sin
==w
wM
Refer to the figure to obtain: L
H=θsin
Substitute to obtain: H
LM ==θsin
1
(b) Express the work done pushing the block up the ramp:
θsinramp mgLFLW ==
Express the work done lifting the block into the truck:
θsinlifting mgLmgHW == and
liftingramp WW =
33 • Picture the Problem We can find the work done per revolution in lifting the weight and the work done in each revolution of the handle and then use the definition of mechanical advantage. Express the mechanical advantage of the jack: F
WM =
Express the work done by the jack in one complete revolution (the weight W is raised a distance p):
WpW =lifting
Express the work done by the force F in one complete revolution:
RFW π2turning =
Work and Energy
387
Equate these expressions to obtain:
RFWp π2=
Solve for the ratio of W to F:
pR
FWM π2
==
Remarks: One does the same amount of work turning as lifting; exerting a smaller force over a greater distance. 34 • Picture the Problem The object whose weight is w
ris supported by two portions of the
rope resulting in what is known as a mechanical advantage of 2. The work that is done in each instance is the product of the force doing the work and the displacement of the object on which it does the work. (a) If w moves through a distance h: hF 2 distance a moves
(b) Assuming that the kinetic energy of the weight does not change, relate the work done on the object to the change in its potential energy to obtain:
whwhUW ==∆= θcos
(c) Because the force you exert on the rope and its displacement are in the same direction:
( ) ( )hFhFW 2cos2 == θ
Determine the tension in the ropes supporting the object:
02vertical =−=∑ wFF and
wF 21=
Substitute for F: ( ) ( ) whhwhFW === 22 2
1
(d) The mechanical advantage of the inclined plane is the ratio of the weight that is lifted to the force required to lift it, i.e.:
221
===w
wFwM
Remarks: Note that the mechanical advantage is also equal to the number of ropes supporting the load.
Chapter 6
388
Dot Products *35 • Picture the Problem Because θcosAB≡⋅ BA
rr we can solve for cosθ and use the fact
that AB−=⋅ BArr
to find θ. Solve for θ :
ABBArr
⋅= −1cosθ
Substitute for BA
rr⋅ and evaluate θ : ( ) °=−= − 1801cos 1θ
36 • Picture the Problem We can use its definition to evaluate BA
rr⋅ .
Express the definition of BA
rr⋅ : θcosAB=⋅ BA
rr
Substitute numerical values and evaluate BA
rr⋅ :
( )( )2m0.18
60cosm6m6
=
°=⋅ BArr
37 • Picture the Problem The scalar product of two-dimensional vectors A
rand B
ris AxBx +
AyBy. (a) For A
r= 3 i − 6 j and
Br
= −4 i + 2 j :
Ar
⋅ Br
= (3)( −4) + (−6)(2) = 24−
(b) For Ar
= 5 i + 5 j and
Br
= 2 i −4 j :
Ar
⋅ Br
= (5)(2) + (5)( −4) = 10−
(c) For Ar
= 6 i + 4 j and Br
= 4 i − 6 j : Ar
⋅ Br
= (6)(4) + (4)( −6) = 0
Work and Energy
389
38 • Picture the Problem The scalar product of two-dimensional vectors A
rand B
ris AB cos θ
= AxBx + AyBy. Hence the angle between vectors Ar
and Br
is given by
.cos 1
ABBABA yyxx +
= −θ
(a) For A
r= 3 i − 6 j and
Br
= −4 i + 2 j :
Ar
⋅ Br
= (3)( −4) + (−6)(2) = −24
( ) ( )( ) ( ) 2024
456322
22
=+−=
=−+=
B
A
and
°=−
= − 1432045
24cos 1θ
(b) For A
r= 5 i + 5 j and
Br
= 2 i − 4 j :
Ar
⋅ Br
= (5)(2) + (5)(−4) = -10
( ) ( )( ) ( ) 2042
505522
22
=−+=
=+=
B
A
and
°=−
= − 1082050
10cos 1θ
(c) For A
r= 6 i + 4 j and B
r= 4 i − 6 j : A
r⋅ Br
= (6)(4) + (4)( −6) = 0
( ) ( )( ) ( ) 5264
524622
22
=−+=
=+=
B
A
and
°== − 0.905252
0cos 1θ
39 • Picture the Problem The work W done by a force F
rduring a displacement ∆ sr for
which it is responsible is given by Fr
⋅∆ .sr
Chapter 6
390
(a) Using the definitions of work and the scalar product, calculate the work done by the given force during the specified displacement:
( )( )
( )( ) ( )( ) ( ) ( )[ ]J00.1
mN213132
ˆm2ˆm3ˆm3
ˆN1ˆN1ˆN2
=
⋅−+−+=−+⋅
+−=
∆⋅=
kji
kji
sF rrW
(b) Using the definition of work that includes the angle between the force and displacement vectors, solve for the component of F
rin the direction
of ∆ :sr
( ) sFsFW ∆=∆= θθ coscos
and
sWF∆
=θcos
Substitute numerical values and evaluate Fcosθ : ( ) ( ) ( )
N213.0
m2m3m3
J1cos222
=
−++=θF
40 •• Picture the Problem The component of a vector that is along another vector is the scalar product of the former vector and a unit vector that is parallel to the latter vector. (a) By definition, the unit vector that is parallel to the vector A
r is: 222
ˆˆˆˆ
zyx
zyxA
AAA
AAAA ++
++==
kjiAur
(b) Find the unit vector parallel to :B
r
( ) ( )jijiBu ˆ
54ˆ
53
43
ˆ4ˆ3ˆ22
+=+
+==
BB
r
The component of A
r along B
ris: ( )
( ) ( ) ( )( )
400.0
01541
532
ˆ54ˆ
53ˆˆˆ2ˆ
=
−+⎟⎠⎞
⎜⎝⎛−+⎟
⎠⎞
⎜⎝⎛=
⎟⎠⎞
⎜⎝⎛ +⋅−−=⋅ jikjiuA B
r
*41 •• Picture the Problem We can use the definitions of the magnitude of a vector and the dot product to show that if BABA
vrvr−=+ , then BA
rr⊥ .
Work and Energy
391
Express 2
BArr
+ : ( )22BABArrrr
+=+
Express BA
vr− :
( )22
BABArrrr
−=−
Equate these expressions to obtain: ( ) ( )22
BABAvrvr
−=+
Expand both sides of the equation to obtain:
2222 22 BABA +⋅−=+⋅+ BABArrrr
Simplify to obtain: 04 =⋅ BArr
or
0=⋅ BArr
From the definition of the dot product we have:
θcosAB=⋅ BArr
where θ is the angle between A
rand .B
v
Because neither Ar
nor Bv
is the zero vector:
0cos =θ ⇒ °= 90θ and .BArr
⊥
42 •• Picture the Problem The diagram shows the unit vectors BA ˆandˆ arbitrarily located in the 1st quadrant. We can express these vectors in terms of the unit vectors i and j and their x and y components. We can then form the dot product of Aand B to show that cos(θ1 − θ2) = cosθ1cosθ2 + sinθ1sinθ2. (a) Express A in terms of the unit vectors i and j :
jiA ˆˆˆyx AA +=
where
1cosθ=xA and 1sinθ=yA
Proceed as above to obtain:
jiB ˆˆˆyx BB +=
where
2cosθ=xB and 2sinθ=yB
(b) Evaluate BA ˆˆ ⋅ :
( )( )
2121
22
11
sinsincoscos
ˆsinˆcos
ˆsinˆcosˆˆ
θθθθθθ
θθ
+=+⋅
+=⋅
ji
jiBA
From the diagram we note that: ( )21cosˆˆ θθ −=⋅ BA
Chapter 6
392
Substitute to obtain: ( )21
2121
sinsincoscoscos
θθθθθθ
+=−
43 • Picture the Problem In (a) we’ll show that it does not follow that CB
rr= by giving a
counterexample. Let iA ˆ=
r, jiB ˆ4ˆ3 +=r
and
.ˆ4ˆ3 jiC −=r
Form BArr
⋅ and :CArr
⋅
( ) 3ˆ4ˆ3ˆ =+⋅=⋅ jiiBArr
and ( ) 3ˆ4ˆ3ˆ =−⋅=⋅ jiiCA
rr
. toequal
ynecessarilnot is that example
-counter aby shown ve We'No.
C
Br
r
44 •• Picture the Problem We can form the dot product of A
rand r
rand require that
1=⋅ rArr
to show that the points at the head of all such vectors rr lie on a straight line. We can use the equation of this line and the components of A
rto find the slope and
intercept of the line. (a) Let jiA ˆˆ
yx aa +=r
. Then: ( ) ( )1
ˆˆˆˆ
=+=
+⋅+=⋅
yaxayxaa
yx
yx jijirArr
Solve for y to obtain:
yy
x
ax
aay 1
+−=
which is of the form bmxy += and hence is the equation of a line.
(b) Given that jiA ˆ3ˆ2 −=r
: 32
32
=−
−=−=y
x
aam
and
31
311
−=−
==ya
b
Work and Energy
393
(c) The equation we obtained in (a) specifies all vectors whose component parallel to A
r has constant magnitude;
therefore, we can write such a vector as
BA
Arr
r
rr
+= 2 , where Br
is any vector
perpendicular to .Ar
This is shown graphically to the right.
Because all possible vectors B
r lie in a
plane, the resultant rr
must lie in a plane as well, as is shown above.
*45 •• Picture the Problem The rules for the differentiation of vectors are the same as those for the differentiation of scalars and scalar multiplication is commutative. (a) Differentiate rr ⋅ rr = r2 = constant: ( )
( ) 0constant
2
==
⋅=⋅+⋅=⋅
dtd
dtd
dtd
dtd rvrrrrrr rrr
rrrrr
Because :0=⋅ rv rr
rv rr⊥
(b) Differentiate v
r⋅ vr
= v2 = constant with respect to time:
( )
( ) 0constant
2
==
⋅=⋅+⋅=⋅
dtd
dtd
dtd
dtd vavvvvvv rrr
rrrrr
Because :0=⋅va rr
va rr⊥
. toel)antiparall(or parallel and and lar toperpendicu is that us tell)( and )( of results The
rra
r
rrba
(c) Differentiate vr ⋅ r
r = 0 with
respect to time: ( )
( ) 002 ==⋅+=
⋅+⋅=⋅
dtdv
dtd
dtd
dtd
ar
vrrvrv
rr
rr
rrrr
Chapter 6
394
Because 02 =⋅+ ar rrv : 2v−=⋅ ar rr (1)
Express ar in terms of θ, where θ is the angle between rr and ar :
θcosr aa =
Express ar rr⋅ : rcos rara ==⋅ θar rr
Substitute in equation (1) to obtain: 2
r vra −=
Solve for ar:
rva
2
r −=
Power 46 •• Picture the Problem The power delivered by a force is defined as the rate at which the
force does work; i.e., .dt
dWP =
Calculate the rate at which force A does work:
W0.5s10J5
==AP
Calculate the rate at which force B does work:
W0.6s5J3
==BP and AB PP >
47 • Picture the Problem The power delivered by a force is defined as the rate at which the
force does work; i.e., .vF rr⋅==
dtdWP
(a) If the box moves upward with a constant velocity, the net force acting it must be zero and the force that is doing work on the box is:
F = mg
The power input of the force is: mgvFvP ==
Substitute numerical values and evaluate P:
( )( )( ) W1.98m/s2m/s81.9kg5 2 ==P
Work and Energy
395
(b) Express the work done by the force in terms of the rate at which energy is delivered:
W = Pt = (98.1 W) (4 s) = J392
48 • Picture the Problem The power delivered by a force is defined as the rate at which the
force does work; i.e., .vF rr⋅==
dtdWP
(a) Using the definition of power, express Fluffy’s speed in terms of the rate at which he does work and the force he exerts in doing the work:
m/s2N3W6
===FPv
(b) Express the work done by the force in terms of the rate at which energy is delivered:
W = Pt = (6 W) (4 s) = J0.24
49 • Picture the Problem We can use Newton’s 2nd law and the definition of acceleration to express the velocity of this object as a function of time. The power input of the force accelerating the object is defined to be the rate at which it does work; i.e.,
.vF rr⋅== dtdWP
(a) Express the velocity of the object as a function of its acceleration and time:
v = at
Apply aF rrm=∑ to the object:
a = F/m
Substitute for a in the expression for v:
( )tttmFv 2
85 m/s
kg8N5
===
(b) Express the power input as a function of F and v and evaluate P:
( )( ) W/s13.3m/sN5 285 ttFvP ===
(c) Substitute t = 3 s: ( )( ) W38.9s3W/s13.3 ==P
Chapter 6
396
50 • Picture the Problem The power delivered by a force is defined as the rate at which the
force does work; i.e., .vF rr⋅==
dtdWP
(a) For F
r= 4 N i + 3 N k and
vr = 6 m/s i : ( ) ( )
W24.0
ˆm/s6ˆN3ˆN4
=
⋅+=⋅= ikivF rrP
(b) For F
r = 6 N i − 5 N j and
vr = − 5 m/s i + 4 m/s j : ( ) ( )W0.50
ˆm/s4ˆm/s5ˆN5ˆN6
−=
+−⋅−=
⋅=
jiji
vF rrP
(c) For F
r= 3 N i + 6 N j
and vr = 2 m/s i + 3 m/s j : ( ) ( )W0.24
ˆm/s3ˆm/s2ˆN6ˆN3
=
+⋅+=
⋅=
jiji
vF rrP
*51 • Picture the Problem Choose a coordinate system in which upward is the positive y direction. We can find inP from the given information that .27.0 inout PP = We can express
outP as the product of the tension in the cable T and the constant speed v of the dumbwaiter. We can apply Newton’s 2nd law to the dumbwaiter to express T in terms of its mass m and the gravitational field g. Express the relationship between the motor’s input and output power:
inout 27.0 PP = or
outin 7.3 PP =
Express the power required to move the dumbwaiter at a constant speed v:
TvP =out
Apply ∑ = yy maF to the dumbwaiter:
ymamgT =− or, because ay = 0,
mgT =
Substitute to obtain: mgvTvP 7.37.3in ==
Substitute numerical values and evaluate Pin:
( )( ) ( )W454
m/s0.35m/s9.81kg357.3 2in
=
=P
Work and Energy
397
52 •• Picture the Problem Choose a coordinate system in which upward is the positive y direction. We can express Pdrag as the product of the drag force Fdrag acting on the skydiver and her terminal velocity vt. We can apply Newton’s 2nd law to the skydiver to express Fdrag in terms of her mass m and the gravitational field g. (a) Express the power due to drag force acting on the skydiver as she falls at her terminal velocity vt:
tdragdrag vFP rrr⋅=
or, because dragFr
and tvr are antiparallel,
tdragdrag vFP −=
Apply ∑ = yy maF to the skydiver: ymamgF =−drag or, because ay = 0,
mgF =drag
Substitute to obtain, for the magnitude of Pdrag:
tdrag mgvP −= (1)
Substitute numerical values and evaluate P:
W1089.2)mi
km1.609s3600
h1h
mi 120()m/s (9.81kg) (55 42drag ×=××−=P
(b) Evaluate equation (1) with v = 15 mi/h:
( ) kW62.3)mi
km1.609s3600
h1h
mi15)m/s (9.81kg55 2drag =××⎟
⎠⎞
⎜⎝⎛−=P
*53 •• Picture the Problem Because, in the absence of air resistance, the acceleration of the cannonball is constant, we can use a constant-acceleration equation to relate its velocity to the time it has been in flight. We can apply Newton’s 2nd law to the cannonball to find the net force acting on it and then form the dot product of F
rand vr to express the rate at
which the gravitational field does work on the cannonball. Integrating this expression over the time-of-flight T of the ball will yield the desired result. Express the velocity of the cannonball as a function of time while it is in the air:
( ) jiv ˆˆ0)( 0 gtvt −+=r
Apply ∑ = aF rrm to the
cannonball to express the force acting on it while it is in the air:
jF ˆmg−=r
Evaluate vF rr⋅ : ( )
tmgmgv
gtvmg2
0
0ˆˆ
+−=
−⋅−=⋅ jjvF rr
Chapter 6
398
Relate vF rr
⋅ to the rate at which work is being done on the cannonball:
tmgmgvdt
dW 20 +−=⋅= vF rr
Separate the variables and integrate over the time T that the cannonball is in the air:
( )
TmgvTmg
dttmgmgvWT
022
21
0
20
−=
+−= ∫ (1)
Using a constant-acceleration equation, relate the speed v of the cannonball when it lands at the bottom of the cliff to its initial speed v0 and the height of the cliff H:
yavv ∆+= 220
2 or, because a = g and ∆y = H,
gHvv 220
2 +=
Solve for v to obtain: gHvv 22
0 +=
Using a constant-acceleration equation, relate the time-of-flight T to the initial and impact speeds of the cannonball:
gTvv −= 0
Solve for T to obtain: g
vvT −= 0
Substitute for T in equation (1) and simplify to evaluate W:
Kmvmv
gvvmgv
gvvvv
mgW
∆=−=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
+−=
202
1221
00
2
20
22
21
20
54 •• Picture the Problem If the particle is acted on by a single force, that force is the net force acting on the particle and is responsible for its acceleration. The rate at which energy is delivered by the force is .vF rr
⋅=P Express the rate at which this force does work in terms of vF rr
and :
vF rr⋅=P
The velocity of the particle, in terms of its acceleration and the time that the force has acted is:
tav rr=
Work and Energy
399
Using Newton’s 2nd law, substitute for a
r: t
mFvr
r=
Substitute for v
r in the expression
for P and simplify to obtain: tmFt
mt
mP
2
=⋅
=⋅=FFFFrrr
r
Potential Energy 55 • Picture the Problem The change in the gravitational potential energy of the earth-man system, near the surface of the earth, is given by ∆U = mg∆h, where ∆h is measured relative to an arbitrarily chosen reference position. Express the change in the man’s gravitational potential energy in terms of his change in elevation:
∆U = mg∆h
Substitute for m, g and ∆h and evaluate ∆U:
( )( ) ( )kJ4.71
m6m/s9.81kg80 2
=
=∆U
56 • Picture the Problem The water going over the falls has gravitational potential energy relative to the base of the falls. As the water falls, the falling water acquires kinetic energy until, at the base of the falls; its energy is entirely kinetic. The rate at which energy is delivered to the base of the falls is given by .// dtdUdtdWP −== Express the rate at which energy is being delivered to the base of the falls; remembering that half the potential energy of the water is converted to electric energy:
( )dtdmghmgh
dtd
dtdU
dtdWP
21
21 −=−=
−==
Substitute numerical values and evaluate P:
( )( )( )
MW879
kg/s101.4
m128m/s9.816
221
=
××
−−=P
Chapter 6
400
57 • Picture the Problem In the absence of friction, the sum of the potential and kinetic energies of the box remains constant as it slides down the incline. We can use the conservation of the mechanical energy of the system to calculate where the box will be and how fast it will be moving at any given time. We can also use Newton’s 2nd law to show that the acceleration of the box is constant and constant-acceleration equations to calculate where the box will be and how fast it will be moving at any given time.
(a) Express and evaluate the gravitational potential energy of the box, relative to the ground, at the top of the incline:
Ui = mgh = (2 kg) (9.81 m/s2) (20 m) = J392
(b) Using a constant-acceleration equation, relate the displacement of the box to its initial speed, acceleration and time-of-travel:
( )221
0 tatvx ∆+∆=∆
or, because v0 = 0, ( )2
21 tax ∆=∆
Apply ∑ = xx maF to the box as it
slides down the incline and solve for its acceleration:
θθ sinsin gamamg =⇒=
Substitute for a and evaluate ∆x(t = 1 s):
( ) ( )( )( )( )( )
m45.2
s1sin30m/s9.81
sins122
21
221
=
°=
∆=∆ tgx θ
Using a constant-acceleration equation, relate the speed of the box at any time to its initial speed and acceleration and solve for its speed when t = 1 s:
0where 00 =+= vatvv
and ( ) ( )
( ) ( )( )m/s91.4
s130sinm/s81.9sins12
=
°=
∆=∆= tgtav θ
Work and Energy
401
(c) Calculate the kinetic energy of the box when it has traveled for 1 s:
( )( )J1.24
m/s4.91kg2 2212
21
=
== mvK
Express the potential energy of the box after it has traveled for 1 s in terms of its initial potential energy and its kinetic energy:
J368
J24.1J923i
=
−=−= KUU
(d) Express the kinetic energy of the box at the bottom of the incline in terms of its initial potential energy and solve for its speed at the bottom of the incline:
J392221 === mvUK i
and
mUv i2
=
Substitute numerical values and evaluate v:
( ) m/s19.8kg2
J3922==v
58 • Picture the Problem The potential energy function U (x) is defined by the equation
( ) ( ) ∫−=−x
x
FdxxUxU0
.0 We can use the given force function to determine U(x) and then
the conditions on U to determine the potential functions that satisfy the given conditions.
(a) Use the definition of the potential energy function to find the potential energy function associated with Fx:
( ) ( )
( ) ( )
( )( )0
0
0
N6
'N60
0
xx
dxxU
dxFxUxU
x
x
x
xx
−−=
−=
−=
∫
∫
because U(x0) = 0.
(b) Use the result obtained in (a) to find U (x) that satisfies the condition that U(4 m) = 0:
( ) ( )( )m40
m4N6m4
0
0
=⇒=−−=
xxU
and ( ) ( )( )
( )xxxU
N6J42
m4N6
−=
−−=
Chapter 6
402
(c) Use the result obtained in (a) to find U that satisfies the condition that U(6 m) = 14 J:
( ) ( )( )m50J14
m6N6m6
0
0
=⇒=−−=
xxU
and
( ) ( )
( )x
xxU
N6J50
m325N6
−=
⎟⎠⎞
⎜⎝⎛ −−=
59 • Picture the Problem The potential energy of a stretched or compressed ideal spring Us is related to its force (stiffness) constant k and stretch or compression ∆x by .2
21
s kxU =
(a) Relate the potential energy stored in the spring to the distance it has been stretched:
221
s kxU =
Solve for x:
kUx s2
=
Substitute numerical values and evaluate x:
( ) m100.0N/m10
J5024 ==x
(b) Proceed as in (a) with Us = 100 J: ( ) m141.0
N/m10J1002
4 ==x
*60 •• Picture the Problem In a simple Atwood’s machine, the only effect of the pulley is to connect the motions of the two objects on either side of it; i.e., it could be replaced by a piece of polished pipe. We can relate the kinetic energy of the rising and falling objects to the mass of the system and to their common speed and relate their accelerations to the sum and difference of their masses … leading to simultaneous equations in m1 and m2.
Use the definition of the kinetic energy of the system to determine the total mass being accelerated:
( ) 2212
1 vmmK +=
and ( )
( )kg0.10
m/s4J8022
2221 ===+vKmm (1)
In Chapter 4, the acceleration of the masses was shown to be:
gmmmma
21
21
+−
=
Work and Energy
403
Because v(t) = at, we can eliminate a in the previous equation to obtain:
( ) gtmmmmtv
21
21
+−
=
Solve for 21 mm − : ( ) ( )gt
tvmmmm 2121
+=−
Substitute numerical values and evaluate 21 mm − :
( )( )( )( ) kg36.1
s3m/s9.81m/s4kg10
221 ==− mm (2)
Solve equations (1) and (2) simultaneously to obtain:
kg68.51 =m and kg32.42 =m
61 ••
Picture the Problem The gravitational potential energy of this system of two objects is the sum of their individual potential energies and is dependent on an arbitrary choice of where, or under what condition(s), the gravitational potential energy is zero. The best choice is one that simplifies the mathematical details of the expression of U. In this problem let’s choose U = 0 where θ = 0.
(a) Express U for the 2-object system as the sum of their gravitational potential energies; noting that because the object whose mass is m2 is above the position we have chosen for U = 0, its potential energy is positive while that of the object whose mass is m1 is negative:
( )
( ) θ
θθθ
sin
sinsin
1122
1122
21
gmm
gmgmUUU
ll
ll
−=
−=+=
(b) Differentiate U with respect toθ and set this derivative equal to zero to identify extreme values:
( ) 0cos1122 =−= θθ
gmmddU
ll
from which we can conclude that cosθ = 0 and θ = cos−10.
To be physically meaningful, :22 πθπ ≤≤−
2πθ ±=∴
Express the 2nd derivative of U with respect to θ and evaluate this derivative at :2πθ ±=
( ) θθ
sin11222
2
gmmd
Udll −−=
Chapter 6
404
If we assume, in the expression for U that we derived in (a), that m2l2 – m1l1 >0, then U(θ) is a sine function and, in the interval of interest, 22 πθπ ≤≤− , takes on
its minimum value when θ = −π/2:
02
2
2
>−πθd
Ud
and 2atminimumais πθ −=U
02
2
2
<πθd
Ud
and 2atmaximumais πθ =U
(c) If m1l1 = m2l2, then (m2l2 − m1l1) = 0
and . oftly independen 0 θ=U
Remarks: An alternative approach to establishing the U is a maximum at θ = π/2 is to plot its graph and note that, in the interval of interest, U is concave downward with its maximum value at θ = π/2. Force, Potential Energy, and Equilibrium 62 • Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is, dxdUFx −= . Consequently, given U as a function of
x, we can find Fx by differentiating U with respect to x.
(a) Evaluate :dxdUFx −= ( ) 34 4AxAx
dxdFx −=−=
(b) Set Fx = 0 and solve for x: 00 =⇒= xFx
63 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is dxdUFx −= . Consequently, given U as a function of
x, we can find Fx by differentiating U with respect to x.
(a) Evaluate :dxdUFx −=
2xC
xC
dxdFx =⎟
⎠⎞
⎜⎝⎛−=
(b) Because C > 0:
.originthefromawaydirected is
thereforeand 0for positive is
Fr
≠xFx
Work and Energy
405
(c) Because U is inversely proportional to x and C > 0:
( ) .increasingwithdecreases xxU
(d) With C < 0:
.origin thefrom towarddirected is
thereforeand 0for negative is
Fr
≠xFx
Because U is inversely proportional to x and C < 0, U(x) becomes less negative as x increases:
( ) .increasingwithincreases xxU
*64 •• Picture the Problem Fy is defined to be the negative of the derivative of the potential function with respect to y, i.e. dydUFy −= . Consequently, we can obtain Fy by
examining the slopes of the graph of U as a function of y.
The table to the right summarizes the information we can obtain from Figure 6-40:
Slope Fy Interval (N) (N) A→B −2 2 B→C transitional −2 → 1.4 C→D 1.4 −1.4
The graph of F as a function of y is shown to the right:
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5 6
y (m)
F (N
)
65 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, i.e. dxdUFx −= . Consequently, given F as a function of x,
we can find U by integrating Fx with respect to x. Evaluate the integral of Fx with respect to x:
( ) ( )
0
2
Uxa
dxxadxxFxU
+=
−=−= ∫∫
where U0 is a constant determined by whatever conditions apply to U.
Chapter 6
406
66 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is, dxdUFx −= . Consequently, given U as a function
of x, we can find Fx by differentiating U with respect to x. To determine whether the object is in stable or unstable equilibrium at a given point, we’ll evaluate 22 dxUd at
the point of interest.
(a) Evaluate :dxdUFx −= ( ) ( )1623 32 −=−−= xxxx
dxdFx
(b) We know that, at equilibrium, Fx = 0:
When Fx =0, 6x(x – 1) = 0. Therefore, the object is in equilibrium at m.1and0 == xx
(c) To decide whether the equilibrium at a particular point is stable or unstable, evaluate the 2nd derivative of the potential energy function at the point of interest:
( ) 232 6623 xxxxdxd
dxdU
−=−=
and
xdx
Ud 1262
2
−=
Evaluate 2
2
dxUd
at x = 0:
0atmequilibriustable
060
2
2
=⇒
>==
x
dxUd
x
Evaluate 2
2
dxUd
at x = 1 m:
m1atmequilibriuunstable
0126m1
2
2
=⇒
<−==
x
dxUd
x
67 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, i.e. dxdUFx −= . Consequently, given U as a function of x,
we can find Fx by differentiating U with respect to x. To determine whether the object is in stable or unstable equilibrium at a given point, we’ll evaluate 22 dxUd at the point of
interest.
Work and Energy
407
(a) Evaluate the negative of the derivative of U with respect to x:
( )( )( )224
1648 342
−+=
−=−−=
−=
xxx
xxxxdxddxdUFx
(b) The object is in equilibrium wherever Fnet = Fx = 0:
( )( ).m2and,0,m2arepoints
mequilibriuthe0224
−=
⇒=−+
x
xxx
(c) To decide whether the equilibrium at a particular point is stable or unstable, evaluate the 2nd derivative of the potential energy function at the point of interest:
( ) 232
2
1216416 xxxdxd
dxUd
−=−=
Evaluate 2
2
dxUd
at x = −2 m:
m2atmequilibriuunstable
032m2
2
2
−=⇒
<−=−=
x
dxUd
x
Evaluate 2
2
dxUd
at x = 0:
0atmequilibriustable
0160
2
2
=⇒
>==
x
dxUd
x
Evaluate 2
2
dxUd
at x = 2 m:
m2atmequilibriuunstable
032m2
2
2
=⇒
<−==
x
dxUd
x
Remarks: You could also decide whether the equilibrium positions are stable or unstable by plotting F(x) and examining the curve at the equilibrium positions. 68 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, i.e. dxdUFx −= . Consequently, given F as a function of x,
we can find U by integrating Fx with respect to x. Examination of 22 dxUd at extreme
points will determine the nature of the stability at these locations.
Chapter 6
408
Determine the equilibrium locations by setting Fnet = F(x) = 0:
F(x) = x3 – 4x = x(x2 – 4) = 0 ∴ the positions of stable and unstable equilibrium are at 2and0,2−=x .
Evaluate the negative of the integral of F(x) with respect to x:
( ) ( )( )
02
4
3
24
4
Uxx
dxxx
xFxU
++−=
−−=
−=
∫∫
where U0 is a constant whose value is determined by conditions on U(x).
Differentiate U(x) twice: xxFdxdU
x 43 +−=−=
and
43 22
2
+−= xdx
Ud
Evaluate 2
2
dxUd
at x = −2:
2 at unstable is mequilibriu the
082
2
2
−=∴
<−=−=
x
dxUd
x
Evaluate 2
2
dxUd
at x = 0:
0 at stable is mequilibriu the
040
2
2
=∴
>==
x
dxUd
x
Evaluate 2
2
dxUd
at x = 2:
2 at unstable is mequilibriu the
082
2
2
=∴
<−==
x
dxUd
x
Thus U(x) has a local minimum at x = 0 and
local maxima at x = ±2. 69 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, i.e. dxdUFx −= . Consequently, given U as a function of x,
we can find Fx by differentiating U with respect to x. To determine whether the object is in stable or unstable equilibrium at a given point, we can examine the graph of U.
Work and Energy
409
(a) Evaluate dxdUFx −= for x ≤ 3 m: ( ) ( )xxxx
dxdFx −=−−= 233 32
Set Fx = 0 to identify those values of x for which the 4-kg object is in equilibrium:
When Fx = 0, 3x(2 – x) = 0. Therefore, the object is in equilibrium at m.2and0 == xx
Evaluate dxdUFx −= for x > 3 m: 0=xF
because U = 0.
m. 3 for mequilibriu neutralin isobject theTherefore,
>x
(b) A graph of U(x) in the interval –1 m ≤ x ≤ 3 m is shown to the right:
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
x (m)
U (J
)
(c) From the graph, U(x) is a minimum at x = 0:
0atmequilibriustable =∴ x
From the graph, U(x) is a maximum at x = 2 m:
m2atmequilibriuunstable =∴ x
(d) Relate the kinetic energy of the object to its total energy and its potential energy:
UEmvK −== 221
Solve for v: ( )m
UEv −=
2
Evaluate U(x = 2 m): ( ) ( ) ( ) J4223m 2 32 =−==xU
Substitute in the equation for v to obtain:
( ) m/s00.2kg4
J4J122=
−=v
Chapter 6
410
70 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is dxdUFx −= . Consequently, given F as a function of
x, we can find U by integrating Fx with respect to x. (a) Evaluate the negative of the integral of F(x) with respect to x:
( ) ( )
02
3
21 U
xA
dxAxxFxU
+=
−=−= ∫∫ −
where U0 is a constant whose value is determined by conditions on U(x).
For x > 0: increases as decreases xU
(b) As x → ∞, 221
xA
→ 0: ∴ U0 = 0 and
( ) 322
3
2 mN4mN821
21
⋅=⋅
==xxx
AxU
(c) The graph of U(x) is shown to the right:
0
50
100
150
200
250
300
350
400
0.0 0.5 1.0 1.5 2.0
x ( m )
*71 ••• Picture the Problem Let L be the total length of one cable and the zero of gravitational potential energy be at the top of the pulleys. We can find the value of y for which the potential energy of the system is an extremum by differentiating U(y) with respect to y and setting this derivative equal to zero. We can establish that this value corresponds to a minimum by evaluating the second derivative of U(y) at the point identified by the first derivative. We can apply Newton’s 2nd law to the clock to confirm the result we obtain by examining the derivatives of U(y). (a) Express the potential energy of the system as the sum of the potential energies of the clock and counterweights:
( ) ( ) ( )yUyUyU weightsclock +=
Substitute to obtain: ( )222)( dyLMgmgyyU +−−−=
Work and Energy
411
(b) Differentiate U(y) with respect to y:
( )[ ]
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−−=
+−+−=
22
22
2
2)(
dyyMgmg
dyLMgmgydyd
dyydU
or
extremafor 0222
=+
−dy'
y'Mgmg
Solve for y′ to obtain: 22
2
4 mMmdy'
−=
Find ( )2
2
dyyUd
:
( ) 2322
2
222
2
2
2)(
dyMgd
dyyMgmg
dyd
dyyUd
+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+−−=
Evaluate ( )2
2
dyyUd
at y = y′: ( )( )
0
14
2
2
23
22
2
2322
2
2
2
>
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−
=
+=
mMm
Mgd
dyMgd
dyyUd
y'y'
and the potential energy is a minimum at
22
2
4 mMmdy
−=
(c) The FBD for the clock is shown to the right:
Apply ∑ = 0yF to the clock: 0sin2 =− mgMg θ
and
Mm
2sin =θ
Chapter 6
412
Express sinθ in terms of y and d: 22
sindy
y+
=θ
Substitute to obtain: 222 dy
yMm
+=
which is equivalent to the first equation in part (b).
it. fromaway displaced isit ifpoint mequilibriu thedback towar pulled be clock will the this,of Because
decreases. cables thefrom forcenet theupward, displaced isclock theif Similarly, clock. on the force upwardlarger a toleading increases,
downward, displaced isclock Ifthe m.equilibriu stable ofpoint a is This θ
Remarks: Because we’ve shown that the potential energy of the system is a minimum at y = y′ (i.e., U(y) is concave upward at that point), we can conclude that this point is one of stable equilibrium. General Problems *72 • Picture the Problem 25 percent of the electrical energy generated is to be diverted to do the work required to change the potential energy of the American people. We can calculate the height to which they can be lifted by equating the change in potential energy to the available energy. Express the change in potential energy of the population of the United States in this process:
∆U = Nmgh
Letting E represent the total energy generated in February 2002, relate the change in potential to the energy available to operate the elevator:
Nmgh = 0.25E
Solve for h:
NmgEh 25.0
=
Work and Energy
413
Substitute numerical values and evaluate h: ( )( )
( )( )( )km323
m/s9.81kg6010287h1
s3600hkW107.6025.0
26
9
=
×
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅×
=h
73 • Picture the Problem We can use the definition of the work done in changing the potential energy of a system and the definition of power to solve this problem. (a) Find the work done by the crane in changing the potential energy of its load:
W = mgh = (6×106 kg) (9.81 m/s2) (12 m) = MJ706
(b) Use the definition of power to find the power developed by the crane:
MW8.11s60MJ706
==≡dt
dWP
74 • Picture the Problem The power P of the engine needed to operate this ski lift is related to the rate at which it changes the potential energy U of the cargo of the gondolas according to P = ∆U/∆t. Because as many empty gondolas are descending as are ascending, we do not need to know their mass. Express the rate at which work is done as the cars are lifted:
tUP
∆∆
=
Letting N represent the number of gondola cars and M the mass of each, express the change in U as they are lifted a vertical displacement ∆h:
∆U = NMg∆h
Substitute to obtain: t
hNMgt
UP∆
∆=
∆∆
≡
Relate ∆h to the angle of ascent θ and the length L of the ski lift:
∆h = Lsinθ
Substitute for ∆h in the expression for P: t
NMgLP∆
=θsin
Chapter 6
414
Substitute numerical values and evaluate P:
( )( )( )( )( ) kW4.50
s/min60min60sin30km5.6m/s9.81kg55012 2
=°
=P
75 • Picture the Problem The application of Newton’s 2nd law to the forces shown in the free-body diagram will allow us to relate R to T. The unknown mass and speed of the object can be eliminated by introducing its kinetic energy.
Apply ∑ = radialradial maF the object
and solve for R: RmvT
2
= and T
mvR2
=
Express the kinetic energy of the object:
221 mvK =
Eliminate mv2 between the two equations to obtain: T
KR 2=
Substitute numerical values and evaluate R:
( ) m0.500N360J902
==R
*76 • Picture the Problem We can solve this problem by equating the expression for the gravitational potential energy of the elevated car and its kinetic energy when it hits the ground. Express the gravitational potential energy of the car when it is at a distance h above the ground:
U = mgh
Express the kinetic energy of the car when it is about to hit the ground:
221 mvK =
Equate these two expressions (because at impact, all the potential energy has been converted to kinetic energy) and solve for h:
gvh2
2
=
Work and Energy
415
Substitute numerical values and evaluate h:
( )( )[ ]( ) m3.39
m/s9.812sh/36001km/h100
2
2
==h
77 ••• Picture the Problem The free-body diagram shows the forces acting on one of the strings at the bridge. The force whose magnitude is F is one-fourth of the force (103 N) the bridge exerts on the strings. We can apply the condition for equilibrium in the y direction to find the tension in each string. Repeating this procedure at the site of the plucking will yield the restoring force acting on the string. We can find the work done on the string as it returns to equilibrium from the product of the average force acting on it and its displacement.
(a) Noting that, due to symmetry, T′ = T, apply 0=∑ yF to the string
at the point of contact with the bridge:
018sin2 =°− TF
Solve for and evaluate T: ( ) N7.4118sin2N103
18sin241
=°
=°
=FT
(b) A free-body diagram showing the forces restoring the string to its equilibrium position just after it has been plucked is shown to the right:
Express the net force acting on the string immediately after it is released:
θcos2net TF =
Use trigonometry to find θ: °=⎟⎟
⎠
⎞⎜⎜⎝
⎛×= − 6.88
cmmm10
mm4cm16.3tan 1θ
Substitute and evaluate Fnet: ( ) N68.1cos88.6N4.432net =°=F
Chapter 6
416
(c) Express the work done on the string in displacing it a distance dx′:
'FdxdW =
If we pull the string out a distance x′, the magnitude of the force pulling it down is approximately:
( ) '42'2 x
LT
LxTF ==
Substitute to obtain:
''4 dxxLTdW =
Integrate to obtain: 2
0
2''4 xLTdxx
LTW
x
== ∫
where x is the final displacement of the string.
Substitute numerical values to obtain: ( ) ( )
mJ09.4
m104m106.32
N7.412 232
=
××
= −−W
78 •• Picture the Problem Fx is defined to be the negative of the derivative of the potential function with respect to x, that is dxdUFx −= . Consequently, given F as a function of
x, we can find U by integrating Fx with respect to x. Evaluate the integral of Fx with respect to x:
( ) ( ) ( )0
331
2
Uax
dxaxdxxFxU
+=
−−=−= ∫∫
Apply the condition that U(0) = 0 to determine U0:
U(0) = 0 + U0 = 0 ⇒ U0 = 0 ( ) 3
31 axxU =∴
The graph of U(x) is shown to the right:
-3
-2
-1
0
1
2
3
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
x (m)
U (J
)
Work and Energy
417
*79 •• Picture the Problem We can use the definition of work to obtain an expression for the position-dependent force acting on the cart. The work done on the cart can be calculated from its change in kinetic energy. (a) Express the force acting on the cart in terms of the work done on it:
( )dx
dWxF =
Because U is constant: ( ) ( ) ( )[ ]xmC
Cxmdxdmv
dxdxF
2
2212
21
=
==
(b) The work done by this force changes the kinetic energy of the cart:
( )21
221
212
1212
1
202
1212
1
0
xmC
Cxmmv
mvmvKW
=
=−=
−=∆=
80 •• Picture the Problem The work done by F
rdepends on whether it causes a displacement
in the direction it acts. (a) Because F
r is along x-axis and
the displacement is along y-axis:
W = ∫ Fr
⋅ srd = 0
(b) Calculate the work done by Fr
during the displacement from x = 2 m to 5 m:
( )
( ) J0.783
N/m2
N/m2
m5
m2
32
m5
m2
22
=⎥⎦
⎤⎢⎣
⎡=
=⋅= ∫∫
x
dxxdW sF rr
81 •• Picture the Problem The velocity and acceleration of the particle can be found by differentiation. The power delivered to the particle can be expressed as the product of its velocity and the net force acting on it, and the work done by the force and can be found from the change in kinetic energy this work causes. In the following, if t is in seconds and m is in kilograms, then v is in m/s, a is in m/s2, P is in W, and W is in J.
Chapter 6
418
(a) The velocity of the particle is given by:
( )( )tt
ttdtd
dtdxv
86
42
2
23
−=
−==
The acceleration of the particle is given by:
( )( )812
86 2
−=
−==
t
ttdtd
dtdva
(b) Express and evaluate the rate at which energy is delivered to this particle as it accelerates:
( )( )( )81898
868122
2
+−=
−−=
==
ttmt
tttmmavFv P
(c) Because the particle is moving in such a way that its potential energy is not changing, the work done by the force acting on the particle equals the change in its kinetic energy:
( )( ) ( )( )[ ]( )[ ]
( )21
21
21
212
1
2212
1
01
432
086
0
−=
−−=
−=
−=∆=
tmt
ttm
vtvm
KKKW
Remarks: We could also find W by integrating P(t) with respect to time. 82 •• Picture the Problem We can calculate the work done by the given force from its definition. The power can be determined from vF rr
⋅=P and v from the change in kinetic energy of the particle produced by the work done on it. (a) Calculate the work done from its definition: ( )
J00.93
32
46
346
m3
0
32
m3
0
2
=⎥⎦
⎤⎢⎣
⎡−+=
−+=⋅= ∫∫
xxx
dxxxdW sF rr
(b) Express the power delivered to the particle in terms of Fx=3 m and its velocity:
vFP x m3==⋅= vF rr
Relate the work done on the particle to its kinetic energy and solve for its velocity:
0since 02
21
final ===∆= vmvKKW
Work and Energy
419
Solve for and evaluate v:
( ) m/s45.2kg3
J922===
mKv
Evaluate Fx=3 m: ( ) ( ) N933346 2
m3 −=−+==xF
Substitute for Fx=3 m and v: ( )( ) W1.22m/s2.45N9 −=−=P
*83 •• Picture the Problem We’ll assume that the firing height is negligible and that the bullet lands at the same elevation from which it was fired. We can use the equation
( ) θ2sin20 gvR = to find the range of the bullet and constant-acceleration equations to
find its maximum height. The bullet’s initial speed can be determined from its initial kinetic energy. Express the range of the bullet as a function of its firing speed and angle of firing:
θ2sin20
gvR =
Rewrite the range equation using the trigonometric identity sin2θ = 2sinθ cosθ:
gv
gvR θθθ cossin22sin 2
020 ==
Express the position coordinates of the projectile along its flight path in terms of the parameter t:
( )tvx θcos0=
and ( ) 2
21
0 sin gttvy −= θ
Eliminate the parameter t and make use of the fact that the maximum height occurs when the projectile is at half the range to obtain:
( )g
vh2sin 2
0 θ=
Equate R and h and solve the resulting equation for θ:
4tan =θ ⇒ °== − 0.764tan 1θ
Relate the bullet’s kinetic energy to its mass and speed and solve for the square of its speed:
mKvmvK 2 and 2
0202
1 ==
Substitute for 20v and θ and evaluate
R:
( )( )( ) ( )
km5.74
76sin2m/s9.81kg0.02
J120022
=
°=R
Chapter 6
420
84 •• Picture the Problem The work done on the particle is the area under the force-versus-displacement curve. Note that for negative displacements, F is positive, so W is negative for x < 0. (a) Use either the formulas for the areas of simple geometric figures or counting squares and multiplying by the work represented by one square to complete the table to the right:
x W (m) (J) −4 −11−3 −10−2 −7 −1 −3 0 0 1 1 2 0 3 −2 4 −3
(b) Choosing U(0) = 0, and using the definition of ∆U = −W, complete the third column of the table to the right:
x W ∆U (m) (J) (J) −4 −11 11 −3 −10 10 −2 −7 7 −1 −3 3 0 0 0 1 1 −1 2 0 0 3 −2 2 4 −3 3
The graph of U as a function of x is shown to the right:
-2
0
2
4
6
8
10
12
-4 -3 -2 -1 0 1 2 3 4
x (m)
U (J
)
Work and Energy
421
85 •• Picture the Problem The work done on the particle is the area under the force-versus-displacement curve. Note that for negative displacements, F is negative, so W is positive for x < 0. (a) Use either the formulas for the areas of simple geometric figures or counting squares and multiplying by the work represented by one square to complete the table to the right:
x W (m) (J) −4 6 −3 4 −2 2 −1 0.5 0 0 1 0.5 2 1.5 3 2.5 4 3
(b) Choosing U(0) = 0, and using the definition of ∆U = −W, complete the third column of the table to the right:
x W ∆U (m) (J) (J) −4 6 −6 −3 4 −4 −2 2 −2 −1 0.5 −0.5 0 0 0 1 0.5 −0.5 2 1.5 −1.5 3 2.5 −2.5 4 3 −3
The graph of U as a function of x is shown to the right:
-6
-5
-4
-3
-2
-1
0-4 -3 -2 -1 0 1 2 3 4
x (m)
U (
J)
Chapter 6
422
86 •• Picture the Problem The pictorial representation shows the box at its initial position 0 at the bottom of the inclined plane and later at position 1. We’ll assume that the block is at position 0. Because the surface is frictionless, the work done by the tension will change both the potential and kinetic energy of the block. We’ll use Newton’s 2nd law to find the acceleration of the block up the incline and a constant-acceleration equation to express v in terms of T, x, M, and θ. Finally, we can express the power produced by the tension in terms of the tension and the speed of the box.
(a) Use the definition of work to express the work the tension T does moving the box a distance x up the incline:
TxW =
(b) Apply xx MaF =∑ to the box: xMaMgT =− θsin
Solve for ax:
θθ sinsin gMT
MMgTax −=
−=
Using a constant-acceleration equation, express the speed of the box in terms of its acceleration and the distance x it has moved up the incline:
xavv x220
2 +=
or, because v0 = 0, xav x2=
Substitute for ax to obtain: xg
MTv ⎟
⎠⎞
⎜⎝⎛ −= θsin2
(c) The power produced by the tension in the string is given by: xg
MTTTvP ⎟
⎠⎞
⎜⎝⎛ −== θsin2
Work and Energy
423
87 ••• Picture the Problem We can use the definition of the magnitude of vector to show that the magnitude of F
ris F0 and the definition of the scalar product to show that its direction
is perpendicular to rr
. The work done as the particle moves in a circular path can be found from its definition.
(a) Express the magnitude of Fr
:
220
20
20
22
yxrF
xrFy
rF
FF yx
+=
⎟⎠⎞
⎜⎝⎛−+⎟
⎠⎞
⎜⎝⎛=
+=Fr
Because 22 yxr += : 0
0220 FrrF
yxrF
==+=Fr
Form the scalar product of F
rand rr : ( ) ( )
( ) 0
ˆˆˆˆ
0
0
=−⎟⎠⎞
⎜⎝⎛=
+⋅−⎟⎠⎞
⎜⎝⎛=⋅
xyxyrF
yxxyrF jijirF
rr
Because F
r⋅ rr
= 0, rF rr⊥
(b) Because F
r⊥ rr
, Fr
is tangential to the circle and constant. At (5 m, 0), F
r points in the j− direction. If
srd is in the j− direction, dW > 0.
The work it does in one revolution is:
( ) ( )( )clockwise is
rotation theif m10m522
0
00
FFrFW
πππ
===
and ( )
ckwise.counterclo isrotation theif m10 0FW π−=
ve.conservatinot is circuit, complete afor 0 Because ckwise.counterclo
isrotation theifm)(10 clockwise, isrotation theif m)(10 00
Fr
≠
−=
W
FFW ππ
*88 ••• Picture the Problem We can substitute for r and ji ˆˆ yx + in F
rto show that the
magnitude of the force varies as the inverse of the square of the distance to the origin, and that its direction is opposite to the radius vector. We can find the work done by this force by evaluating the integral of F with respect to x from an initial position x = 2 m, y = 0 m to a final position x = 5 m, y = 0 m. Finally, we can apply Newton’s 2nd law to the particle to relate its speed to its radius, mass, and the constant b.
Chapter 6
424
(a) Substitute for r and ji ˆˆ yx + in F
rto obtain: ( ) rF ˆ22
2322yx
yxb
+⎟⎟⎠
⎞⎜⎜⎝
⎛
+−=
r
where r is a unit vector pointing from the origin toward the point of application of Fr
.
Simplify to obtain: rrF ˆˆ1
222 rb
yxb −=⎟⎟
⎠
⎞⎜⎜⎝
⎛+
−=r
i.e., the magnitude of the force varies as the inverse of the square of the distance to the origin, and its direction is antiparallel (opposite) to the radius vector .ˆˆ jir yx +=
r
(b) Find the work done by this force by evaluating the integral of F with respect to x from an initial position x = 2 m, y = 0 m to a final position x = 5 m, y = 0 m:
J 900.0m 21
m 51mN 3
'1'
'
2
m5
m2
m5
m22
−=⎟⎠⎞
⎜⎝⎛ −⋅=
⎥⎦⎤
⎢⎣⎡=−= ∫ x
bdxxbW
(c) velocity. thelar toperpendicu is force theas done is work No
(d) Because the particle is moving in a circle, the force on the particle must be supplying the centripetal acceleration keeping it moving in the circle. Apply ∑ = cr maF to the particle:
rvm
rb 2
2 =
Solve for v:
mrbv =
Substitute numerical values and evaluate v: ( )( ) m/s 463.0
m7kg2mN3 2
=⋅
=v
89 ••• Picture the Problem A spreadsheet program to calculate the potential is shown below. The constants used in the potential function and the formula used to calculate the ″6-12″ potential are as follows:
Cell Content/Formula Algebraic FormB2 1.09×10−7 a B3 6.84×10−5 b
Work and Energy
425
D8 $B$2/C8^12−$B$3/C8^6612 r
bra
−
C9 C8+0.1 rr ∆+ (a)
A B C D 1 2 a = 1.09E-07 3 b = 6.84E-05 4 5 6 7 r U 8 3.00E-01 1.11E-01 9 3.10E-01 6.13E-02 10 3.20E-01 3.08E-02 11 3.30E-01 1.24E-02 12 3.40E-01 1.40E-03 13 3.50E-01 −4.95E-03
45 6.70E-01 −7.43E-0446 6.80E-01 −6.81E-0447 6.90E-01 −6.24E-0448 7.00E-01 −5.74E-04
The graph shown below was generated from the data in the table shown above. Because the force between the atomic nuclei is given by ( )drdUF −= , we can conclude that the shape of the potential energy function supports Feynman’s claim.
"6-12" Potential
-0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
r (nm)
U (e
V)
(b) The minimum value is about −0.0107 eV, occurring at a separation of approximately 0.380 nm. Because the function is concave upward (a potential ″well″) at this separation,
Chapter 6
426
this separation is one of stable equilibrium, although very shallow. (c) Relate the force of attraction between two argon atoms to the slope of the potential energy function: 713
612
612rb
ra
rb
ra
drd
drdUF
−=
⎥⎦⎤
⎢⎣⎡ −−=−=
Substitute numerical values and evaluate F(5 Å):
( )( )
( )( )
N1069.6
m10nm1
eVJ101.6
nmeV1018.4
nm5.01084.66
nm5.01009.112
12
9
192
7
5
13
7
−
−
−−
−−
×−=
××
××−=×
−×
=F
where the minus sign means that the force is attractive.
Substitute numerical values and evaluate F(3.5 Å):
( )( )
( )( )
N1049.7
m10nm1
eVJ101.6
nmeV1068.4
nm35.01084.66
nm35.01009.112
11
9
191
7
5
13
7
−
−
−−
−−
×=
××
××=×
−×
=F
where the plus sign means that the force is repulsive. *90 ••• Picture the Problem A spreadsheet program to plot the Yukawa potential is shown below. The constants used in the potential function and the formula used to calculate the Yukawa potential are as follows:
Cell Content/Formula Algebraic FormB1 4 U0 B2 2.5 a D8 −$B$1*($B$2/C9)*EXP(−C9/$B$2) are
raU /
0−⎟
⎠⎞
⎜⎝⎛−
C10 C9+0.1 rr ∆+ (a)
A B C D
1 U0= 4 pJ 2 a= 2.5 fm 3
7 8 r U 9 0.5 −16.37
10 0.6 −13.11
Work and Energy
427
11 0.7 −10.80 12 0.8 −9.08 13 0.9 −7.75 14 1 −6.70
64 6 −0.15 65 6.1 −0.14 66 6.2 −0.14 67 6.3 −0.13 68 6.4 −0.12 69 6.5 −0.11 70 6.6 −0.11
U as a function of r is shown below.
-18
-16
-14
-12
-10
-8
-6
-4
-2
00 1 2 3 4 5 6 7
r (fm)
U (p
J)
(b) Relate the force between the nucleons to the slope of the potential energy function:
( ) ( )
⎟⎠⎞
⎜⎝⎛ +−=
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−−=
−=
−
−
rraeU
eraU
drd
drrdUrF
ar
ar
12
/0
0
(c) Evaluate F(2a): ( )
( )
⎟⎠⎞
⎜⎝⎛−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
−
−
aeU
aaaeUaF aa
43
21
22
20
2/2
0
Chapter 6
428
Evaluate F(a): ( )
( )
⎟⎠⎞
⎜⎝⎛−=⎟
⎠⎞
⎜⎝⎛ +−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
−−
−
aeU
aaeU
aaaeUaF aa
211
1
10
10
2/
0
Express the ratio F(2a)/F(a):
( )( )
138.0
83
243
2 1
10
20
=
=⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
= −
−
−
e
aeU
aeU
aFaF
(d) Evaluate F(5a): ( )
( )
⎟⎠⎞
⎜⎝⎛−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
−
−
aeU
aaaeUaF aa
256
51
55
50
2/5
0
Express the ratio F(5a)/F(a):
( )( )
3
4
10
50
1020.2
253
225
65
−
−
−
−
×=
=⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛−
= e
aeU
aeU
aFaF
437
Chapter 7 Conservation of Energy Conceptual Problems *1 • Determine the Concept Because the peg is frictionless, mechanical energy is conserved as this system evolves from one state to another. The system moves and so we know that ∆K > 0. Because ∆K + ∆U = constant, ∆U < 0. correct. is )( a
2 • Determine the Concept Choose the zero of gravitational potential energy to be at ground level. The two stones have the same initial energy because they are thrown from the same height with the same initial speeds. Therefore, they will have the same total energy at all times during their fall. When they strike the ground, their gravitational potential energies will be zero and their kinetic energies will be equal. Thus, their speeds at impact will be equal. The stone that is thrown at an angle of 30° above the horizontal has a longer flight time due to its initial upward velocity and so they do not strike the ground at the same time. correct. is )( c
3 • (a) False. Forces that are external to a system can do work on the system to change its energy. (b) False. In order for some object to do work, it must exert a force over some distance. The chemical energy stored in the muscles of your legs allows your muscles to do the work that launches you into the air. 4 • Determine the Concept Your kinetic energy increases at the expense of chemical energy. *5 • Determine the Concept As she starts pedaling, chemical energy inside her body is converted into kinetic energy as the bike picks up speed. As she rides it up the hill, chemical energy is converted into gravitational potential and thermal energy. While freewheeling down the hill, potential energy is converted to kinetic energy, and while braking to a stop, kinetic energy is converted into thermal energy (a more random form of kinetic energy) by the frictional forces acting on the bike. *6 • Determine the Concept If we define the system to include the falling body and the earth, then no work is done by an external agent and ∆K + ∆Ug + ∆Etherm= 0. Solving for the change in the gravitational potential energy we find ∆Ug = −(∆K + friction energy).
Chapter 7
438
correct. is )( b
7 •• Picture the Problem Because the constant friction force is responsible for a constant acceleration, we can apply the constant-acceleration equations to the analysis of these statements. We can also apply the work-energy theorem with friction to obtain expressions for the kinetic energy of the car and the rate at which it is changing. Choose the system to include the earth and car and assume that the car is moving on a horizontal surface so that ∆U = 0. (a) A constant frictional force causes a constant acceleration. The stopping distance of the car is related to its speed before the brakes were applied through a constant-acceleration equation.
0. where220
2 =∆+= vsavv
0. where2
20 <
−=∆∴ a
avs
Thus, ∆s ∝ 20v and statement (a) is false.
(b) Apply the work-energy theorem with friction to obtain:
smgWK ∆−=−=∆ kf µ
Express the rate at which K is dissipated: t
smgtK
∆∆
−=∆∆
kµ
Thus, v
tK
∝∆∆
and therefore not constant.
Statement (b) is false.
(c) In part (b) we saw that: K ∝ ∆s
Because ∆s ∝ ∆t: K ∝ ∆t and statement (c) is false.
Because none of the above are correct: correct. is )( d
8 • Picture the Problem We’ll let the zero of potential energy be at the bottom of each ramp and the mass of the block be m. We can use conservation of energy to predict the speed of the block at the foot of each ramp. We’ll consider the distance the block travels on each ramp, as well as its speed at the foot of the ramp, in deciding its descent times. Use conservation of energy to find the speed of the blocks at the bottom of each ramp:
0=∆+∆ UK or
0topbottopbot =−+− UUKK
Conservation of Energy
439
Because Ktop = Ubot = 0:
0topbot =−UK
Substitute to obtain:
02bot2
1 =− mgHmv
Solve for vbot: gHv 2bot = independently of the shape of the ramp.
Because the block sliding down the circular arc travels a greater distance (an arc length is greater than the length of the chord it defines) but arrives at the bottom of the ramp with the same speed that it had at the bottom of the inclined plane, it will require more time to arrive at the bottom of the arc. correct. is )(b
9 •• Determine the Concept No. From the work-kinetic energy theorem, no total work is being done on the rock, as its kinetic energy is constant. However, the rod must exert a tangential force on the rock to keep the speed constant. The effect of this force is to cancel the component of the force of gravity that is tangential to the trajectory of the rock. Estimation and Approximation *10 •• Picture the Problem We’ll use the data for the "typical male" described above and assume that he spends 8 hours per day sleeping, 2 hours walking, 8 hours sitting, 1 hour in aerobic exercise, and 5 hours doing moderate physical activity. We can approximate his energy utilization using activityactivityactivity tAPE ∆= , where A is the surface area of his body, Pactivity is the rate of energy consumption in a given activity, and ∆tactivity is the time spent in the given activity. His total energy consumption will be the sum of the five terms corresponding to his daily activities. (a) Express the energy consumption of the hypothetical male: act. aerobicact. mod.
sittingwalkingsleeping
EE
EEEE
++
++=
Evaluate Esleeping:
( )( )( )( )J1030.2
s/h3600h8W/m40m26
22
sleepingsleepingsleeping
×=
=
∆= tAPE
Evaluate Ewalking:
( )( )( )( )J1030.2
s/h3600h2W/m160m26
22
walkingwalkingwalking
×=
=
∆= tAPE
Evaluate Esitting:
( )( )( )( )J1046.3
s/h3600h8W/m60m26
22
sittingsittingsitting
×=
=
∆= tAPE
Chapter 7
440
Evaluate Emod. act.:
( )( )( )( )J1030.6
s/h3600h5W/m175m26
22act. mod.act. mod.act. mod.
×=
=
∆= tAPE
Evaluate Eaerobic act.:
( )( )( )( )J1016.2
s/h3600h1W/m300m26
22act. aerobicact. aerobicact. aerobic
×=
=
∆= tAPE
Substitute to obtain:
J105.16
J1016.2J1030.6J1046.3J1030.2J1030.2
6
66
666
×=
×+×+
×+×+×=E
Express the average metabolic rate represented by this energy consumption:
( )( ) W191s/h3600h24J1016.5 6
av =×
=∆
=t
EP
or about twice that of a 100 W light bulb.
(b) Express his average energy consumption in terms of kcal/day: kcal/day3940
J/kcal4190J/day1016.5 6
=×
=E
(c) kcal/lb22.5lb175kcal3940
= is higher than the estimate given in the statement of the
problem. However, by adjusting the day's activities, the metabolic rate can vary by more than a factor of 2. 11 • Picture the Problem The rate at which you expend energy, i.e., do work, is defined as power and is the ratio of the work done to the time required to do the work. Relate the rate at which you can expend energy to the work done in running up the four flights of stairs and solve for your running time:
PWt
tWP ∆
=∆⇒∆
∆=
Express the work done in climbing the stairs:
mghW =∆
Substitute for ∆W to obtain: P
mght =∆
Conservation of Energy
441
Assuming that your weight is 600 N, evaluate ∆t:
( )( ) s6.33W250
m3.54N600=
×=∆t
12 • Picture the Problem The intrinsic rest energy in matter is related to the mass of matter through Einstein’s equation .2
0 mcE =
(a) Relate the rest mass consumed to the energy produced and solve for and evaluate m:
202
0 cEmmcE =⇒= (1)
( ) kg1011.1m/s10998.2
J1 1728
−×=×
=m
(b) Express the energy required as a function of the power of the light bulb and evaluate E:
( )( )
J1047.9
hs3600
dh24
yd365.24
y10W10033
10×=
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛×
== PtE
Substitute in equation (1) to obtain:
( ) g05.1m/s10998.2
J1047.928
10
µ=×
×=m
*13 • Picture the Problem There are about 3×108 people in the United States. On the assumption that the average family has 4 people in it and that they own two cars, we have a total of 1.5×108 automobiles on the road (excluding those used for industry). We’ll assume that each car uses about 15 gal of fuel per week. Calculate, based on the assumptions identified above, the total annual consumption of energy derived from gasoline:
( ) J/y103.04galJ102.6weeks52
weekautogal15auto101.5 1988 ×=⎟⎟
⎠
⎞⎜⎜⎝
⎛×⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⋅×
y
Express this rate of energy use as a fraction of the total annual energy use by the US:
%6J/y105
J/y103.0420
19
≈×
×
Remarks: This is an average power expenditure of roughly 9x1011 watt, and a total cost (assuming $1.15 per gallon) of about 140 billion dollars per year. 14 • Picture the Problem The energy consumption of the U.S. works out to an average power consumption of about 1.6×1013 watt. The solar constant is roughly 103 W/m2 (reaching
Chapter 7
442
the ground), or about 120 W/m2 of useful power with a 12% conversion efficiency. Letting P represent the daily rate of energy consumption, we can relate the power available at the surface of the earth to the required area of the solar panels using IAP = . Relate the required area to the electrical energy to be generated by the solar panels:
IAP = where I is the solar intensity that reaches the surface of the Earth.
Solve for and evaluate A: ( )
211
2
13
m1067.2W/m120
W101.62
×=
×==
IPA
where the factor of 2 comes from the fact that the sun is only up for roughly half the day.
Find the side of a square with this area: km516m1067.2 211 =×=s
Remarks: A more realistic estimate that would include the variation of sunlight over the day and account for latitude and weather variations might very well increase the area required by an order of magnitude. 15 • Picture the Problem We can relate the energy available from the water in terms of its mass, the vertical distance it has fallen, and the efficiency of the process. Differentiation of this expression with respect to time will yield the rate at which water must pass through its turbines to generate Hoover Dam’s annual energy output. Assuming a total efficiencyη, use the expression for the gravitational potential energy near the earth’s surface to express the energy available from the water when it has fallen a distance h:
mghE η=
Differentiate this expression with respect to time to obtain:
[ ]dtdVgh
dtdmghmgh
dtdP ηρηη ===
Solve for dV/dt: gh
PdtdV
ηρ= (1)
Using its definition, relate the dam’s annual power output to the energy produced: t
EP∆∆
=
Substitute numerical values to obtain: ( )( ) W1057.4
h/d24d365.24hkW104 8
9
×=⋅×
=P
Conservation of Energy
443
Substitute in equation (1) and evaluate dV/dt: ( )( )( )
L/s1010.1
m211m/s9.81kg/L12.0W1057.4
6
2
8
×=
×=
dtdV
The Conservation of Mechanical Energy 16 • Picture the Problem The work done in compressing the spring is stored in the spring as potential energy. When the block is released, the energy stored in the spring is transformed into the kinetic energy of the block. Equating these energies will give us a relationship between the compressions of the spring and the speeds of the blocks.
Let the numeral 1 refer to the first case and the numeral 2 to the second case. Relate the compression of the spring in the second case to its potential energy, which equals its initial kinetic energy when released:
( )( )211
2112
1
2222
1222
1
18
34
vm
vm
vmkx
=
=
=
Relate the compression of the spring in the first case to its potential energy, which equals its initial kinetic energy when released:
2112
1212
1 vmkx =
or 21
211 kxvm =
Substitute to obtain: 21
222
1 18kxkx =
Solve for x2: 12 6xx =
17 • Picture the Problem Choose the zero of gravitational potential energy to be at the foot of the hill. Then the kinetic energy of the woman on her bicycle at the foot of the hill is equal to her gravitational potential energy when she has reached her highest point on the hill.
Equate the kinetic energy of the rider at the foot of the incline and her gravitational potential energy when she has reached her highest point on the hill and solve for h:
gvhmghmv2
22
21 =⇒=
Relate her displacement along the d = h/sinθ
Chapter 7
444
incline d to h and the angle of the incline: Substitute for h to obtain:
gvd2
sin2
=θ
Solve for d:
θsin2
2
gvd =
Substitute numerical values and evaluate d:
( )( ) m4.97
sin3m/s9.812m/s10
2
2
=°
=d
and correct. is )( c
*18 • Picture the Problem The diagram shows the pendulum bob in its initial position. Let the zero of gravitational potential energy be at the low point of the pendulum’s swing, the equilibrium position. We can find the speed of the bob at it passes through the equilibrium position by equating its initial potential energy to its kinetic energy as it passes through its lowest point.
Equate the initial gravitational potential energy and the kinetic energy of the bob as it passes through its lowest point and solve for v:
hgv
mvhmg
∆=
=∆
2
and
221
Express ∆h in terms of the length L of the pendulum: 4
Lh =∆
Substitute and simplify:
2gLv =
19 • Picture the Problem Choose the zero of gravitational potential energy to be at the foot of the ramp. Let the system consist of the block, the earth, and the ramp. Then there are
Conservation of Energy
445
no external forces acting on the system to change its energy and the kinetic energy of the block at the foot of the ramp is equal to its gravitational potential energy when it has reached its highest point.
Relate the gravitational potential energy of the block when it has reached h, its highest point on the ramp, to its kinetic energy at the foot of the ramp:
221 mvmgh =
Solve for h: g
vh2
2
=
Relate the displacement d of the block along the ramp to h and the angle the ramp makes with the horizontal:
d = h/sinθ
Substitute for h: g
vd2
sin2
=θ
Solve for d:
θsin2
2
gvd =
Substitute numerical values and evaluate d:
( )( ) m89.3
sin40m/s9.812m/s7
2
2
=°
=d
20 • Picture the Problem Let the system consist of the earth, the block, and the spring. With this choice there are no external forces doing work to change the energy of the system. Let Ug = 0 at the elevation of the spring. Then the initial gravitational potential energy of the 3-kg object is transformed into kinetic energy as it slides down the ramp and then, as it compresses the spring, into potential energy stored in the spring. (a) Apply conservation of energy to relate the distance the spring is compressed to the initial potential energy of the block:
0ext =∆+∆= UKW
and, because ∆K = 0, 02
21 =+− kxmgh
Solve for x:
kmghx 2
=
Chapter 7
446
Substitute numerical values and evaluate x:
( )( )( )
m858.0
N/m400m5m/s9.81kg32 2
=
=x
(b) The energy stored in the compressed spring will accelerate the block, launching it back up the incline:
m. 5 ofheight a torising path, its retrace block will The
21 • Picture the Problem With Ug chosen to be zero at the uncompressed level of the spring, the ball’s initial gravitational potential energy is negative. The difference between the initial potential energy of the spring and the gravitational potential energy of the ball is first converted into the kinetic energy of the ball and then into gravitational potential energy as the ball rises and slows … eventually coming momentarily to rest.
Apply the conservation of energy to the system as it evolves from its initial to its final state:
mghkxmgx =+− 221
Solve for h: x
mgkxh −=2
2
Substitute numerical values and evaluate h:
( )( )( )( )
m05.5
m05.0m/s9.81kg0.0152m0.05N/m600
2
2
=
−=h
22 • Picture the Problem Let the system include the earth and the container. Then the work done by the crane is done by an external force and this work changes the energy of the system. Because the initial and final speeds of the container are zero, the initial and final kinetic energies are zero and the work done by the crane equals the change in the gravitational potential energy of the container. Choose Ug = 0 to be at the level of the deck of the freighter.
Apply conservation of energy to the system:
UKEW ∆+∆=∆= sysext
Because ∆K = 0:
hmgUW ∆=∆=ext
Conservation of Energy
447
Evaluate the work done by the crane: ( )( )( )
kJ314
m8m/s9.81kg4000 2
ext
−=
−=
∆= hmgW
23 • Picture the Problem Let the system consist of the earth and the child. Then Wext = 0. Choose Ug,i = 0 at the child’s lowest point as shown in the diagram to the right. Then the child’s initial energy is entirely kinetic and its energy when it is at its highest point is entirely gravitational potential. We can determine h from energy conservation and then use trigonometry to determine θ.
Using the diagram, relate θ to h and L:
⎟⎠⎞
⎜⎝⎛ −=⎟
⎠⎞
⎜⎝⎛ −
= −−
Lh
LhL 1coscos 11θ
Apply conservation of energy to the system to obtain:
02i2
1 =− mghmv
Solve for h: g
vh2
2i=
Substitute to obtain:
⎟⎟⎠
⎞⎜⎜⎝
⎛−= −
gLv
21cos
2i1θ
Substitute numerical values and evaluate θ :
( )( )( )
°=
⎟⎟⎠
⎞⎜⎜⎝
⎛−= −
6.25
m6m/s9.812m/s3.41cos 2
21θ
*24 •• Picture the Problem Let the system include the two objects and the earth. Then Wext = 0. Choose Ug = 0 at the elevation at which the two objects meet. With this choice, the initial potential energy of the 3-kg object is positive and that of the 2-kg object is negative. Their sum, however, is positive. Given our choice for Ug = 0, this initial potential energy is transformed entirely into kinetic energy.
Apply conservation of energy: 0gext =∆+∆= UKW
or, because Wext = 0,
Chapter 7
448
∆K = −∆Ug
Substitute for ∆K and solve for vf; noting that m represents the sum of the masses of the objects as they are both moving in the final state:
g2i2
12f2
1 Umvmv ∆−=−
or, because vi = 0,
mU
v gf
2∆−=
Express and evaluate ∆Ug:
( )( )( )
J91.4m/s9.81
m0.5kg2kg302
ig,fg,g
−=×
−−=
−=∆ UUU
Substitute and evaluate vf: ( ) m/s1.40
kg5J4.912
f =−−
=v
25 •• Picture the Problem The free-body diagram shows the forces acting on the block when it is about to move. Fsp is the force exerted by the spring and, because the block is on the verge of sliding, fs = fs,max. We can use Newton’s 2nd law, under equilibrium conditions, to express the elongation of the spring as a function of m, k and θ and then substitute in the expression for the potential energy stored in a stretched or compressed spring.
Express the potential energy of the spring when the block is about to move:
221 kxU =
Apply ,m∑ = aF rrunder equilibrium
conditions, to the block: ∑
∑
=−=
=−−=
0cosand
0sin
n
maxs,sp
θ
θ
mgFF
mgfFF
y
x
Using fs,max = µsFn and Fsp = kx, eliminate fs,max and Fsp from the x equation and solve for x:
( )k
mgx θµθ cossin s+=
Conservation of Energy
449
Substitute for x in the expression for U:
( )
( )[ ]k
mg
kmgkU
2cossin
cossin
2s
2s
21
θµθ
θµθ
+=
⎥⎦⎤
⎢⎣⎡ +
=
26 •• Picture the Problem The mechanical energy of the system, consisting of the block, the spring, and the earth, is initially entirely gravitational potential energy. Let Ug = 0 where the spring is compressed 15 cm. Then the mechanical energy when the compression of the spring is 15 cm will be partially kinetic and partially stored in the spring. We can use conservation of energy to relate the initial potential energy of the system to the energy stored in the spring and the kinetic energy of block when it has compressed the spring 15 cm. Apply conservation of energy to the system:
0=∆+∆ KU or
0ifis,fs,ig,fg, =−+−+− KKUUUU
Because Ug,f = Us,I = Ki = 0:
0ffs,ig, =++− KUU
Substitute to obtain:
( ) 02212
21 =+++− mvkxxhmg
Solve for v:
( )m
kxxhgv2
2 −+=
Substitute numerical values and evaluate v:
( )( ) ( )( ) m/s00.8kg2.4
m0.15N/m3955m0.15m5m/s9.8122
2 =−+=v
Chapter 7
450
*27 •• Picture the Problem The diagram represents the ball traveling in a circular path with constant energy. Ug has been chosen to be zero at the lowest point on the circle and the superimposed free-body diagrams show the forces acting on the ball at the top and bottom of the circular path. We’ll apply Newton’s 2nd law to the ball at the top and bottom of its path to obtain a relationship between TT and TB and the conservation of mechanical energy to relate the speeds of the ball at these two locations. Apply ∑ = radialradial maF to the ball
at the bottom of the circle and solve for TB:
RvmmgT
2B
B =−
and
RvmmgT
2B
B += (1)
Apply ∑ = radialradial maF to the ball
at the top of the circle and solve for TT:
RvmmgT
2T
T =+
and
RvmmgT
2T
T +−= (2)
Subtract equation (2) from equation (1) to obtain:
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−
+=−
Rvmmg
RvmmgTT
2T
2B
TB
mgRvm
Rvm 2
2T
2B +−= (3)
Using conservation of energy, relate the mechanical energy of the ball at the bottom of its path to its mechanical energy at the top of the
circle and solve for Rvm
Rvm
2T
2B − :
( )Rmgmvmv 22T2
12B2
1 +=
mgRvm
Rvm 4
2T
2B =−
Substitute in equation (3) to obtain: mgTT 6TB =−
Conservation of Energy
451
28 •• Picture the Problem Let Ug = 0 at the lowest point in the girl’s swing. Then we can equate her initial potential energy to her kinetic energy as she passes through the low point on her swing to relate her speed v to R. The FBD show the forces acting on the girl at the low point of her swing. Applying Newton’s 2nd law to her will allow us to establish the relationship between the tension T and her speed.
Apply ∑ = radialradial maF to the girl
at her lowest point and solve for T:
RvmmgT
RvmmgT
2
2
and
+=
=−
Equate the girl’s initial potential energy to her final kinetic energy
and solve for Rv2
:
gRvmvRmg =⇒=
22
21
2
Substitute for v2/R2 and simplify to obtain:
mgmgmgT 2=+=
29 •• Picture the Problem The free-body diagram shows the forces acting on the car when it is upside down at the top of the loop. Choose Ug = 0 at the bottom of the loop. We can express Fn in terms of v and R by apply Newton’s 2nd law to the car and then obtain a second expression in these same variables by applying the conservation of mechanical energy. The simultaneous solution of these equations will yield an expression for Fn in terms of known quantities.
Apply ∑ = radialradial maF to the car
at the top of the circle and solve for Fn:
RvmmgF
2
n =+
and
mgRvmF −=
2
n (1)
Chapter 7
452
Using conservation of energy, relate the energy of the car at the beginning of its motion to its energy when it is at the top of the loop:
( )RmgmvmgH 2221 +=
Solve for Rvm
2
: ⎟⎠⎞
⎜⎝⎛ −= 22
2
RHmg
Rvm (2)
Substitute equation (2) in equation (1) to obtain:
⎟⎠⎞
⎜⎝⎛ −=
−⎟⎠⎞
⎜⎝⎛ −=
52
22n
RHmg
mgRHmgF
Substitute numerical values and evaluate Fn:
( ) ( ) ( ) N1067.15m7.5m232m/s9.81kg1500 42
n ×=⎥⎦
⎤⎢⎣
⎡−=F ⇒ correct. is )( c
30 • Picture the Problem Let the system include the roller coaster, the track, and the earth and denote the starting position with the numeral 0 and the top of the second hill with the numeral 1. We can use the work-energy theorem to relate the energies of the coaster at its initial and final positions.
(a) Use conservation of energy to relate the work done by external forces to the change in the energy of the system:
UKEW ∆+∆=∆= sysext
Because the track is frictionless, Wext = 0:
0=∆+∆ UK and
00101 =−+− UUKK
Substitute to obtain:
001202
1212
1 =−+− mghmghmvmv
Solve for v0: ( )01210 2 hhgvv −+=
If the coaster just makes it to the top of the second hill, v1 = 0 and:
( )010 2 hhgv −=
Conservation of Energy
453
Substitute numerical values and evaluate v0:
( )( )m/s9.40
m5m9.5m/s9.812 20
=
−=v
(b) hills. two theof heights in the
difference on theonly depends speed required that theNote No.
31 •• Picture the Problem Let the radius of the loop be R and the mass of one of the riders be m. At the top of the loop, the centripetal force on her is her weight (the force of gravity). The two forces acting on her at the bottom of the loop are the normal force exerted by the seat of the car, pushing up, and the force of gravity, pulling down. We can apply Newton’s 2nd law to her at both the top and bottom of the loop to relate the speeds at those locations to m and R and, at b, to F, and then use conservation of energy to relate vt and vb. Apply radialradial maF =∑ to the rider at the bottom of the circular arc:
RvmmgF
2b=−
Solve for F to obtain: R
vmmgF2b+= (1)
Apply radialradial maF =∑ to the rider at the top of the circular arc:
Rvmmg
2t=
Solve for 2t v : gRv =2
t
Use conservation of energy to relate the energies of the rider at the top and bottom of the arc:
0tbtb =−+− UUKK or, because Ub = 0,
0ttb =−− UKK
Substitute to obtain:
022t2
12b2
1 =−− mgRmvmv
Solve for 2bv : gRvb 52 =
Substitute in equation (1) to obtain: mg
RgRmmgF 65
=+=
i.e., the rider will feel six times heavier than her normal weight.
Chapter 7
454
*32 •• Picture the Problem Let the system consist of the stone and the earth and ignore the influence of air resistance. Then Wext = 0. Choose Ug = 0 as shown in the figure. Apply the law of the conservation of mechanical energy to describe the energy transformations as the stone rises to the highest point of its trajectory.
Apply conservation of energy:
0ext =∆+∆= UKW
and 00101 =−+− UUKK
Because U0 = 0:
0101 =+− UKK
Substitute to obtain:
02212
21 =+− mgHmvmvx
In the absence of air resistance, the horizontal component of vr is constant and equal to vx = vcosθ. Hence:
( ) 0cos 2212
21 =+− mgHmvvm θ
Solve for v:
θ2cos12
−=
gHv
Substitute numerical values and evaluate v:
( )( ) m/s2.2753cos1
m24m/s9.8122
2
=°−
=v
33 •• Picture the Problem Let the system consist of the ball and the earth. Then Wext = 0. The figure shows the ball being thrown from the roof of a building. Choose Ug = 0 at ground level. We can use the conservation of mechanical energy to determine the maximum height of the ball and its speed at impact with the ground. We can use the definition of the work done by gravity to calculate how much work was done by gravity as the ball rose to its maximum height. (a) Apply conservation of energy: 0ext =∆+∆= UKW
Conservation of Energy
455
or 01212 =−+− UUKK
Substitute for the energies to obtain:
012212
1222
1 =−+− mghmghmvmv
Note that, at point 2, the ball is moving horizontally and:
θcos12 vv =
Substitute for v2 and h2:
( )0
cos
1
212
1212
1
=−
+−
mghmgHmvvm θ
Solve for H: ( )1cos
22
2
1 −−= θg
vhH
Substitute numerical values and evaluate H:
( )( )( )
m0.31
140cosm/s9.812
m/s30m21 22
2
=
−°−=H
(b) Using its definition, express the work done by gravity:
( )( ) ( )ii
g i
hHmgmghmgH
UUUW hH
−−=−−=
−−=∆−=
Substitute numerical values and evaluate Wg:
( )( )( )J7.31
m12m31m/s9.81kg0.17 2g
−=
−−=W
(c) Relate the initial mechanical energy of the ball to its just-before-impact energy:
2f2
1i
2i2
1 mvmghmv =+
Solve for vf: i2if 2ghvv +=
Substitute numerical values and evaluate vf
( ) ( )( )m/s7.33
m12m/s9.812m/s30 22f
=
+=v
Chapter 7
456
34 •• Picture the Problem The figure shows the pendulum bob in its release position and in the two positions in which it is in motion with the given speeds. Choose Ug = 0 at the low point of the swing. We can apply the conservation of mechanical energy to relate the two angles of interest to the speeds of the bob at the intermediate and low points of its trajectory. (a) Apply conservation of energy: 0ext =∆+∆= UKW
or
.zeroequalandwhere0
if
ifif
KUUUKK =−+−
0if =−∴ UK
Express Ui: ( )0i cos1 θ−== mgLmghU
Substitute for Kf and Ui: ( ) 0cos1 0
2f2
1 =−− θmgLmv
Solve for θ0:
⎟⎟⎠
⎞⎜⎜⎝
⎛−= −
gLv
21cos
21
0θ
Substitute numerical values and evaluate θ0:
( )( )( )
°=
⎥⎦
⎤⎢⎣
⎡−= −
0.60
m0.8m/s9.812m/s2.81cos 2
21
0θ
(b) Letting primed quantities describe the indicated location, use the law of the conservation of mechanical energy to relate the speed of the bob at this point to θ :
.0where0
i
ifif
==−+−
KU'UK'K
0iff =−+∴ U'U'K
Express 'U f : ( )θcos1f −== mgLmgh'U '
Substitute for iff and, U'U'K : ( ) ( )
( ) 0cos1cos1
0
2f2
1
=−−−+
θθ
mgLmgL'vm
Conservation of Energy
457
Solve for θ : ( )⎥⎦
⎤⎢⎣
⎡+= −
0
2f1 cos
2'cos θθ
gLv
Substitute numerical values and evaluate θ :
( )( )( )
°=
⎥⎦
⎤⎢⎣
⎡°+= −
3.51
60cosm0.8m/s9.812
m/s4.1cos 2
21θ
*35 •• Picture the Problem Choose Ug = 0 at the bridge, and let the system be the earth, the jumper and the bungee cord. Then Wext = 0. Use the conservation of mechanical energy to relate to relate her initial and final gravitational potential energies to the energy stored in the stretched bungee, Us cord. In part (b), we’ll use a similar strategy but include a kinetic energy term because we are interested in finding her maximum speed. (a) Express her final height h above the water in terms of L, d and the distance x the bungee cord has stretched:
h = L – d − x (1)
Use the conservation of mechanical energy to relate her gravitational potential energy as she just touches the water to the energy stored in the stretched bungee cord:
0ext =∆+∆= UKW
Because ∆K = 0 and ∆U = ∆Ug + ∆Us, ,02
21 =+− kxmgL
where x is the maximum distance the bungee cord has stretched.
Solve for k: 2
2xmgLk =
Find the maximum distance the bungee cord stretches:
x = 310 m – 50 m = 260 m.
Evaluate k: ( )( )( )( )
N/m40.5m260
m310m/s9.81kg6022
2
=
=k
Chapter 7
458
Express the relationship between the forces acting on her when she has finally come to rest and solve for x:
0net =−= mgkxF
and
kmgx =
Evaluate x: ( )( ) m109
N/m5.40m/s9.81kg60 2
==x
Substitute in equation (1) and evaluate h:
m151m109m50m310 =−−=h
(b) Using conservation of energy, express her total energy E:
0isg ==++= EUUKE
Because v is a maximum when K is a maximum, solve for K::
( ) 221
sg
kxxdmg
UUK
−+=
−−= (1)
Use the condition for an extreme value to obtain:
valuesextremefor 0=−= kxmgdxdK
Solve for and evaluate x: ( )( ) m109N/m5.40
m/s9.81kg60 2
===k
mgx
From equation (1) we have: ( ) 2
212
21 kxxdmgmv −+=
Solve for v to obtain:
( )m
kxxdgv2
2 −+=
Substitute numerical values and evaluate v for x = 109 m:
( )( ) ( )( ) m/s3.45kg60
m109N/m5.4m109m50m/s9.8122
2 =−+=v
Because ,02
2
<−= kdx
Kd x = 109 m corresponds to Kmax and so v is a maximum.
Conservation of Energy
459
36 •• Picture the Problem Let the system be the earth and pendulum bob. Then Wext = 0. Choose Ug = 0 at the low point of the bob’s swing and apply the law of the conservation of mechanical energy to its motion. When the bob reaches the 30° position its energy will be partially kinetic and partially potential. When it reaches its maximum height, its energy will be entirely potential. Applying Newton’s 2nd law will allow us to express the tension in the string as a function of the bob’s speed and its angular position.
(a) Apply conservation of energy to relate the energies of the bob at points 1 and 2: 0
or0
1212
ext
=−+−
=∆+∆=
UUKK
UKW
Because U1 = 0:
02212
1222
1 =+− Umvmv
Express U2: ( )θcos12 −= mgLU
Substitute for U2 to obtain:
( ) 0cos1212
1222
1 =−+− θmgLmvmv
Solve for v2: ( )θcos12212 −−= gLvv
Substitute numerical values and evaluate v2:
( ) ( )( )( ) m/s52.3cos301m3m/s9.812m/s4.5 222 =°−−=v
(b) From (a) we have:
( )θcos12 −= mgLU
Substitute numerical values and evaluate U2:
( )( )( )( )J89.7
cos301m3m/s9.81kg2 22
=
°−=U
(c) Apply ∑ = radialradial maF to the bob to
obtain:
LvmmgT
22cos =− θ
Solve for T: ⎟⎟⎠
⎞⎜⎜⎝
⎛+=
LvgmT
22cosθ
Chapter 7
460
Substitute numerical values and evaluate T:
( ) ( ) ( ) N3.25m3m/s3.52cos30m/s9.81kg2
22 =⎥
⎦
⎤⎢⎣
⎡+°=T
(d) When the bob reaches its greatest height:
( )
0and
cos1
max1
maxmax
=+
−==
UK
mgLUU θ
Substitute for K1 and Umax ( ) 0cos1 max
212
1 =−+− θmgLmv
Solve for θmax:
⎟⎟⎠
⎞⎜⎜⎝
⎛−= −
gLv
21cos
211
maxθ
Substitute numerical values and evaluate θmax:
( )( )( )
°=
⎥⎦
⎤⎢⎣
⎡−= −
0.49
m3m/s9.812m/s4.51cos 2
21
maxθ
37 •• Picture the Problem Let the system consist of the earth and pendulum bob. Then Wext = 0. Choose Ug = 0 at the bottom of the circle and let points 1, 2 and 3 represent the bob’s initial point, lowest point and highest point, respectively. The bob will gain speed and kinetic energy until it reaches point 2 and slow down until it reaches point 3; so it has its maximum kinetic energy when it is at point 2. We can use Newton’s 2nd law at points 2 and 3 in conjunction with the law of the conservation of mechanical energy to find the maximum kinetic energy of the bob and the tension in the string when the bob has its maximum kinetic energy.
(a) Apply ∑ = radialradial maF to the
bob at the top of the circle and solve for 2
3v :
Lvmmg
23=
and gLv =2
3
Conservation of Energy
461
Use conservation of energy to express the relationship between K2, K3 and U3 and solve for K2:
0where0 22323 ==−+− UUUKK
Therefore,
( )Lmgmv
UKKK
2232
1
33max2
+=
+==
Substitute for 2
3v and simplify to
obtain:
( ) mgLmgLgLmK 25
21
max 2 =+=
(b) Apply ∑ = radialradial maF to the
bob at the bottom of the circle and solve for T2:
LvmmgTF
22
2net =−=
and
LvmmgT
22
2 += (1)
Use conservation of energy to relate the energies of the bob at points 2 and 3 and solve for K2:
0where0 22323 ==−+− UUUKK
( )Lmgmv
UKK
2232
1
332
+=
+=
Substitute for 2
3v and K2 and solve
for 22v :
( ) ( )LmggLmmv 2212
221 +=
and gLv 52
2 =
Substitute in equation (1) to obtain: mgT 62 =
38 •• Picture the Problem Let the system consist of the earth and child. Then Wext = 0. In the figure, the child’s initial position is designated with the numeral 1; the point at which the child releases the rope and begins to fall with a 2, and its point of impact with the water is identified with a 3. Choose Ug = 0 at the water level. While one could use the law of the conservation of energy between points 1 and 2 and then between points 2 and 3, it is more direct to consider the energy transformations between points 1 and 3. Given our choice of the zero of gravitational potential energy, the initial potential energy at point 1 is transformed into kinetic energy at point 3.
Chapter 7
462
Apply conservation of energy to the energy transformations between points 1 and 3:
0ext =∆+∆= UKW
zero.areandwhere0
13
1313
KUUUKK =−+−
Substitute for K3 and U1; ( )[ ] 0cos1232
1 =−+− θLhmgmv
Solve for v3: ( )[ ]θcos123 −+= Lhgv
Substitute numerical values and evaluate v3:
( ) ( )( )[ ] m/s91.8cos231m10.6m3.2m/s9.812 23 =°−+=v
*39 •• Picture the Problem Let the system consist of you and the earth. Then there are no external forces to do work on the system and Wext = 0. In the figure, your initial position is designated with the numeral 1, the point at which you release the rope and begin to fall with a 2, and your point of impact with the water is identified with a 3. Choose Ug = 0 at the water level. We can apply Newton’s 2nd law to the forces acting on you at point 2 and apply conservation of energy between points 1 and 2 to determine the maximum angle at which you can begin your swing and then between points 1 and 3 to determine the speed with which you will hit the water.
(a) Use conservation of energy to relate your speed at point 2 to your potential energy there and at point 1:
0ext =∆+∆= UKW
or 01212 =−+− UUKK
Because K1 = 0: ( )[ ] 0cos1
222
1
=+−−+
mghmgLmghmv
θ
Solve this equation for θ :
⎥⎦
⎤⎢⎣
⎡−= −
gLv
21cos
221θ (1)
Apply ∑ = radialradial maF to
LvmmgT
22=−
Conservation of Energy
463
yourself at point 2 and solve for T: and
LvmmgT
22+=
Because you’ve estimated that the rope might break if the tension in it exceeds your weight by 80 N, it must be that: ( )
mLv
Lvm
N80or
N80
22
22
=
=
Let’s assume your weight is 650 N. Then your mass is 66.3 kg and:
( )( ) 2222 /sm55.5
66.3kgm4.6N80
==v
Substitute numerical values in equation (1) to obtain: ( )( )
°=
⎥⎦
⎤⎢⎣
⎡−= −
2.20
m4.6m/s9.812/sm5.551cos 2
221θ
(b) Apply conservation of energy to the energy transformations between points 1 and 3:
0ext =∆+∆= UKW
zeroareandwhere0
1
31313
KUUUKK =−+−
Substitute for K3 and U1 to obtain: ( )[ ] 0cos1232
1 =−+− θLhmgmv
Solve for v3:
( )[ ]θcos123 −+= Lhgv
Substitute numerical values and evaluate v3:
( ) ( )( )[ ] m/s39.6cos20.21m4.6m8.1m/s9.812 23 =°−+=v
Chapter 7
464
40 •• Picture the Problem Choose Ug = 0 at point 2, the lowest point of the bob’s trajectory and let the system consist of the bob and the earth. Given this choice, there are no external forces doing work on the system. Because θ << 1, we can use the trigonometric series for the sine and cosine functions to approximate these functions. The bob’s initial energy is partially gravitational potential and partially potential energy stored in the stretched spring. As the bob swings down to point 2 this energy is transformed into kinetic energy. By equating these energies, we can derive an expression for the speed of the bob at point 2.
Apply conservation of energy to the system as the pendulum bob swings from point 1 to point 2:
( )θcos12212
221 −+= mgLkxmv
Note, from the figure, that x ≈ Lsinθ when θ << 1:
( ) ( )θθ cos1sin 2212
221 −+= mgLLkmv
Also, when θ << 1:
2211cosandsin θθθθ −≈≈
Substitute, simplify and solve for v2:
Lg
mkLv += θ2
Conservation of Energy
465
41 ••• Picture the Problem Choose Ug = 0 at point 2, the lowest point of the bob’s trajectory and let the system consist of the earth, ceiling, spring, and pendulum bob. Given this choice, there are no external forces doing work to change the energy of the system. The bob’s initial energy is partially gravitational potential and partially potential energy stored in the stretched spring. As the bob swings down to point 2 this energy is transformed into kinetic energy. By equating these energies, we can derive an expression for the speed of the bob at point 2.
Apply conservation of energy to the system as the pendulum bob swings from point 1 to point 2:
( )θcos12212
221 −+= mgLkxmv (1)
Apply the Pythagorean theorem to the lower triangle in the diagram to obtain: ( ) ( )[ ] [ ] ( )θθθθθθ cos3coscos3sincossin 4
132249222
23222
21 −=+−+=+=+ LLLLx
Take the square root of both sides of the equation to obtain:
( )θcos3413
21 −=+ LLx
Solve for x: ( )[ ]21
413 cos3 −−= θLx
Substitute for x in equation (1):
( )[ ] ( )θθ cos1cos32
21
4132
212
221 −+−−= mgLkLmv
Solve for 2
2v to obtain:
( ) [ ]( ) ( ) ⎥⎦
⎤⎢⎣⎡ −−+−=
−−+−=
2
21
4132
2
21
41322
2
cos3cos12
cos3cos12
θθ
θθ
mk
LgL
LmkgLv
Chapter 7
466
Finally, solve for v2:
( ) ( ) 2
21
413
2 cos3cos12 −−+−= θθmk
LgLv
The Conservation of Energy 42 • Picture the Problem The energy of the eruption is initially in the form of the kinetic energy of the material it thrusts into the air. This energy is then transformed into gravitational potential energy as the material rises. (a) Express the energy of the eruption in terms of the height ∆h to which the debris rises:
hmgE ∆=
Relate the density of the material to its mass and volume:
Vm
=ρ
Substitute for m to obtain: hVgE ∆= ρ
Substitute numerical values and evaluate E:
( )( )( )( ) J1014.3m500m/s9.81km4kg/m1600 16233 ×==E
(b) Convert 3.13×1016 J to megatons of TNT:
TNTMton48.7J104.2
TNTMton1J1014.3J1014.3 151616 =
×××=×
43 •• Picture the Problem The work done by the student equals the change in his/her gravitational potential energy and is done as a result of the transformation of metabolic energy in the climber’s muscles. (a) The increase in gravitational potential energy is: ( )( )( )
kJ2.94
m120m/s9.81kg80 2
=
=
∆=∆ hmgU
Conservation of Energy
467
(b) body. the
in storedenergy chemical from comes work thisdo torequiredenergy The
(c) Relate the chemical energy expended by the student to the change in his/her potential energy and solve for E:
UE ∆=2.0 and
( ) kJ471kJ94.255 ==∆= UE
Kinetic Friction 44 • Picture the Problem Let the car and the earth be the system. As the car skids to a stop on a horizontal road, its kinetic energy is transformed into internal (i.e., thermal) energy. Knowing that energy is transformed into heat by friction, we can use the definition of the coefficient of kinetic friction to calculate its value. (a) The energy dissipated by friction is given by:
thermEsf ∆=∆
Apply the work-energy theorem for problems with kinetic friction:
sfEEEW ∆+∆=∆+∆= mechthermmechext or, because imech KKE −=∆=∆ and
Wext = 0, sfmv ∆+−= 2
i210
Solve for f∆s to obtain:
2i2
1 mvsf =∆
Substitute numerical values and evaluate f∆s:
( )( ) kJ625m/s25kg2000 221 ==∆sf
(b) Relate the kinetic friction force to the coefficient of kinetic friction and the weight of the car and solve for the coefficient of kinetic friction:
mgfmgf k
kkk =⇒= µµ
Express the relationship between the energy dissipated by friction and the kinetic friction force and solve fk:
sEfsfE∆
∆=⇒∆=∆ therm
kktherm
Substitute to obtain: smg
E∆
∆= therm
kµ
Chapter 7
468
Substitute numerical values and evaluate µk: ( )( )( )
531.0
m60m/s9.81kg2000kJ625
2k
=
=µ
45 • Picture the Problem Let the system be the sled and the earth. Then the 40-N force is external to the system. The free-body diagram shows the forces acting on the sled as it is pulled along a horizontal road. The work done by the applied force can be found using the definition of work. To find the energy dissipated by friction, we’ll use Newton’s 2nd law to determine fk and then use it in the definition of work. The change in the kinetic energy of the sled is equal to the net work done on it. Finally, knowing the kinetic energy of the sled after it has traveled 3 m will allow us to solve for its speed at that location.
(a) Use the definition of work to calculate the work done by the applied force:
( )( ) J10430cosm3N40
cosext
=°=
=⋅≡ θFsW sF rr
(b) Express the energy dissipated by friction as the sled is dragged along the surface:
xFxfE ∆=∆=∆ nktherm µ
Apply ∑ = yy maF to the sled and
solve for Fn:
0sinn =−+ mgFF θ
and θsinn FmgF −=
Substitute to obtain: ( )θµ sinktherm FmgxE −∆=∆
Substitute numerical values and evaluate ∆Etherm:
( )( ) ( )( )[( ) ]
J2.70
sin30N40m/s9.81kg8m34.0 2
therm
=
°−=∆E
(c) Apply the work-energy theorem sfEEEW ∆+∆=∆+∆= mechthermmechext
Conservation of Energy
469
for problems with kinetic friction:
or, because UKE ∆+∆=∆ mech and
∆U = 0, thermext EKW ∆+∆=
Solve for and evaluate ∆K to obtain:
J33.8
J70.2J041thermext
=
−=∆−=∆ EWK
(d) Because Ki = 0: 2
f21
f mvKK =∆=
Solve for vf:
mKv ∆
=2
f
Substitute numerical values and evaluate vf:
( ) m/s2.91kg8
J33.82f ==v
*46 • Picture the Problem Choose Ug = 0 at the foot of the ramp and let the system consist of the block, ramp, and the earth. Then the kinetic energy of the block at the foot of the ramp is equal to its initial kinetic energy less the energy dissipated by friction. The block’s kinetic energy at the foot of the incline is partially converted to gravitational potential energy and partially dissipated by friction as the block slides up the incline. The free-body diagram shows the forces acting on the block as it slides up the incline. Applying Newton’s 2nd law to the block will allow us to determine fk and express the energy dissipated by friction.
(a) Apply conservation of energy to the system while the block is moving horizontally:
sfUKEEW
∆+∆+∆=∆+∆= thermmechext
or, because ∆U = Wext = 0, sfKKsfK ∆+−=∆+∆= if 0
Solve for Kf: sfKK ∆−= if
Chapter 7
470
Substitute for Kf, Ki, and f∆s to obtain:
xmgmvmv ∆−= k2i2
12f2
1 µ
Solving for vf yields: xgvv ∆−= k2if 2µ
Substitute numerical values and evaluate vf:
( ) ( )( )( )m/s6.10
m2m/s9.810.32m/s7 22f
=
−=v
(b) Apply conservation of energy to the system while the block is on the incline:
sfUKEEW
∆+∆+∆=∆+∆= thermmechext
or, because Kf = Wext = 0, sfUK ∆+∆+−= i0
Apply ∑ = yy maF to the block
when it is on the incline:
θθ cos0cos nn mgFmgF =⇒=−
Express f∆s: θµµ cosknkk mgLLFLfsf ===∆
The final potential energy of the block is:
θsinf mgLU =
Substitute for Uf, Ui, and f∆s to obtain:
θµθ cossin0 ki mgLmgLK ++−=
Solving for L yields: ( )θµθ cossin k
2i2
1
+=
gvL
Substitute numerical values and evaluate L:
( )( ) ( )( )
m17.2
cos400.3sin40m/s9.81m/s10.6
2
221
=
°+°=L
47 • Picture the Problem Let the system include the block, the ramp and horizontal surface, and the earth. Given this choice, there are no external forces acting that will change the energy of the system. Because the curved ramp is frictionless, mechanical energy is conserved as the block slides down it. We can calculate its speed at the bottom of the ramp by using the law of the conservation of energy. The potential energy of the block at the top of the ramp or, equivalently, its kinetic energy at the bottom of the ramp is
Conservation of Energy
471
converted into thermal energy during its slide along the horizontal surface. (a) Choosing Ug = 0 at point 2 and letting the numeral 1 designate the initial position of the block and the numeral 2 its position at the foot of the ramp, use conservation of energy to relate the block’s potential energy at the top of the ramp to its kinetic energy at the bottom:
thermmechext EEW ∆+∆=
or, because Wext = Ki = Uf = ∆Etherm = 0, 00 2
221 =∆−= hmgmv
Solve for v2 to obtain: hgv ∆= 22
Substitute numerical values and evaluate v2:
( )( ) m/s67.7m3m/s9.812 22 ==v
(b) The energy dissipated by friction is responsible for changing the thermal energy of the system:
0thermf =∆+∆+∆=∆+∆+ UKEUKW
Because ∆K = 0 for the slide: ( ) 112f UUUUW =−−=∆−=
Substitute numerical values and evaluate Wf:
( )( )( )J9.58
m3m/s9.81kg2 2f
=
=∆= hmgW
(c) The energy dissipated by friction is given by:
xmgsfE ∆=∆=∆ ktherm µ
Solve for µk: xmg
E∆
∆= therm
kµ
Substitute numerical values and evaluate µk: ( )( )( ) 333.0
m9m/s9.81kg2J58.9
2k ==µ
48 •• Picture the Problem Let the system consist of the earth, the girl, and the slide. Given this choice, there are no external forces doing work to change the energy of the system. By the time she reaches the bottom of the slide, her potential energy at the top of the slide has been converted into kinetic and thermal energy. Choose Ug = 0 at the bottom of the slide and denote the top and bottom of the slide as shown in
Chapter 7
472
the figure. We’ll use the work-energy theorem with friction to relate these quantities and the forces acting on her during her slide to determine the friction force that transforms some of her initial potential energy into thermal energy.
(a) Express the work-energy theorem:
0thermext =∆+∆+∆= EUKW
Because U2 = K1 = Wext = 0:
222
121therm
therm12
or00
mvhmgKUE
EUK
−∆=−=∆
=∆+−=
Substitute numerical values and evaluate ∆Etherm:
( )( )( ) ( )( ) J611m/s1.3kg20m3.2m/s9.81kg20 2212
therm =−=∆E
(b) Relate the energy dissipated by friction to the kinetic friction force and the distance over which this force acts and solve for µk:
sFsfE ∆=∆=∆ nktherm µ
and
sFE
∆∆
=n
thermkµ
Apply ∑ = yy maF to the girl and
solve for Fn:
0cosn =− θmgF ⇒ θcosn mgF =
Referring to the figure, relate ∆h to ∆s and θ: θsin
hs ∆=∆
Substitute for ∆s and Fn to obtain:
hmgE
hmg
E∆
∆=
∆∆
=θ
θθ
µ tan
cossin
thermthermk
Substitute numerical values and evaluate µk:
Conservation of Energy
473
( )( )( )( ) 354.0
m3.2m/s9.81kg20tan20J611
2k =°
=µ
49 •• Picture the Problem Let the system consist of the two blocks, the shelf, and the earth. Given this choice, there are no external forces doing work to change the energy of the system. Due to the friction between the 4-kg block and the surface on which it slides, not all of the energy transformed during the fall of the 2-kg block is realized in the form of kinetic energy. We can find the energy dissipated by friction and then use the work-energy theorem with kinetic friction to find the speed of either block when they have moved the given distance. (a) The energy dissipated by friction when the 2-kg block falls a distance y is given by:
gymsfE 1ktherm µ=∆=∆
Substitute numerical values and evaluate ∆Etherm:
( )( )( )( )y
yE
N7.13
m/s9.81kg435.0 2therm
=
=∆
(b) From the work-energy theorem with kinetic friction we have:
thermmechext EEW ∆+∆=
or, because Wext = 0, ( )yEE N7.13thermmech −=∆−=∆
(c) Express the total mechanical energy of the system:
( ) therm22
2121 Egymvmm ∆−=−+
Solve for v to obtain: ( )21
therm22mm
Egymv+
∆−= (1)
Substitute numerical values and evaluate v:
( )( )( ) ( )( )[ ] m/s98.1kg2kg4
m2N73.13m2m/s9.81kg22 2
=+
−=v
*50 •• Picture the Problem Let the system consist of the particle, the table, and the earth. Then Wext = 0 and the energy dissipated by friction during one revolution is the change in the thermal energy of the system. (a) Apply the work-energy theorem thermext EUKW ∆+∆+∆=
Chapter 7
474
with kinetic friction to obtain: or, because ∆U = Wext = 0, therm0 EK ∆+∆=
Substitute for ∆Kf and simplify to obtain:
( )( ) ( )[ ]208
3
202
1202
121
2i2
12f2
1therm
mv
vmvm
mvmvE
=
−−=
−−=∆
(b) Relate the energy dissipated by friction to the distance traveled and the coefficient of kinetic friction:
( )rmgsmgsfE πµµ 2kktherm =∆=∆=∆
Substitute for ∆E and solve for µk to obtain: gr
vmgrmv
mgrE
πππµ
163
22
20
208
3therm
k ==∆
=
(c) .remaining thelose torevolution
1/3another requireonly it will ,revolution onein lost it Because
i41
i43
KK
51 •• Picture the Problem The box will slow down and stop due to the dissipation of thermal energy. Let the system be the earth, the box, and the inclined plane and apply the work-energy theorem with friction. With this choice of the system, there are no external forces doing work to change the energy of the system. The free-body diagram shows the forces acting on the box when it is moving up the incline.
Apply the work-energy theorem with friction to the system: therm
thermmechext
EUKEEW
∆+∆+∆=∆+∆=
Substitute for ∆K, ∆U, and ∆Etherm to obtain:
LFhmgmvmv nk202
1212
10 µ+∆+−= (1)
Referring to the FBD, relate the normal force to the weight of the box and the angle of the incline:
θcosn mgF =
Relate ∆h to the distance L along the θsinLh =∆
Conservation of Energy
475
incline: Substitute in equation (1) to obtain: 0sin
cos 202
1212
1k
=+
−+
θθµ
mgLmvmvmgL
(2)
Solving equation (2) for L yields: ( )θθµ sincos2 k
20
+=
gvL
Substitute numerical values and evaluate L:
( )( ) ( )[ ]
m875.0
sin37cos370.3m/s9.812m/s3.8
2
2
=
°+°=L
Let vf represent the box’s speed as it passes its starting point on the way down the incline. For the block’s descent, equation (2) becomes:
0sincos 2
1212
f21
k
=−
−+
θθµ
mgLmvmvmgL
Set v1 = 0 (the block starts from rest at the top of the incline) and solve for vf :
( )θµθ cossin2 kf −= gLv
Substitute numerical values and evaluate vf:
( )( ) ( )[ ]] m/s2.49cos370.3sin37m 0.875m/s9.812 2f =°−°=v
52 •••
Picture the Problem Let the system consist of the earth, the block, the incline, and the spring. With this choice of the system, there are no external forces doing work to change the energy of the system. The free-body diagram shows the forces acting on the block just before it begins to move. We can apply Newton’s 2nd law to the block to obtain an expression for the extension of the spring at this instant. We’ll apply the work-energy theorem with friction to the second part of the problem.
(a) Apply ∑ = aF rr
m to the block ∑ =−−= 0sinmaxs,spring θmgfFFx
Chapter 7
476
when it is on the verge of sliding: and
∑ =−= 0cosn θmgFFy
Eliminate Fn, fs,max, and Fspring between the two equations to obtain:
0sincoss =−− θθµ mgmgkd
Solve for and evaluate d: ( )θµθ cossin s+=k
mgd
(b) Begin with the work-energy theorem with friction and no work being done by an external force:
therm
thermmechext
EUUKEEW
sg ∆+∆+∆+∆=∆+∆=
Because the block is at rest in both its initial and final states, ∆K = 0 and:
0therm =∆+∆+∆ EUU sg (1)
Let Ug = 0 at the initial position of the block. Then: θsin
0initialg,finalg,g
mgdmghUUU
=
−=−=∆
Express the change in the energy stored in the spring as it relaxes to its unstretched length:
221
221
initials,finals,s 0
kd
kdUUU
−=
−=−=∆
The energy dissipated by friction is:
θµµ
cosk
nkktherm
mgddFdfsfE
−=−=−=∆=∆
Substitute in equation (1) to obtain:
0cossin k2
21 =−− θµθ mgdkdmgd
Finally, solve for µk: ( )s2
1k tan µθµ −=
Mass and Energy 53 • Picture the Problem The intrinsic rest energy in matter is related to the mass of matter through Einstein’s equation .2
0 mcE =
Conservation of Energy
477
(a) Relate the rest mass consumed to the energy produced and solve for and evaluate m:
( )( )J1000.9
m/s103kg10113
283
20
×=
××=
=−
mcE
(b) Express kW⋅h in joules: ( )( )( )
J1060.3s/h3600h1J/s101hkW1
6
3
×=
×=⋅
Convert 9×1013 J to kW⋅h: ( )
hkW1050.2
J103.60hkW1J109J109
7
61313
⋅×=
⎟⎟⎠
⎞⎜⎜⎝
⎛×
⋅×=×
Determine the price of the electrical energy:
( )6
7
105.2$
hkW$0.10hkW102.50Price
×=
⎟⎠⎞
⎜⎝⎛
⋅⋅×=
(c) Relate the energy consumed to its rate of consumption and the time and solve for the latter:
PtE = and
y28,500s109
W100J109
11
13
=×=
×==
PEt
54 • Picture the Problem We can use the equation expressing the equivalence of energy and matter, E = mc2, to find the mass equivalent of the energy from the explosion. Solve E = mc2 for m:
2cEm =
Substitute numerical values and evaluate m: ( )
kg1056.5
m/s102.998J105
5
28
12
−×=
×
×=m
55 • Picture the Problem The intrinsic rest energy in matter is related to the mass of matter through Einstein’s equation .2
0 mcE =
Relate the rest mass of a muon to its rest energy: 20 c
Em =
Chapter 7
478
Express 1 MeV in joules: 1 MeV = 1.6×10−13 J
Substitute numerical values and evaluate m0:
( )( )( )
kg1088.1
m/s103J/MeV101.6MeV105.7
28
28
13
0
−
−
×=
×
×=m
*56 • Picture the Problem We can differentiate the mass-energy equation to obtain an expression for the rate at which the black hole gains energy. Using the mass-energy relationship, express the energy radiated by the black hole:
201.0 mcE =
Differentiate this expression to obtain an expression for the rate at which the black hole is radiating energy:
[ ]dtdmcmc
dtd
dtdE 22 01.001.0 ==
Solve for dm/dt: 201.0 c
dtdEdtdm
=
Substitute numerical values and evaluate dm/dt: ( )( )
kg/s1045.4
m/s10998.201.0watt104
16
28
31
×=
×
×=
dtdm
57 • Picture the Problem The number of reactions per second is given by the ratio of the power generated to the energy released per reaction. The number of reactions that must take place to produce a given amount of energy is the ratio of the energy per second (power) to the energy released per second. In Example 7-15 it is shown that the energy per reaction is 17.59 MeV. Convert this energy to joules:
( )( )
J1028.1J/eV101.6
MeV17.59MeV59.17
13
19
−
−
×=
××
=
The number of reactions per second is:
sreactions/103.56
J/reaction1028.1J/s1000
14
13
×=
× −
Conservation of Energy
479
58 • Picture the Problem The energy required for this reaction is the difference between the rest energy of 4He and the sum of the rest energies of 3He and a neutron. Express the reaction:
nHeHe 34 +→
The rest energy of a neutron (Table 7-1) is:
939.573 MeV
The rest energy of 4He (Example 7-15) is:
3727.409 MeV
The rest energy of 3He is:
2808.432 MeV
Substitute numerical values to find the difference in the rest energy of 4He and the sum of the rest energies of 3He and n:
( )[ ] MeV574.20MeV573.93941.2808409.3727 =+−=E
59 • Picture the Problem The energy required for this reaction is the difference between the rest energy of a neutron and the sum of the rest energies of a proton and an electron. The rest energy of a proton (Table 7-1) is:
938.280 MeV
The rest energy of an electron (Table 7-1) is:
0.511 MeV
The rest energy of a neutron (Table 7-1) is:
939.573 MeV
Substitute numerical values to find the difference in the rest energy of a neutron and the sum of the rest energies of a positron and an electron:
( )[ ]MeV.7820
MeV511.0280.938573.939
=
+−=E
60 •• Picture the Problem The reaction is E+→+ HeHH 422 . The energy released in this reaction is the difference between twice the rest energy of 2H and the rest energy of 4He.
Chapter 7
480
The number of reactions that must take place to produce a given amount of energy is the ratio of the energy per second (power) to the energy released per reaction. (a) The rest energy of 4He (Example 7-14) is:
3727.409 MeV
The rest energy of a deuteron, 2H, (Table 7-1) is:
1875.628 MeV
The energy released in the reaction is:
( )[ ]J103.816MeV847.23
MeV409.3727628.1875212−×==
−=E
(b) The number of reactions per second is:
sreactions/1062.2
J/reaction10816.3J/s1000
14
12
×=
× −
61 •• Picture the Problem The annual consumption of matter by the fission plant is the ratio of its annual energy output to the square of the speed of light. The annual consumption of coal in a coal-burning power plant is the ratio of its annual energy output to energy per unit mass of the coal. (a) Express m in terms of E:
2cEm =
Assuming an efficiency of 33 percent, find the energy produced annually:
( )( )( )( )
( )( )J1084.2
d365.24h/d24s/h3600J/s1033
y1J/s10333
17
9
9
×=
××=
×=∆= tPE
Substitute to obtain:
( ) kg16.3m/s103
J1084.228
17
=×
×=m
(b) Assuming an efficiency of 38 percent, express the mass of coal required in terms of the annual energy production and the energy released per kilogram:
( ) ( )kg1004.8
J/kg103.138.0J109.47
/38.09
7
16annual
coal
×=
××
==mE
Em
Conservation of Energy
481
General Problems *62 •• Picture the Problem Let the system consist of the block, the earth, and the incline. Then the tension in the string is an external force that will do work to change the energy of the system. Because the incline is frictionless; the work done by the tension in the string as it displaces the block on the incline is equal to the sum of the changes in the kinetic and gravitational potential energies.
Relate the work done by the tension force to the changes in the kinetic and gravitational potential energies of the block:
KUWW ∆+∆== extforcetension
Referring to the figure, express the change in the potential energy of the block as it moves from position 1 to position 2:
θsinmgLhmgU =∆=∆
Because the block starts from rest:
221
2 mvKK ==∆
Substitute to obtain:
221
forcetension sin mvmgLW += θ
and correct. is (c)
63 •• Picture the Problem Let the system include the earth, the block, and the inclined plane. Then there are no external forces to do work on the system and Wext = 0. Apply the work-energy theorem with friction to find an expression for the energy dissipated by friction.
Express the work-energy theorem with friction:
0thermext =∆+∆+∆= EUKW
Chapter 7
482
Because the velocity of the block is constant, ∆K = 0 and:
hmgUE ∆−=∆−=∆ therm
In time ∆t the block slides a distance tv∆ . From the figure:
θsintvh ∆=∆
Substitute to obtain: θsintherm tmgvE ∆−=∆
and correct. is )( b
64 • Picture the Problem Let the system include the earth and the box. Then the applied force is external to the system and does work on the system in compressing the spring. This work is stored in the spring as potential energy. Express the work-energy theorem: thermsgext EUUKW ∆+∆+∆+∆=
Because :0thermg =∆=∆=∆ EUK
sext UW ∆=
Substitute for Wext and ∆Us: 221 kxFx =
Solve for x:
kFx 2
=
Substitute numerical values and evaluate x:
( ) cm06.2N/m6800N702
==x
*65 • Picture the Problem The solar constant is the average energy per unit area and per unit time reaching the upper atmosphere. This physical quantity can be thought of as the power per unit area and is known as intensity. Letting Isurface represent the intensity of the solar radiation at the surface of the earth, express Isurface as a function of power and the area on which this energy is incident:
AtE
API ∆∆
==/
surface
Solve for ∆E: tAIE ∆=∆ surface
Conservation of Energy
483
Substitute numerical values and evaluate ∆E:
( )( )( )( )MJ6.57
s/h3600h8m2kW/m1 22
=
=∆E
66 •• Picture the Problem The luminosity of the sun (or of any other object) is the product of the power it radiates per unit area and its surface area. If we let L represent the sun’s luminosity, I the power it radiates per unit area (also known as the solar constant or the intensity of its radiation), and A its surface area, then L = IA. We can estimate the solar lifetime by dividing the number of hydrogen nuclei in the sun by the rate at which they are being transformed into energy. (a) Express the total energy the sun radiates every second in terms of the solar constant:
IAL =
Letting R represent its radius, express the surface area of the sun:
24 RA π=
Substitute to obtain:
IRL 24π=
Substitute numerical values and evaluate L:
( ) ( )watt1082.3
kW/m1.35m101.5426
2211
×=
×= πL
Note that this result is in good agreement with the value given in the text of 3.9×1026 watt.
(b) Express the solar lifetime in terms of the mass of the sun and the rate at which its mass is being converted to energy:
tnmM
tnNt
∆∆=
∆∆= nuclei H
solar
where M is the mass of the sun, m the mass of a hydrogen nucleus, and n is the number of nuclei used up.
Substitute numerical values to obtain:
tn
tnt
∆∆×
=
∆∆×
×
=−
nucleiH1019.1
nucleuskg/H101.67kg1099.1
57
27
30
solar
For each reaction, 4 hydrogen nuclei are "used up"; so:
( )
138
12
26
s1057.3J104.27J/s103.824
−
−
×=
××
=∆∆
tn
Chapter 7
484
Because we’ve assumed that the sun will continue burning until roughly 10% of its hydrogen fuel is used up, the total solar lifetime should be: y101.06s1033.3
s1057.3nucleiH1019.11.0
1017
138
57
solar
×=×=
⎟⎟⎠
⎞⎜⎜⎝
⎛×
×= −t
67 • Picture the Problem Let the system include the earth and the Spirit of America. Then there are no external forces to do work on the car and Wext = 0. We can use the work-energy theorem to relate the coefficient of kinetic friction to the given information. A constant-acceleration equation will yield the car’s velocity when 60 s have elapsed. (a) Apply the work-energy theorem with friction to relate the coefficient of kinetic friction µk to the initial and final kinetic energies of the car:
0k202
1221 =∆+− smgmvmv µ
or, because v = 0, 0k
202
1 =∆+− smgmv µ
Solve for µk:
sgv∆
=2
2
kµ
Substitute numerical values and evaluate µk:
( )( )[ ]( )( ) 208.0
km9.5m/s9.812sh/36001km/h708
2
2
k ==µ
(b) Express the kinetic energy of the car:
221 mvK = (1)
Using a constant-acceleration equation, relate the speed of the car to its acceleration, initial speed, and the elapsed time:
tavv ∆+= 0
Express the braking force acting on the car:
mamgfF =−=−= kknet µ
Solve for a:
ga kµ−=
Substitute for a to obtain: tgvv ∆−= k0 µ
Substitute in equation (1) to obtain: ( )2
k021 tgvmK ∆−= µ
Substitute numerical values and evaluate K:
Conservation of Energy
485
( )[ ( )( )( )] MJ45.3s60m/s9.810.208m/h10708kg1250 22321 =−×=K
68 •• Picture the Problem The free-body diagram shows the forces acting on the skiers as they are towed up the slope at constant speed. Because the power required to move them is ,vF rr
⋅ we need to find F as a function of mtot, θ, and µk. We can apply Newton’s 2nd law to obtain such a function. Express the power required as a function of force on the skiers and their speed:
FvP = (1)
Apply ∑ = aF rrm to the skiers:
∑ =−−= 0sintotk θgmfFFx
and
∑ =−= 0costotn θgmFFy
Eliminate fk = µkFn and Fn between the two equations and solve for F:
θµθ cossin totktot gmgmF +=
Substitute in equation (1) to obtain: ( )( )θµθ
θµθcossin
cossin
ktot
totktot
+=+=
gvmvgmgmP
Substitute numerical values and evaluate P:
( )( )( ) ( )[ ] kW6.4615cos06.015sinm/s2.5m/s9.81kg7580 2 =°+°=P
Chapter 7
486
69 •• Picture the Problem The free-body diagram for the box is superimposed on the pictorial representation shown to the right. The work done by friction slows and momentarily stops the box as it slides up the incline. The box’s speed when it returns to bottom of the incline will be less than its speed when it started up the incline due to the energy dissipated by friction while it was in motion. Let the system include the box, the earth, and the incline. Then Wext = 0. We can use the work-energy theorem with friction to solve the several parts of this problem.
(a) earth. by the exerted box) theof weight (the force nalgravitatio the
and force,friction kinetic a plane, inclined by the exerted force normal thearebox on the acting forces that theseecan weFBD theFrom
(b) Apply the work-energy theorem with friction to relate the distance ∆x the box slides up the incline to its initial kinetic energy, its final potential energy, and the work done against friction:
0cosk212
1 =∆+∆+− θµ xmghmgmv
Referring to the figure, relate ∆h to ∆x to obtain:
θsinxh ∆=∆
Substitute for ∆h to obtain:
0cossin
k
212
1
=∆+
∆+−
θµθ
xmgxmgmv
Solve for ∆x:
( )θµθ cossin2 k
21
+=∆
gvx
Substitute numerical values and evaluate ∆x:
( )( ) ( )[ ]
m0.451
cos600.3sin60m/s9.812m/s3
2
2
=
°+°=∆x
Conservation of Energy
487
(c) Express and evaluate the energy dissipated by friction:
( )( )( )( ) J1.33cos60m0.451m/s9.81kg20.3
cos2
kktherm
=°=
∆=∆=∆ θµ xmgxfE
(d) Use the work-energy theorem with friction to obtain:
0thermext =∆+∆+∆= EUKW
or 0therm2121 =∆+−+− EUUKK
Because K2 = U1 = 0 we have: 0therm21 =∆+− EUK
or
0cossin
k
212
1
=∆+
∆−
θµθ
xmgxmgmv
Solve for v1: ( )θµθ cossin2 k1 −∆= xgv
Substitute numerical values and evaluate v1:
( )( ) ( )[ ] m/s2.52cos600.3sin60m0.451m/s9.812 21 =°−°=v
*70 • Picture the Problem The power provided by a motor that is delivering sufficient energy to exert a force F on a load which it is moving at a speed v is Fv. The power provided by the motor is given by:
P = Fv
Because the elevator is ascending with constant speed, the tension in the support cable(s) is:
( )gmmF loadelev +=
Substitute for F to obtain: ( )gvmmP loadelev +=
Substitute numerical values and evaluate P:
( )( )( )kW45.1
m/s2.3m/s9.81kg2000 2
=
=P
Chapter 7
488
71 •• Picture the Problem The power a motor must provide to exert a force F on a load that it is moving at a speed v is Fv. The counterweight does negative work and the power of the motor is reduced from that required with no counterbalance. The power provided by the motor is given by:
P = Fv
Because the elevator is counterbalanced and ascending with constant speed, the tension in the support cable(s) is:
( )gmmmF cwloadelev −+=
Substitute and evaluate P: ( )gvmmmP cwloadelev −+=
Substitute numerical values and evaluate P:
( )( )( )kW3.11
m/s2.3m/s9.81kg005 2
=
=P
Without a load: ( )gmmF cwelev −=
and ( )( )( )( )
kW6.77
m/s2.3m/s9.81kg003 2cwelev
−=
−=
−= gvmmP
72 •• Picture the Problem We can use the work-energy theorem with friction to describe the energy transformation within the dart-spring-air-earth system. With this choice of the system, there are no external forces to do work on the system; i.e., Wext = 0. Choose Ug = 0 at the elevation of the dart on the compressed spring. The energy initially stored in the spring is transformed into gravitational potential energy and thermal energy. During the dart’s descent, its gravitational potential energy is transformed into kinetic energy and thermal energy. Apply conservation of energy during the dart’s ascent:
0thermext =∆+∆+∆= EUKW
or, because ∆K = 0, 0thermis,fs,ig,fg, =∆+−+− EUUUU
Because 0fs,ig, == UU :
0thermis,fg, =∆+− EUU
Conservation of Energy
489
Substitute for Ug,i and Us,f and solve for ∆Etherm:
mghkxUUE −=−=∆ 221
fg,is,therm
Substitute numerical values and evaluate ∆Etherm:
( )( )( )( )( )
J0.602
m24m/s9.81kg0.007
m0.03N/m50002
221
therm
=
−
=∆E
Apply conservation of energy during the dart’s descent:
0thermext =∆+∆+∆= EUKW
or, because Ki = Ug,f = 0, 0thermig,f =∆+− EUK
Substitute for Kf and Ug,i to obtain: 0therm
2f2
1 =∆+− Emghmv
Solve for vf: ( )
mEmghv therm
f2 ∆−
=
Substitute numerical values and evaluate vf:
( )( )( )[ ] m/s3.17kg007.0
J602.0m24m/s81.9kg007.02 2
f =−
=v
*73 •• Picture the Problem Let the system consist of the earth, rock and air. Given this choice, there are no external forces to do work on the system and Wext = 0. Choose Ug = 0 to be where the rock begins its upward motion. The initial kinetic energy of the rock is partially transformed into potential energy and partially dissipated by air resistance as the rock ascends. During its descent, its potential energy is partially transformed into kinetic energy and partially dissipated by air resistance. (a) Using the definition of kinetic energy, calculate the initial kinetic energy of the rock:
( )( )kJ1.60
m/s40kg2 2212
i21
i
=
== mvK
(b) Apply the work-energy theorem with friction to relate the energies of the system as the rock ascends:
0therm =∆+∆+∆ EUK
Because Kf = 0: 0thermi =∆+∆+− EUK
and UKE ∆−=∆ itherm
Chapter 7
490
Substitute numerical values and evaluate ∆Etherm:
( )( )( )J619
m50m/s9.81kg2J6001 2therm
=
−=∆E
(c) Apply the work-energy theorem with friction to relate the energies of the system as the rock descends:
07.0 therm =∆+∆+∆ EUK
Because Ki = Uf = 0: 07.0 thermif =∆+− EUK
Substitute for the energies to obtain: 07.0 therm
2f2
1 =∆+− Emghmv
Solve for vf:
mEghv therm
f4.12 ∆
−=
Substitute numerical values and evaluate vf:
( )( ) ( )
m/s23.4
kg2J6191.4m50m/s9.812 2
f
=
−=v
74 •• Picture the Problem Let the distance the block slides before striking the spring be L. The pictorial representation shows the block at the top of the incline (1), just as it strikes the spring (2), and the block against the fully compressed spring (3). Let the block, spring, and the earth comprise the system. Then Wext = 0. Let Ug = 0 where the spring is at maximum compression. We can apply the work-energy theorem to relate the energies of the system as it evolves from state 1 to state 3.
Express the work-energy theorem: 0sg =∆+∆+∆ UUK
or 0s,1s,3g,1g,3 =−+−+∆ UUUUK
Conservation of Energy
491
Because ∆K = Ug,3 = Us,1 = 0: 0s,3g,1 =+− UU
Substitute for each of these energy terms to obtain:
0221
1 =+− kxmgh
Substitute for h3 and h1:
( ) 0sin 221 =++− kxxLmg θ
Rewrite this equation explicitly as a quadratic equation:
0sin2sin22 =−−k
mgLxk
mgx θθ
Solve this quadratic equation to obtain:
θθθ sin2sinsin 22
kmgL
kmg
kmgx +⎟
⎠⎞
⎜⎝⎛+=
Note that the negative sign between the two terms leads to a non-physical solution. *75 • Picture the Problem We can find the work done by the girder on the slab by calculating the change in the potential energy of the slab. (a) Relate the work the girder does on the slab to the change in potential energy of the slab:
hmgUW ∆=∆=
Substitute numerical values and evaluate W:
( )( )( )J147
m0.001m/s9.81kg101.5 24
=
×=W
(b)
expansion. sgirder' thecauses which ,separation averagelarger a toleading energy, kinetic averagegreater a with rategirder vib in the
atoms therises,girder theof re temperatu theAs girder. n thewarmer thaare which gs,surroundin its fromgirder the toed transferrisenergy The
76 •• Picture the Problem The average power delivered by the car’s engine is the rate at which it changes the car’s energy. Because the car is slowing down as it climbs the hill, its potential energy increases and its kinetic energy decreases. Express the average power delivered by the car’s engine: t
EP∆∆
=av
Chapter 7
492
Express the increase in the car’s mechanical energy:
( )hgvvm
hmgmvmv
UUKKUKE
∆+−=
∆+−=
−+−=∆+∆=∆
22bot
2top2
1
2bot2
12top2
1
bottopbottop
Substitute numerical values and evaluate ∆E:
( ) ( ) ( ) ( )( )[ ] MJ41.1m120m/s9.812m/s24m/s10kg1500 22221 =+−=∆E
Assuming that the acceleration of the car is constant, find its average speed during this climb:
m/s172
bottopav =
+=
vvv
Using the vav, find the time it takes the car to climb the hill:
s118m/s17
m2000
av
==∆
=∆v
st
Substitute to determine Pav: kW11.9s118
MJ1.41av ==P
*77 •• Picture the Problem Given the potential energy function as a function of y, we can find the net force acting on a given system from dydUF /−= . The maximum extension of
the spring; i.e., the lowest position of the mass on its end, can be found by applying the work-energy theorem. The equilibrium position of the system can be found by applying the work-energy theorem with friction … as can the amount of thermal energy produced as the system oscillates to its equilibrium position. (a) The graph of U as a function of y is shown to the right. Because k and m are not specified, k has been set equal to 2 and mg to 1. The spring is unstretched when y = y0 = 0. Note that the minimum value of U (a position of stable equilibrium) occurs near y = 5 m.
Conservation of Energy
493
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
y (m)
U (
J)
(b) Evaluate the negative of the derivative of U with respect to y:
( )
mgky
mgykydyd
dydUF
+−=
−−=−= 221
(c) Apply conservation of energy to the movement of the mass from y = 0 to y = ymax:
0therm =∆+∆+∆ EUK
Because ∆K = 0 (the object starts from rest and is momentarily at rest at y = ymax) and ∆Etherm = 0 (no friction), it follows that:
∆U = U(ymax) – U(0) = 0
Because U(0) = 0: U(ymax) = 0 ⇒ 0max2max2
1 =− mgyky
Solve for ymax:
kmgy 2
max =
(d) Express the condition of F at equilibrium and solve for yeq:
00 eqeq =+−⇒= mgkyF
and
kmgy =eq
(e) Apply the conservation of energy to the movement of the mass from y = 0 to y = yeq and solve for ∆Etherm:
0therm =∆+∆+∆ EUK
or, because ∆K = 0. fitherm UUUE −=∆−=∆
Chapter 7
494
Because ( ) :00i == UU ( )eq
2eq2
1ftherm mgykyUE −−=−=∆
Substitute for yeq and simplify to obtain: k
gmE2
22
therm =∆
78 •• Picture the Problem The energy stored in the compressed spring is initially transformed into the kinetic energy of the signal flare and then into gravitational potential energy and thermal energy as the flare climbs to its maximum height. Let the system contain the earth, the air, and the flare so that Wext = 0. We can use the work-energy theorem with friction in the analysis of the energy transformations during the motion of the flare. (a) The work done on the spring in compressing it is equal to the kinetic energy of the flare at launch. Therefore:
202
1flarei,s mvKW ==
(b) Ignoring changes in gravitational potential energy (i.e., assume that the compression of the spring is small compared to the maximum elevation of the flare), apply the conservation of energy to the transformation that takes place as the spring decompresses and gives the flare its launch speed:
0s =∆+∆ UK
or 0is,fs,if =−+− UUKK
Because Ki = ∆Ug = Us,f:
0is,f =−UK
Substitute for is,f and UK :
02212
021 =− kdmv
Solve for k to obtain: 2
20
dmvk =
(c) Apply the work-energy theorem with friction to the upward trajectory of the flare:
0thermg =∆+∆+∆ EUK
Conservation of Energy
495
Solve for ∆Etherm:
fifi
gtherm
UUKKUKE
−+−=
∆−∆−=∆
Because Kf = Ui = 0: mghmvE −=∆ 202
1therm
79 •• Picture the Problem Let UD = 0. Choose the system to include the earth, the track, and the car. Then there are no external forces to do work on the system and change its energy and we can use Newton’s 2nd law and the work-energy theorem to describe the system’s energy transformations to point G … and then the work-energy theorem with friction to determine the braking force that brings the car to a stop. The free-body diagram for point C is shown to the right.
The free-body diagram for point D is shown to the right.
The free-body diagram for point F is shown to the right.
(a) Apply the work-energy theorem to the system’s energy transformations between A and B:
0=∆+∆ UK or
0ABAB =−+− UUKK
If we assume that the car arrives at point B with vB = 0, then:
02A2
1 =∆+− hmgmv
where ∆h is the difference in elevation between A and B.
Chapter 7
496
Solve for and evaluate ∆h: ( )( ) m34.7
m/s9.812m/s12
2 2
22A ===∆g
vh
The height above the ground is: m17.3m7.34m10 =+=∆+ hh
(b) If the car just makes it to point B; i.e., if it gets there with vB = 0, then the force exerted by the track on the car will be the normal force:
( )( )kN4.91
m/s9.81kg500 2
ncarontrack
=
=
== mgFF
(c) Apply ∑ = xx maF to the car at
point C (see the FBD) and solve for a:
mamg =θsin
and ( )2
2
m/s4.91
sin30m/s9.81sin
=
°== θga
(d) Apply ∑ = yy maF to the car at
point D (see the FBD) and solve for Fn:
RvmmgF
2D
n =−
and
R
2D
nvmmgF +=
Apply the work-energy theorem to the system’s energy transformations between B and D:
0=∆+∆ UK or
0BDBD =−+− UUKK
Because KB = UD = 0:
0BD =−UK
Substitute to obtain: ( ) 02D2
1 =∆+− hhmgmv
Solve for 2
Dv :
( )hhgv ∆+= 22D
Substitute to find Fn:
( )
( )⎥⎦⎤
⎢⎣⎡ ∆+
+=
∆++=
+=
Rhhmg
Rhhgmmg
vmmgF
21
2R
2D
n
Conservation of Energy
497
Substitute numerical values and evaluate Fn:
( )( ) ( )
upward.directedkN,13.4
m20m17.321m/s9.81kg500 2
n
=
⎥⎦
⎤⎢⎣
⎡+=F
(e) F has two components at point F; one horizontal (the inward force that the track exerts) and the other vertical (the normal force). Apply
∑ = aF rrm to the car at point F:
∑ =⇒=−= mgFmgFFy nn 0
and
∑ ==RvmFFx
2F
c
Express the resultant of these two forces:
( )
22
4F
222
F
2n
2c
gRvm
mgRvm
FFF
+=
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
+=
Substitute numerical values and evaluate F: ( ) ( )
( )( )
kN46.5
m/s9.81m30
m/s12kg500 222
4
=
+=F
Find the angle the resultant makes with the x axis:
( )( )( )
°=⎥⎦
⎤⎢⎣
⎡=
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−
−−
9.63m/s12
m30m/s9.81tan
tantan
2
21
2F
1
c
n1
vgR
FFθ
(f) Apply the work-energy theorem with friction to the system’s energy transformations between F and the car’s stopping position:
0thermG =∆+− EK
and 2G2
1Gtherm mvKE ==∆
The work done by friction is also given by:
dFsfE braketherm =∆=∆
where d is the stopping distance.
Equate the two expressions for ∆Etherm and solve for Fbrake: d
mvF2
2F
brake =
Chapter 7
498
Substitute numerical values and evaluate Fbrake:
( )( )( ) kN1.44
m252m/s12kg500 2
brake ==F
*80 • Picture the Problem The rate of conversion of mechanical energy can be determined from .vF rr
⋅=P The pictorial representation shows the elevator moving downward just as it goes into freefall as state 1. In state 2 the elevator is moving faster and is about to strike the relaxed spring. The momentarily at rest elevator on the compressed spring is shown as state 3. Let Ug = 0 where the spring has its maximum compression and the system consist of the earth, the elevator, and the spring. Then Wext = 0 and we can apply the conservation of mechanical energy to the analysis of the falling elevator and compressing spring.
(a) Express the rate of conversion of mechanical energy to thermal energy as a function of the speed of the elevator and braking force acting on it:
0brakingvFP =
Because the elevator is moving with constant speed, the net force acting on it is zero and:
MgF =braking
Substitute for Fbraking and evaluate P: ( )( )( )
kW29.4
m/s1.5m/s9.81kg2000 20
=
=
= MgvP
(b) Apply the conservation of energy to the falling elevator and compressing spring:
0sg =∆+∆+∆ UUK
or 0s,1s,3g,1g,313 =−+−+− UUUUKK
Because K3 = Ug,3 = Us,1 = 0: ( ) ( ) 02212
021 =∆+∆+−− ykydMgMv
Conservation of Energy
499
Rewrite this equation as a quadratic equation in ∆y, the maximum compression of the spring:
( ) ( ) 022 20
2 =+−∆⎟⎠⎞
⎜⎝⎛−∆ vgd
kMy
kMgy
Solve for ∆y to obtain: ( )202
22
2 vgdkM
kgM
kMgy ++±=∆
Substitute numerical values and evaluate ∆y:
( )( )
( ) ( )( ) ( )( ) ( )[ ]
m19.5
m/s5.1m5m/s81.92N/m105.1kg2000
N/m105.1m/s81.9kg2000
N/m105.1m/s81.9kg2000
22424
222
4
2
=
+×
+×
+
×=∆y
81 • Picture the Problem We can use Newton’s 2nd law to determine the force of friction as a function of the angle of the hill for a given constant speed. The power output of the engine is given by vF
rr⋅= fP .
FBD for (a):
FBD for (b):
(a) Apply ∑ = xx maF to the car: 0sin f =− Fmg θ ⇒ θsinf mgF =
Evaluate Ff for the two speeds: ( )( )
( )( )N981
sin5.74m/s9.81kg1000and
N491
sin2.87m/s9.81kg1000
230
220
=
°=
=
°=
F
F
(b) Express the power an engine must deliver on a level road in order ( )( ) kW9.82m/s20N49120 ==
=
P
vFP f
Chapter 7
500
to overcome friction loss and evaluate this expression for v = 20 m/s and 30 m/s:
and ( )( ) kW29.4m/s30N98130 ==P
(c) Apply ∑ = xx maF to the car: ∑ =−−= 0sin fx FmgFF θ
Relate F to the power output of the engine and the speed of the car:
vPFFvP == ,Since
Substitute for F and solve for θ :
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −= −
mg
FvP
201sinθ
Substitute numerical values and evaluate θ :
( )( )
°=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −= −
85.8
m/s9.81kg1000
N491m/s20kW40
sin 21θ
(d) Express the equivalence of the work done by the engine in driving the car at the two speeds:
( ) ( )30302020engine sFsFW ∆=∆=
Let ∆V represent the volume of fuel consumed by the engine driving the car on a level road and divide both sides of the work equation by ∆V to obtain:
( ) ( )Vs
FVs
F∆∆
=∆∆ 30
3020
20
Solve for ( )
Vs
∆∆ 30 :
( ) ( )Vs
FF
Vs
∆∆
=∆∆ 20
30
2030
Substitute numerical values and
evaluate ( )
Vs
∆∆ 30 :
( ) ( )
km/L6.36
km/L12.7N981N49130
=
=∆∆
Vs
82 •• Picture the Problem Let the system include the earth, block, spring, and incline. Then Wext = 0. The pictorial representation to the left shows the block sliding down the incline
Conservation of Energy
501
and compressing the spring. Choose Ug = 0 at the elevation at which the spring is fully compressed. We can use the conservation of mechanical energy to determine the maximum compression of the spring. The pictorial representation to the right shows the block sliding up the rough incline after being accelerated by the fully compressed spring. We can use the work-energy theorem with friction to determine how far up the incline the block slides before stopping.
(a) Apply conservation of mechanical energy to the system as it evolves from state 1 to state 3:
0sg =∆+∆+∆ UUK
or
0s,1s,3
g,1g,313
=−+
−+−
UU
UUKK
Because
0s,1g,313 ==== UUKK : 0s,3g,1 =+− UU
or 02
21 =+∆− kxhmg
Relate ∆h to L + x and θ and substitute to obtain:
( ) θsinxLh +=∆ ( ) 0sin2
21 =+−∴ θxLmgkx
Rewrite this equation in the form of an explicit quadratic equation:
( ) 0sinsin221 =−− θθ mgLxmgkx
Substitute for k, m, g, θ and L to obtain:
( ) 0J24.39N81.9mN50 2 =−−⎟
⎠⎞
⎜⎝⎛ xx
Solve for the physically meaningful (i.e., positive) root:
m989.0=x
(b) Proceed as in (a) but include energy dissipated by friction:
0therms,3g,1 =∆++− EUU
The mechanical energy transformed to thermal energy is given by:
( ) ( )( )xLmg
xLFxLFE+=
+=+=∆θµ
µcosk
nkftherm
Chapter 7
502
Substitute for ∆h and ∆Etherm to obtain:
( )( ) 0cos
sin
k
221
=++
++−
xLmgkxxLmg
θµθ
Substitute for k, m, g, θ, µk and L to obtain:
( ) 0J65.25N41.6mN50 2 =−−⎟
⎠⎞
⎜⎝⎛ xx
Solve for the positive root:
m783.0=x
(c) Apply the work-energy theorem with friction to the system as it evolves from state 3 to state 4:
0therms,3s,4
g,3g,434
=∆+−+
−+−
EUUUUKK
Because 0s,4g,314 ==== UUKK :
0therms,3g,4 =∆+− EUU
or 0' therm
221 =∆++∆− Ekxhmg
Substitute for ∆h′ and ∆Etherm to obtain:
( )( ) 0'cos
sin'
k
221
=++
++−
xLmgkxxLmg
θµθ
Solve for L′ with x = 0.783 m: m54.1'=L
83 •• Picture the Problem The work done by the engines maintains the kinetic energy of the cars and overcomes the work done by frictional forces. Let the system include the earth, track, and the cars but not the engines. Then the engines will do external work on the system and we can use this work to find the power output of the train’s engines. (a) Use the definition of kinetic energy:
( )
MJ17.4
s3600h1
hkm15kg102
26
21
221
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅×=
= mvK
(b) The change in potential energy of the train is: ( )( )( )
J101.39
m707m/s9.81kg10210
26
×=
×=
∆=∆ hmgU
(c) Express the energy dissipated by kinetic friction:
sfE ∆=∆ therm
Conservation of Energy
503
Express the frictional force:
mgf 008.0=
Substitute for f and evaluate ∆Etherm:
( )( )( ) J109.73km62m/s9.81kg1020.008008.0 926therm ×=×=∆=∆ smgE
(d) Express the power output of the train’s engines in terms of the work done by them:
tWP
∆∆
=
Use the work-energy theorem with friction to find the work done by the train’s engines:
thermext EUKW ∆+∆+∆=
or, because ∆K = 0, thermext EUW ∆+∆=
Find the time during which the engines do this work:
vst ∆
=∆
Substitute in the expression for P to obtain:
( )s
vEUP∆∆+∆
= therm
Substitute numerical values and evaluate P:
( ) MW1.59km62
s3600h1
hkm15
J109.73J101.39 910 =⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
×+×=P
*84 •• Picture the Problem While on a horizontal surface, the work done by an automobile engine changes the kinetic energy of the car and does work against friction. These energy transformations are described by the work-energy theorem with friction. Let the system include the earth, the roadway, and the car but not the car’s engine. (a) The required energy equals the change in the kinetic energy of the car: ( )
kJ116
s3600h1
hkm50kg1200
2
21
221
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
=∆ mvK
(b) The required energy equals the sfE ∆=∆ therm
Chapter 7
504
work done against friction: Substitute numerical values and evaluate ∆Etherm:
( )( ) kJ90.0m300N300therm ==∆E
(c) Apply the work-energy theorem with friction to express the required energy:
EKEKWE
75.0' thermext
+∆=∆+∆==
Divide both sides of the equation by E to express the ratio of the two energies:
75.0'+
∆=
EK
EE
Substitute numerical values and evaluate E′/E:
04.20.75kJ90kJ116'
=+=EE
*85 ••• Picture the Problem Assume that the bob is moving with speed v as it passes the top vertical point when looping around the peg. There are two forces acting on the bob: the tension in the string (if any) and the force of gravity, Mg; both point downward when the ball is in the topmost position. The minimum possible speed for the bob to pass the vertical occurs when the tension is 0; from this, gravity must supply the centripetal force required to keep the ball moving in a circle. We can use conservation of energy to relate v to L and R.
Express the condition that the bob swings around the peg in a full circle:
2
MgRvM >
Simplify to obtain: g
Rv
>2
Use conservation of energy to relate the kinetic energy of the bob at the bottom of the loop to its potential energy at the top of its swing:
( )2221 RLMgMv −=
Solve for v2: ( )RLgv 222 −=
Substitute to obtain: ( ) gR
RLg>
− 22
Conservation of Energy
505
Solve for R:
LR52 <
86 •• Picture the Problem If the wood exerts an average force F on the bullet, the work it does has magnitude FD. This must be equal to the change in the kinetic energy of the bullet, or because the final kinetic energy of the bullet is zero, to the negative of the initial kinetic energy. We’ll let m be the mass of the bullet and v its initial speed and apply the work-kinetic energy theorem to relate the penetration depth to v. Apply the work-kinetic energy theorem to relate the penetration depth to the change in the kinetic energy of the bullet:
iftotal KKKW −=∆= or, because Kf = 0,
itotal KW −=
Substitute for Wtotal and Ki to obtain: 221 mvFD −=
Solve for D to obtain:
FmvD2
2
−=
For an identical bullet with twice the speed we have:
( )221 2' vmFD −=
Solve for D′ to obtain: D
FmvD 42
4'2
=⎟⎟⎠
⎞⎜⎜⎝
⎛−=
and correct. is )(c
87 •• Picture the Problem For part (a), we’ll let the system include the glider, track, weight, and the earth. The speeds of the glider and the falling weight will be the same while they are in motion. Let their common speed when they have moved a distance Y be v and let the zero of potential energy be at the elevation of the weight when it has fallen the distance Y. We can use conservation of energy to relate the speed of the glider (and the weight) to the distance the weight has fallen. In part (b), we’ll let the direction of motion be the x direction, the tension in the connecting string be T, and apply Newton’s 2nd law to the glider and the weight to find their common acceleration. Because this acceleration is constant, we can use a constant-acceleration equation to find their common speed when they have moved a distance Y. (a) Use conservation of energy to relate the kinetic and potential energies of the system:
0=∆+∆ UK or
0ifif =−+− UUKK
Because the system starts from rest and Uf = 0:
0if =−UK
Substitute to obtain: 02212
21 =−+ mgYMvmv
Chapter 7
506
Solve for v:
mMmgYv
+=
2
(b) The free-body diagrams for the glider and the weight are shown to the right:
Apply Newton’s 3rd law to obtain: T== 21 TT
rr
Apply maFx =∑ to the glider:
MaT =
Apply maFx =∑ to the weight:
maTmg =−
Add these equations to eliminate T and obtain:
maMamg +=
Solve for a to obtain:
Mmmga+
=
Using a constant-acceleration equation, relate the speed of the glider to its initial speed and to the distance that the weight has fallen:
aYvv 220
2 += or, because v0 = 0,
aYv 22 =
Substitute for a and solve for v to obtain:
mMmgYv
+=
2, the same result we
obtained in part (a). *88 •• Picture the Problem We’re given dtdWP /= and are asked to evaluate it under the assumed conditions. Express the rate of energy expenditure by the man:
( )( )W270
m/s3kg1033 22
=== mvP
Express the rate of energy expenditure P′ assuming that his
P'P 51=
Conservation of Energy
507
muscles have an efficiency of 20%: Solve for and evaluate P′: ( ) kW1.35W27055 === PP'
89 •• Picture the Problem The pictorial representation shows the bob swinging through an angle θ before the thread is cut and it is launched horizontally. Let its speed at position 1 be v. We can use conservation of energy to relate v to the change in the potential energy of the bob as it swings through the angle θ . We can find its flight time ∆t from a constant-acceleration equation and then express D as the product of v and ∆t.
Relate the distance D traveled horizontally by the bob to its launch speed v and time of flight ∆t:
tvD ∆= (1)
Use conservation of energy to relate its launch speed v to the length of the pendulum L and the angle θ :
00101 =−+− UUKK or, because U1 = K0 = 0,
001 =−UK
Substitute to obtain:
( ) 0cos1221 =−− θmgLmv
Solving for v yields:
( )θcos12 −= gLv
In the absence of air resistance, the horizontal and vertical motions of the bob are independent of each other and we can use a constant-acceleration equation to express the time of flight (the time to fall a distance H):
( )221
0 tatvy yy ∆+∆=∆ or, because ∆y = −H, ay = −g, and v0y = 0,
( )221 tgH ∆−=−
Solve for ∆t to obtain: gHt /2=∆
Substitute in equation (1) and simplify to obtain: ( )
( )θ
θ
cos12
2cos12
−=
−=
HL
gHgLD
which shows that, while D depends on θ, it is independent of g.
Chapter 7
508
90 •• Picture the Problem The pictorial representation depicts the block in its initial position against the compressed spring (1), as it separates from the spring with its maximum kinetic energy (2), and when it has come to rest after moving a distance x + d. Let the system consist of the earth, the block, and the surface on which the block slides. With this choice, Wext = 0. We can use the work-energy theorem with friction to determine how far the block will slide before coming to rest.
(a) The work done by the spring on the block is given by:
221
springspring kxUW =∆=
Substitute numerical values and evaluate Wspring:
( )( ) J0.900cm3N/cm20 221
spring ==W
(b) The energy dissipated by friction is given by:
xmgxFsfE ∆=∆=∆=∆ knktherm µµ
Substitute numerical values and evaluate ∆Etherm:
( )( )( )( )J0.294
m0.03m/s9.81kg50.2 2therm
=
=∆E
(c) Apply the conservation of energy between points 1 and 2:
0therms,1s,212 =∆+−+− EUUKK
Because K1 = Us,2 = 0:
0therms,12 =∆+− EUK
Substitute to obtain: 0therm2
212
221 =∆+− Ekxmv
Solve for v2:
mEkxv therm
2
22∆−
=
Substitute numerical values and evaluate v2:
( )( ) ( )
m/s0.492
kg5J0.2942cm3N/cm20 2
2
=
−=v
Conservation of Energy
509
(d) Apply the conservation of energy between points 1 and 3:
0therms,1s,3 =∆+−+∆ EUUK
Because ∆K = Us,3 = 0: 0therms,1 =∆+− EU
or ( ) 0k
221 =++− dxmgkx µ
Solve for d:
xmg
kxd −=k
2
2µ
Substitute numerical values and evaluate d:
( )( )( )( )( )
cm6.17
m0.03m/s9.81kg50.22
cm3N/cm202
2
=
−=d
91 •• Picture the Problem The pictorial representation shows the block initially at rest at point 1, falling under the influence of gravity to point 2, partially compressing the spring as it continues to gain kinetic energy at point 3, and finally coming to rest at point 4 with the spring fully compressed. Let the system consist of the earth, the block, and the spring so that Wext = 0. Let Ug = 0 at point 3 for part (a) and at point 4 for part (b). We can use the work-energy theorem to express the kinetic energy of the system as a function of the block’s position and then use this function to maximize K as well as determine the maximum compression of the spring and the location of the block when the system has half its maximum kinetic energy.
(a) Apply conservation of mechanical energy to describe the energy transformations between state 1 and state 3:
0sg =∆+∆+∆ UUK
or 0s,1s,3g,1g,313 =−+−+− UUUUKK
Because K1 = Ug,3 = Us,1 = 0: 0s,3g,13 =+− UUK
Chapter 7
510
and ( ) 2
21
3 kxxhmgKK −+==
Differentiate K with respect to x and set this derivative equal to zero to identify extreme values:
.valuesextremefor0=−= kxmgdxdK
Solve for x: k
mgx =
Evaluate the second derivative of K with respect to x:
.maximizes
02
2
Kk
mgx
kdt
Kd
=⇒
<−=
Evaluate K for x = mg/k:
kgmmgh
kmgk
kmgmgmghK
2
22
2
21
max
+=
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+=
(b) The spring will have its maximum compression at point 4 where K = 0:
( )
022or
0
max2max
2max2
1max
=−−
=−+
kmghx
kmgx
kxxhmg
Solve for x and keep the physically meaningful root:
kmgh
kgm
kmgx 2
2
22
max ++=
(c) Apply conservation of mechanical energy to the system as it evolves from state 1 to the state in which max2
1 KK = :
0sg =∆+∆+∆ UUK
or 0s,1s,3g,1g,31 =−+−+− UUUUKK
Because K1 = Ug,3 = Us,1 = 0: 0s,3g,1 =+− UUK
and ( ) 2
21 kxxhmgK −+=
Conservation of Energy
511
Substitute for K to obtain: ( ) 221
22
21
2kxxhmg
kgmmgh −+=⎟⎟
⎠
⎞⎜⎜⎝
⎛+
Express this equation in quadratic form:
02
22
222 =⎟⎟
⎠
⎞⎜⎜⎝
⎛−+−
kmgh
kgmx
kmgx
Solve for the positive value of x:
kmgh
kgm
kmgx 42
2
22
++=
92 ••• Picture the Problem The free-body diagram shows the forces acting on the pendulum bob. The application of Newton’s 2nd law leads directly to the required expression for the tangential acceleration. Recall that, provided θ is in radian measure, s = Lθ. Differentiation with respect to time produces the result called for in part (b). The remaining parts of the problem simply require following the directions for each part.
(a) Apply ∑ = xx maF to the bob: tantan sin mamgF =−= θ
Solve for atan: θsin/tan gdtdva −==
(b) Relate the arc distance s to the length of the pendulum L and the angle θ :
θLs =
Differentiate with respect to time: dtLdvdtds // θ==
(c) Multiply θθ
dd
dtdv by and
substitute for dtdθ
from part (b): ⎟⎠⎞
⎜⎝⎛=
==
Lv
ddv
dtd
ddv
dd
dtdv
dtdv
θ
θθθ
θ
Chapter 7
512
(d) Equate the expressions for dv/dt from (a) and (c):
θθ
singLv
ddv
−=⎟⎠⎞
⎜⎝⎛
Separate the variables to obtain: θθ dgLvdv sin−=
(e) Integrate the left side of the equation in part (d) from v = 0 to the final speed v and the right side from θ = θ0 to θ = 0:
∫∫ −=0
0 0
''sin''θ
θθ dgLdvvv
Evaluate the limits of integration to obtain:
( )02
21 cos1 θ−= gLv
Note, from the figure, that h = L(1 − cosθ0). Substitute and solve for v:
ghv 2=
93 ••• Picture the Problem The potential energy of the climber is the sum of his gravitational potential energy and the potential energy stored in the spring-like bungee cord. Let θ be the angle which the position of the rock climber on the cliff face makes with a vertical axis and choose the zero of gravitational potential energy to be at the bottom of the cliff. We can use the definitions of Ug and Uspring to express the climber’s total potential energy. (a) Express the total potential energy of the climber:
( ) gcord bungee UUsU +=
Substitute to obtain: ( )( )
( ) ⎟⎠⎞
⎜⎝⎛+−=
+−=
+−=
HsMgHLsk
MgHLsk
MgyLsksU
cos
cos
)(
221
221
221
θ
A spreadsheet solution is shown below. The constants used in the potential energy function and the formulas used to calculate the potential energy are as follows:
Cell Content/Formula Algebraic Form B3 300 H B4 5 k B5 60 L B6 85 M B7 9.81 g
D11 60 s D12 D11+1 s + 1
Conservation of Energy
513
E11 0.5*$B$4*(D11−$B$5)^2 +$B$6*$B$7*$B$3*(cos(D11/$B$3)) ( ) ⎟
⎠⎞
⎜⎝⎛+−
HsMgHLsk cos2
21
G11 E11−E61 ( ) ( )m110m60 UU −
A B C D E 1 2 3 H = 300 m 4 k = 5 N/m 5 L = 60 m 6 m = 85 kg 7 g = 9.81 m/s^2 8 9
10 s U(s) 11 60 2.45E+05 12 61 2.45E+05 13 62 2.45E+05 14 63 2.45E+05 15 64 2.45E+05
147 196 2.45E+05 148 197 2.45E+05 149 198 2.45E+05 150 199 2.45E+05 151 200 2.46E+05
The following graph was plotted using the data from columns D (s) and E (U(s)).
238
239
240
241
242
243
244
245
246
50 70 90 110 130 150 170 190 210
s (m)
U (k
J)
Chapter 7
514
*94 ••• Picture the Problem The diagram shows the forces each of the springs exerts on the block. The change in the potential energy stored in the springs is due to the elongation of both springs when the block is displaced a distance x from its equilibrium position and we can find ∆U using ( )2
21 Lk ∆ . We can find the magnitude of the force pulling the block
back toward its equilibrium position by finding the sum of the magnitudes of the y components of the forces exerted by the springs. In Part (d) we can use conservation of energy to find the speed of the block as it passes through its equilibrium position.
(a) Express the change in the potential energy stored in the springs when the block is displaced a distance x:
( )[ ] ( )22212 LkLkU ∆=∆=∆
where ∆L is the change in length of a spring.
Referring to the force diagram, express ∆L:
LxLL −+=∆ 22
Substitute to obtain: ( )222 LxLkU −+=∆
(b) Sum the forces acting on the block to express Frestoring:
22
restoring
2
cos2cos2
xLxLk
LkFF
+∆=
∆== θθ
Substitute for ∆L to obtain: ( )
⎟⎟⎠
⎞⎜⎜⎝
⎛
+−=
+−+=
22
22
22restoring
12
2
xLLkx
xLxLxLkF
(c) A spreadsheet program to calculate U(x) is shown below. The constants used in the potential energy function and the formulas used to calculate the potential energy are as follows:
Cell Content/Formula Algebraic Form B1 1 L B2 1 k B3 1 M C8 C7+0.01 x D7 $B$2*((C7^2+$B$1^2)^0.5−$B$1)^2 U(x)
Conservation of Energy
515
A B C D 1 L = 0.1 m 2 k = 1 N/m 3 M = 1 kg 4 5 6 x U(x) 7 0 0 8 0.01 2.49E−07 9 0.02 3.92E−06 10 0.03 1.94E−05 11 0.04 5.93E−05 12 0.05 1.39E−04
23 0.16 7.86E−03 24 0.17 9.45E−03 25 0.18 1.12E−02 26 0.19 1.32E−02 27 0.20 1.53E−02
The following graph was plotted using the data from columns C (x) and D (U(x)).
0
2
4
6
8
10
12
14
16
0.00 0.05 0.10 0.15 0.20
x (m)
U (m
J)
(d) Use conservation of energy to relate the kinetic energy of the block as it passes through the equilibrium position to the change in its potential energy as it returns to its equilibrium position:
UK ∆=mequilibriu or
UMv ∆=221
Chapter 7
516
Solve for v to obtain: ( )
( )MkLxL
MLxLk
MUv
2
22
22
222
−+=
−+=
∆=
Substitute numerical values and evaluate v:
( ) ( ) ( ) cm/s86.5kg1N/m12m1.0m1.0m1.0 22 =⎟
⎠⎞⎜
⎝⎛ −+=v
517
Chapter 8 Systems of Particles and Conservation of Momentum Conceptual Problems 1 • Determine the Concept A doughnut. The definition of the center of mass of an object does not require that there be any matter at its location. Any hollow sphere (such as a basketball) or an empty container with any geometry are additional examples of three-dimensional objects that have no mass at their center of mass. *2 • Determine the Concept The center of mass is midway between the two balls and is in free-fall along with them (all forces can be thought to be concentrated at the center of mass.) The center of mass will initially rise, then fall. Because the initial velocity of the center of mass is half of the initial velocity of the ball thrown upwards, the mass thrown upwards will rise for twice the time that the center of mass rises. Also, the center of mass will rise until the velocities of the two balls are equal but opposite. correct. is )(b
3 • Determine the Concept The acceleration of the center of mass of a system of particles is described by ,cm
iexti,extnet, aFF
rrrM== ∑ where M is the total mass of the system.
Express the acceleration of the center of mass of the two pucks: 21
1extnet,cm mm
FM
Fa
+==
and correct. is )(b
4 • Determine the Concept The acceleration of the center of mass of a system of particles is described by ,cm
iexti,extnet, aFF
rrrM== ∑ where M is the total mass of the system.
Express the acceleration of the center of mass of the two pucks: 21
1extnet,cm mm
FM
Fa
+==
because the spring force is an internal force.
correct. is )( b
Chapter 8
518
*5 • Determine the Concept No. Consider a 1-kg block with a speed of 1 m/s and a 2- kg block with a speed of 0.707 m/s. The blocks have equal kinetic energies but momenta of magnitude 1 kg·m /s and 1.414 kg·m/s, respectively. 6 • (a) True. The momentum of an object is the product of its mass and velocity. Therefore, if we are considering just the magnitudes of the momenta, the momentum of a heavy object is greater than that of a light object moving at the same speed. (b) True. Consider the collision of two objects of equal mass traveling in opposite directions with the same speed. Assume that they collide inelastically. The mechanical energy of the system is not conserved (it is transformed into other forms of energy), but the momentum of the system is the same after the collision as before the collision, i.e., zero. Therefore, for any inelastic collision, the momentum of a system may be conserved even when mechanical energy is not. (c) True. This is a restatement of the expression for the total momentum of a system of particles. 7 • Determine the Concept To the extent that the system in which the rifle is being fired is an isolated system, i.e., the net external force is zero, momentum is conserved during its firing. Apply conservation of momentum to the firing of the rifle:
0bulletrifle =+ pp rr
or
bulletrifle pp rr−=
*8 • Determine the Concept When she jumps from a boat to a dock, she must, in order for momentum to be conserved, give the boat a recoil momentum, i.e., her forward momentum must be the same as the boat’s backward momentum. The energy she imparts to the boat is .2 boat
2boatboat mpE =
zero.y essentiall is them toimparts sheenergy that thelarge so isearth theplusdock theof mass theanother, dock to one from jumps sheWhen
*9 ••
Determine the Concept Conservation of momentum requires only that the net external force acting on the system be zero. It does not require the presence of a medium such as air.
Systems of Particles and Conservation of Momentum
519
10 • Determine the Concept The kinetic energy of the sliding ball is 2
cm21 mv . The kinetic
energy of the rolling ball is rel2cm2
1 Kmv + , where its kinetic energy relative to its center of mass is relK . Because the bowling balls are identical and have the same velocity, the
rolling ball has more energy. 11 • Determine the Concept Think of someone pushing a box across a floor. Her push on the box is equal but opposite to the push of the box on her, but the action and reaction forces act on different objects. You can only add forces when they act on the same object. 12 • Determine the Concept It’s not possible for both to remain at rest after the collision, as that wouldn't satisfy the requirement that momentum is conserved. It is possible for one to remain at rest: This is what happens for a one-dimensional collision of two identical particles colliding elastically. 13 • Determine the Concept It violates the conservation of momentum! To move forward requires pushing something backwards, which Superman doesn’t appear to be doing when flying around. In a similar manner, if Superman picks up a train and throws it at Lex Luthor, he (Superman) ought to be tossed backwards at a pretty high speed to satisfy the conservation of momentum. *14 •• Determine the Concept There is only one force which can cause the car to move forward−the friction of the road! The car’s engine causes the tires to rotate, but if the road were frictionless (as is closely approximated by icy conditions) the wheels would simply spin without the car moving anywhere. Because of friction, the car’s tire pushes backwards against the road−from Newton’s third law, the frictional force acting on the tire must then push it forward. This may seem odd, as we tend to think of friction as being a retarding force only, but true. 15 •• Determine the Concept The friction of the tire against the road causes the car to slow down. This is rather subtle, as the tire is in contact with the ground without slipping at all times, and so as you push on the brakes harder, the force of static friction of the road against the tires must increase. Also, of course, the brakes heat up, and not the tires. 16 • Determine the Concept Because ∆p = F∆t is constant, a safety net reduces the force acting on the performer by increasing the time ∆t during which the slowing force acts. 17 • Determine the Concept Assume that the ball travels at 80 mi/h ≈ 36 m/s. The ball stops in a distance of about 1 cm. So the distance traveled is about 2 cm at an average speed of
Chapter 8
520
about 18 m/s. The collision time is ms1m/s18
m0.02≈ .
18 • Determine the Concept The average force on the glass is less when falling on a carpet because ∆t is longer. 19 • (a) False. In a perfectly inelastic collision, the colliding bodies stick together but may or may not continue moving, depending on the momentum each brings to the collision. (b) True. In a head-on elastic collision both kinetic energy and momentum are conserved and the relative speeds of approach and recession are equal. (c) True. This is the definition of an elastic collision. *20 •• Determine the Concept All the initial kinetic energy of the isolated system is lost in a perfectly inelastic collision in which the velocity of the center of mass is zero. 21 •• Determine the Concept We can find the loss of kinetic energy in these two collisions by finding the initial and final kinetic energies. We’ll use conservation of momentum to find the final velocities of the two masses in each perfectly elastic collision. (a) Letting V represent the velocity of the masses after their perfectly inelastic collision, use conservation of momentum to determine V:
afterbefore pp =
or 02 =⇒=− VmVmvmv
Express the loss of kinetic energy for the case in which the two objects have oppositely directed velocities of magnitude v/2:
4
220
2
2
21
if
mv
vmKKK
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛−=−=∆
Letting V represent the velocity of the masses after their perfectly inelastic collision, use conservation of momentum to determine V:
afterbefore pp =
or vVmVmv 2
12 =⇒=
Systems of Particles and Conservation of Momentum
521
Express the loss of kinetic energy for the case in which the one object is initially at rest and the other has an initial velocity v:
( )42
22
221
2
21
if
mvmvvm
KKK
−=−⎟⎠⎞
⎜⎝⎛=
−=∆
cases.both in same theisenergy kinetic of loss The
(b) Express the percentage loss for the case in which the two objects have oppositely directed velocities of magnitude v/2:
%100241
241
before
==∆
mvmv
KK
Express the percentage loss for the case in which the one object is initially at rest and the other has an initial velocity v:
%50221
241
before
==∆
mvmv
KK
/2. magnitude ofs velocitiedirected oppositely have
objects twohein which t case thefor greatest is loss percentage The
v
*22 •• Determine the Concept A will travel farther. Both peas are acted on by the same force, but pea A is acted on by that force for a longer time. By the impulse-momentum theorem, its momentum (and, hence, speed) will be higher than pea B’s speed on leaving the shooter. 23 •• Determine the Concept Refer to the particles as particle 1 and particle 2. Let the direction particle 1 is moving before the collision be the positive x direction. We’ll use both conservation of momentum and conservation of mechanical energy to obtain an expression for the velocity of particle 2 after the collision. Finally, we’ll examine the ratio of the final kinetic energy of particle 2 to that of particle 1 to determine the condition under which there is maximum energy transfer from particle 1 to particle 2. Use conservation of momentum to obtain one relation for the final velocities:
f2,2f1,1i1,1 vmvmvm += (1)
Use conservation of mechanical energy to set the velocity of
( ) i1,i1,i2,f1,f2, vvvvv =−−=− (2)
Chapter 8
522
recession equal to the negative of the velocity of approach: To eliminate v1,f, solve equation (2) for v1,f, and substitute the result in equation (1):
i1,f2,f1, vvv +=
( ) f2,2i1,f2,1i1,1 vmvvmvm +−=
Solve for v2,f:
i1,21
1f2,
2v
mmm
v+
=
Express the ratio R of K2,f to K1,i in terms of m1 and m2:
( )221
21
1
2
2i,112
1
2i,1
2
21
122
1
i1,
f2,
4
2
mmm
mm
vm
vmm
mm
KK
R
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
==
Differentiate this ratio with respect to m2, set the derivative equal to zero, and obtain the quadratic equation:
0121
22 =+−
mm
Solve this equation for m2 to determine its value for maximum energy transfer:
12 mm =
.when 2 toed transferrisenergy kinetic s1' of all becausecorrect is )(
12 mmb
=
24 • Determine the Concept In the center-of-mass reference frame the two objects approach with equal but opposite momenta and remain at rest after the collision. 25 • Determine the Concept The water is changing direction when it rounds the corner in the nozzle. Therefore, the nozzle must exert a force on the stream of water to change its direction, and, from Newton’s 3rd law, the water exerts an equal but opposite force on the nozzle. 26 • Determine the Concept The collision usually takes place in such a short period of time that the impulse delivered by gravity or friction is negligible.
Systems of Particles and Conservation of Momentum
523
27 • Determine the Concept No. dtdpF rr
=netext, defines the relationship between the net
force acting on a system and the rate at which its momentum changes. The net external force acting on the pendulum bob is the sum of the force of gravity and the tension in the string and these forces do not add to zero. *28 •• Determine the Concept We can apply conservation of momentum and Newton’s laws of motion to the analysis of these questions. (a) Yes, the car should slow down. An easy way of seeing this is to imagine a "packet" of grain being dumped into the car all at once: This is a completely inelastic collision, with the packet having an initial horizontal velocity of 0. After the collision, it is moving with the same horizontal velocity that the car does, so the car must slow down. (b) When the packet of grain lands in the car, it initially has a horizontal velocity of 0, so it must be accelerated to come to the same speed as the car of the train. Therefore, the train must exert a force on it to accelerate it. By Newton’s 3rd law, the grain exerts an equal but opposite force on the car, slowing it down. In general, this is a frictional force which causes the grain to come to the same speed as the car. (c) No it doesn’t speed up. Imagine a packet of grain being "dumped" out of the railroad car. This can be treated as a collision, too. It has the same horizontal speed as the railroad car when it leaks out, so the train car doesn’t have to speed up or slow down to conserve momentum. *29 •• Determine the Concept Think of the stream of air molecules hitting the sail. Imagine that they bounce off the sail elastically−their net change in momentum is then roughly twice the change in momentum that they experienced going through the fan. Another way of looking at it: Initially, the air is at rest, but after passing through the fan and bouncing off the sail, it is moving backward−therefore, the boat must exert a net force on the air pushing it backward, and there must be a force on the boat pushing it forward. Estimation and Approximation 30 •• Picture the Problem We can estimate the time of collision from the average speed of the car and the distance traveled by the center of the car during the collision. We’ll assume a car length of 6 m. We can calculate the average force exerted by the wall on the car from the car’s change in momentum and it’s stopping time. (a) Relate the stopping time to the assumption that the center of the car travels halfway to the wall with constant deceleration:
( )av
car41
av
car21
21
av
stopping
vL
vL
vd
t ===∆
Chapter 8
524
Express and evaluate vav:
m/s5.122
kmm1000
s3600h1
hkm900
2fi
av
=
××+=
+=
vvv
Substitute for vav and evaluate ∆t: ( )
s120.0m/s12.5m64
1
==∆t
(b) Relate the average force exerted by the wall on the car to the car’s change in momentum:
( )kN417
s0.120km
m1000s3600
h1h
km90kg2000
av =⎟⎟⎠
⎞⎜⎜⎝
⎛××
=∆∆
=tpF
31 •• Picture the Problem Let the direction the railcar is moving be the positive x direction and the system include the earth, the pumpers, and the railcar. We’ll also denote the railcar with the letter c and the pumpers with the letter p. We’ll use conservation of momentum to relate the center of mass frame velocities of the car and the pumpers and then transform to the earth frame of reference to find the time of fall of the car.
(a) Relate the time of fall of the railcar to the distance it falls and its velocity as it leaves the bank:
cvyt ∆
=∆
Use conservation of momentum to find the speed of the car relative to the velocity of its center of mass: 0
or
ppcc
fi
=+
=
umum
pp rr
Relate uc to up and solve for uc:
m/s4
m/s4
cp
pc
−=∴
=−
uu
uu
Substitute for up to obtain: ( ) 0m/s4cpcc =−+ umum
Systems of Particles and Conservation of Momentum
525
Solve for and evaluate uc:
( )
m/s85.1
kg754kg3501
m/s4
1
m/s4
p
cc =
+=
+=
mmu
Relate the speed of the car to its speed relative to the center of mass of the system:
m/s74.10km
m1000s3600
h1h
km32sm.851
cmcc
=
××+=
+= vuv
Substitute and evaluate ∆t: s2.33
m/s10.74m25
==∆t
(b) Find the speed with which the pumpers hit the ground: m/s6.74
m/s4m/s10.74pcp
=
−=−= uvv
injured. bemay theyspeed, at this ground theHitting
*32 •• Picture the Problem The diagram depicts the bullet just before its collision with the melon and the motion of the melon-and-bullet-less-jet and the jet just after the collision. We’ll assume that the bullet stays in the watermelon after the collision and use conservation of momentum to relate the mass of the bullet and its initial velocity to the momenta of the melon jet and the melon less the plug after the collision.
Apply conservation of momentum to the collision to obtain:
( ) 332f1321i1 2 Kmvmmmvm ++−=
Solve for v2f:
132
331i12f
2mmmKmvm
v+−
−=
Express the kinetic energy of the jet of melon in terms of the initial kinetic energy of the bullet:
( ) 21i120
121i12
1101
1101
3 vmvmKK ===
Chapter 8
526
Substitute and simplify to obtain: ( )
( )132
3111i
132
21i120
131i1
2f
1.0
2
mmmmmmv
mmmvmmvm
v
+−−
=
+−−
=
Substitute numerical values and evaluate v2f:
( )( )( )
ft/s1.27
m/s386.0kg0.0104kg0.14kg2.50
kg0.14kg0.01040.1kg0.0104ft3.281
m1sft18002f
−=
−=+−
−⎟⎟⎠
⎞⎜⎜⎝
⎛×=v
Note that this result is in reasonably good agreement with experimental results. Finding the Center of Mass 33 • Picture the Problem We can use its definition to find the center of mass of this system. Apply its definition to find xcm:
( )( ) ( )( ) ( )( ) m233.0kg2kg2kg2
m0.5kg2m0.2kg20kg2
321
332211cm =
++++
=++++
=mmm
xmxmxmx
Because the point masses all lie along the x axis:
0cm =y and the center of mass of this
system of particles is at ( )0,m233.0 .
*34 • Picture the Problem Let the left end of the handle be the origin of our coordinate system. We can disassemble the club-ax, find the center of mass of each piece, and then use these coordinates and the masses of the handle and stone to find the center of mass of the club-ax. Express the center of mass of the handle plus stone system:
stonestick
stonecm,stonestickcm,stickcm mm
xmxmx
+
+=
Assume that the stone is drilled and the stick passes through it. Use symmetry considerations to locate the center of mass of the stick:
cm0.45stickcm, =x
Systems of Particles and Conservation of Momentum
527
Use symmetry considerations to locate the center of mass of the stone:
cm0.89stonecm, =x
Substitute numerical values and evaluate xcm:
( )( ) ( )( )
cm5.78
kg8kg2.5cm89kg8cm54kg2.5
cm
=
++
=x
35 • Picture the Problem We can treat each of balls as though they are point objects and apply the definition of the center of mass to find (xcm, ycm). Use the definition of xcm:
( )( ) ( )( ) ( )( )
m00.2kg1kg1kg3
m3kg1m1kg1m2kg3
cm
=++
++=
++++
=CBA
CCBBAA
mmmxmxmxm
x
Use the definition of ycm:
( )( ) ( )( ) ( )( )
m40.1kg1kg1kg3
0kg1m1kg1m2kg3
cm
=++
++=
++++
=CBA
CCBBAA
mmmymymymy
The center of mass of this system of particles is at:
( )m40.1,m00.2
36 • Picture the Problem The figure shows an equilateral triangle with its y-axis vertex above the x axis. The bisectors of the vertex angles are also shown. We can find x coordinate of the center-of-mass by inspection and the y coordinate using trigonometry. From symmetry considerations:
0cm =x
Chapter 8
528
Express the trigonometric relationship between a/2, 30°, and ycm:
230tan cm
ay
=°
Solve for ycm: aay 289.030tan21
cm =°=
The center of mass of an equilateral
triangle oriented as shown above is at ( )a289.0,0 .
*37 •• Picture the Problem Let the subscript 1 refer to the 3-m by 3-m sheet of plywood before the 2-m by 1-m piece has been cut from it. Let the subscript 2 refer to 2-m by 1-m piece that has been removed and let σ be the area density of the sheet. We can find the center-of-mass of these two regions; treating the missing region as though it had negative mass, and then finding the center-of-mass of the U-shaped region by applying its definition. Express the coordinates of the center of mass of the sheet of plywood:
21
2,cm21cm,1cm mm
xmxmx
−−
=
21
2,cm21cm,1cm mm
ymymy
−−
=
Use symmetry to find xcm,1, ycm,1, xcm,2, and ycm,2:
m0.2,m5.1and
m5.1m,5.1
cm,2cm,2
cm,1cm,1
==
==
yx
yx
Determine m1 and m2:
kgAm
Am
σσ
σσ
2and
kg9
22
11
==
==
Substitute numerical values and evaluate xcm:
( )( ) ( )( )
m50.1kg2kg9
kg5.1kg2m5.1kg9cm
=−−
=σσσσx
Substitute numerical values and evaluate ycm:
( )( ) ( )( )
m36.1kg2kg9
m2kg2m5.1kg9cm
=−−
=σσ
σσy
The center of mass of the U-shaped sheet of plywood is at ( )m1.36m,1.50 .
Systems of Particles and Conservation of Momentum
529
38 •• Picture the Problem We can use its definition to find the center of mass of the can plus water. By setting the derivative of this function equal to zero, we can find the value of x that corresponds to the minimum height of the center of mass of the water as it drains out and then use this extreme value to express the minimum height of the center of mass. (a) Using its definition, express the location of the center of mass of the can + water: mM
xmHMx
+
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
= 22cm
Let the cross-sectional area of the cup be A and use the definition of density to relate the mass m of water remaining in the can at any given time to its depth x:
Axm
AHM
==ρ
Solve for m to obtain: M
Hxm =
Substitute to obtain:
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
+
⎟⎠⎞
⎜⎝⎛+
=
+
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
=
Hx
Hx
H
MHxM
xMHxHM
x
1
1
2
22
2
cm
(b) Differentiate xcm with respect to x and set the derivative equal to zero for extrema:
021
12
1
21
121
2
21
12
1
21
211
21
21
2cm
=
+
+
−
+
+=
+
++
−
+
++
=+
+=
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎪⎪
⎩
⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛
Hx
HHx
Hx
HHx
Hx
H
Hx
Hx
dxd
Hx
Hx
Hx
dxd
Hx
H
Hx
Hx
dxdH
dxdx
Simplify this expression to obtain:
0122
=−⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛
Hx
Hx
Chapter 8
530
Solve for x/H to obtain:
( ) HHx 414.012 ≈−= where we’ve kept the positive solution because a negative value for x/H would make no sense.
Use your graphing calculator to convince yourself that the graph of xcm as a function of x is concave upward at Hx 414.0≈ and that, therefore, the minimum value of xcm occurs at .414.0 Hx ≈ Evaluate xcm at ( )12 −= Hx to obtain:
( )
( )
( )
( )12
121
121
2
2
12cm
−=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−+
⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
=−=
H
HH
HH
HxHx
Finding the Center of Mass by Integration *39 •• Picture the Problem A semicircular disk and a surface element of area dA is shown in the diagram. Because the disk is a continuous object, we’ll use
∫= dmM rrrr
cm and symmetry to find its
center of mass.
Express the coordinates of the center of mass of the semicircular disk:
symmetry.by0cm =x
M
dAyy ∫=
σcm
Express y as a function of r and θ : θsinry =
Express dA in terms of r and θ : drdrdA θ=
Express M as a function of r and θ : 2
21
diskhalf RAM σπσ ==
Systems of Particles and Conservation of Momentum
531
Substitute and evaluate ycm:
RRM
drrMM
drdry
R
R
πσ
σθθσ
π
34
32
2sin
3
0
20 0
2
cm
==
== ∫∫ ∫
40 ••• Picture the Problem Because a solid hemisphere is a continuous object, we’ll use
∫= dmM rrrr
cm to find its center of mass. The volume element for a sphere is
dV = r2 sinθ dθ dφ dr, where θ is the polar angle and φ the azimuthal angle. Let the base of the hemisphere be the xy plane and ρ be the mass density. Then:
θcosrz =
Express the z coordinate of the center of mass:
∫∫=
dV
dVrz
ρ
ρcm
Evaluate ∫= dVM ρ :
( ) 3323
34
21
sphere21
RR
VdVM
πρπρ
ρρ
==
== ∫
Evaluate ∫ dVrρ :
[ ]4
sin2
cossin
42/
02
21
40
2/
0
2
0
3
RR
drddrdVrR
πρθπρ
φθθθρ
π
π π
==
=∫ ∫ ∫ ∫
Substitute and simplify to find zcm:
RRRz 8
33
32
441
cm ==πρπρ
41 ••• Picture the Problem Because a thin hemisphere shell is a continuous object, we’ll use
∫= dmM rrrr
cm to find its center of mass. The element of area on the shell is dA = 2πR2
sinθ dθ, where R is the radius of the hemisphere. Let σ be the surface mass density and express the z coordinate of the center of mass: ∫
∫=dA
dAzz
σ
σcm
Chapter 8
532
Evaluate ∫= dAM σ :
( ) 2221
shellspherical21
24 RR
AdAM
πσπσ
σσ
==
== ∫
Evaluate ∫ dAzσ :
σπ
θθσπ
θθθσπσ
π
π
3
2/
0
3
2/
0
3
2sin
cossin2
R
dR
dRdAz
=
=
=
∫
∫ ∫
Substitute and simplify to find zcm:
RR
Rz 21
2
3
cm 2==
πσσπ
42 ••• Picture the Problem The parabolic sheet is shown to the right. Because the area of the sheet is distributed symmetrically with respect to the y axis, xcm = 0. We’ll integrate the element of area dA (= xdy) to obtain the total area of the sheet and yxdy to obtain the numerator of the definition of the center of mass. Express ycm:
∫
∫= b
b
xdy
xydyy
0
0cm
Evaluate ∫b
xydy0
:
25
0
23
0
21
0
52
1
ba
dyya
ydya
yxydybbb
=
== ∫∫∫
Evaluate ∫b
xdy0
:
23
0
21
0
21
0
32
1
ba
dyya
dya
yxdybbb
=
== ∫∫∫
Systems of Particles and Conservation of Momentum
533
Substitute and simplify to determine ycm:
bb
a
bay 5
3
23
25
cm
32
52
==
Note that, by symmetry:
xcm = 0
The center of mass of the parabolic sheet is at:
( )b53,0
Motion of the Center of Mass 43 • Picture the Problem The velocity of the center of mass of a system of particles is related to the total momentum of the system through cm
iii vvP
rrrMm == ∑ .
Use the expression for the total momentum of a system to relate the velocity of the center of mass of the two-particle system to the momenta of the individual particles:
21
2211iii
cm mmmm
M
m
++
==∑ vv
vv
rrr
r
Substitute numerical values and evaluate cmvr :
( )( ) ( )
( ) ( )[ ]( ) ( ) ji
ji
vvvvv
ˆm/s5.1ˆm/s3
ˆm/s3ˆm/s6
kg6kg3
21
212121
cm
−=
−=
+=+
=rr
rrr
*44 • Picture the Problem Choose a coordinate system in which east is the positive x direction and use the relationship cm
iii vvP
rrrMm == ∑ to determine the velocity of the
center of mass of the system. Use the expression for the total momentum of a system to relate the velocity of the center of mass of the two-vehicle system to the momenta of the individual vehicles:
ct
ccttiii
cm mmmm
M
m
++
==∑ vv
vv
rrr
r
Express the velocity of the truck: ( ) iv ˆm/s16t =r
Chapter 8
534
Express the velocity of the car: ( )iv ˆm/s20c −=r
Substitute numerical values and evaluate cmvr :
( )( ) ( )( ) ( ) iiiv ˆm/s00.4
kg1500kg3000
ˆm/s20kg1500ˆm/s16kg3000cm =
+−+
=r
45 • Picture the Problem The acceleration of the center of mass of the ball is related to the net external force through Newton’s 2nd law: cmextnet, aF rr
M= .
Use Newton’s 2nd law to express the acceleration of the ball:
Mextnet,
cm
Fa
rr
=
Substitute numerical values and evaluate cmar :
( ) ( )iia ˆm/s4.2kg1kg1kg3
ˆN12 2cm =
++=
r
46 •• Picture the Problem Choose a coordinate system in which upward is the positive y direction. We can use Newton’s 2nd law cmextnet, aF rr
M= to find the acceleration of the
center of mass of this two-body system.
(a) . is reading
thefall, freein is while;) ( reads scale theinitially Yes;Mg
mgmM +
(b) Using Newton’s 2nd law, express the acceleration of the center of mass of the system:
tot
extnet,cm m
Fa
rr
=
Substitute to obtain:
ja ˆcm mM
mg+
−=r
(c) Use Newton’s 2nd law to express the net force acting on the scale while the object of mass m is falling:
( ) cmextnet, )( amMgmMF +−+=
Substitute and simplify to obtain: ( )
Mg
mMmgmMgmMF
=
⎟⎠⎞
⎜⎝⎛
++−+= )(extnet,
Systems of Particles and Conservation of Momentum
535
as expected, given our answer to part (a).
*47 •• Picture the Problem The free-body diagram shows the forces acting on the platform when the spring is partially compressed. The scale reading is the force the scale exerts on the platform and is represented on the FBD by Fn. We can use Newton’s 2nd law to determine the scale reading in part (a) and the work-energy theorem in conjunction with Newton’s 2nd law in parts (b) and (c).
(a) Apply ∑ = yy maF to the
spring when it is compressed a distance d:
∑ =−−= 0springonballpn FgmFFy
Solve for Fn:
( )gmmgmgm
kgmkgmkdgm
FgmF
bpbp
bpp
springonballpn
+=+=
⎟⎠⎞
⎜⎝⎛+=+=
+=
(b) Use conservation of mechanical energy, with Ug = 0 at the position at which the spring is fully compressed, to relate the gravitational potential energy of the system to the energy stored in the fully compressed spring:
0sg =∆+∆+∆ UUK
Because ∆K = Ug,f = Us,i = 0, 0fs,ig, =−UU
or 02
21
b =− kdgdm
Solve for d: k
gmd b2
=
Evaluate our force equation in (a)
with k
gmd b2
= :
( )gmmgmgm
kgmkgmkdgm
FgmF
bpbp
bpp
springonballpn
22
2
+=+=
⎟⎠⎞
⎜⎝⎛+=+=
+=
Chapter 8
536
(c) When the ball is in its original position, the spring is relaxed and exerts no force on the ball. Therefore:
gm
F
p
n readingscale
=
=
*48 •• Picture the Problem Assume that the object whose mass is m1 is moving downward and take that direction to be the positive direction. We’ll use Newton’s 2nd law for a system of particles to relate the acceleration of the center of mass to the acceleration of the individual particles. (a) Relate the acceleration of the center of mass to m1, m2, mc and their accelerations:
cc2211cm aaaa rrrr mmmM ++=
Because m1 and m2 have a common acceleration a and ac = 0:
c21
21cm mmm
mmaa
++−
=
From Problem 4-81 we have:
21
21
mmmm
ga+−
=
Substitute to obtain:
( )( )( ) g
mmmmmmm
mmmmmg
mmmma
c2121
221
c21
21
21
21cm
+++−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛++
−⎟⎟⎠
⎞⎜⎜⎝
⎛+−
=
(b) Use Newton’s 2nd law for a system of particles to obtain:
cmMaMgF −=−
where M = m1 + m2 + mc and F is positive upwards.
Solve for F and substitute for acm from part (a):
( )
gmmmmm
gmm
mmMg
MaMgF
⎥⎦
⎤⎢⎣
⎡+
+=
+−
−=
−=
c21
21
21
221
cm
4
(c) From Problem 4-81: g
mmmmT
21
212+
=
Systems of Particles and Conservation of Momentum
537
Substitute in our result from part (b) to obtain:
gmTgmgT
gmmmmmF
cc
c21
21
22
22
+=⎥⎦
⎤⎢⎣
⎡+=
⎥⎦
⎤⎢⎣
⎡+
+=
49 •• Picture the Problem The free-body diagram shows the forces acting on the platform when the spring is partially compressed. The scale reading is the force the scale exerts on the platform and is represented on the FBD by Fn. We can use Newton’s 2nd law to determine the scale reading in part (a) and the result of Problem 7-96 part (b) to obtain the scale reading when the ball is dropped from a height h above the cup.
(a) Apply ∑ = yy maF to the spring
when it is compressed a distance d:
∑ =−−= 0springonballpn FgmFFy
Solve for Fn:
( )gmmgmgm
kgmkgmkdgm
FgmF
bpbp
bpp
springonballpn
+=+=
⎟⎠⎞
⎜⎝⎛+=+=
+=
(b) From Problem 7-96, part (b):
⎟⎟⎠
⎞⎜⎜⎝
⎛++=
gmkh
kgm
xb
bmax
211
From part (a):
⎟⎟⎠
⎞⎜⎜⎝
⎛+++=
+=+=
gmkhgmgm
kxgmFgmF
bbp
maxpspringonballpn
211
The Conservation of Momentum 50 • Picture the Problem Let the system include the woman, the canoe, and the earth. Then the net external force is zero and linear momentum is conserved as she jumps off the
Chapter 8
538
canoe. Let the direction she jumps be the positive x direction. Apply conservation of momentum to the system:
0canoecanoegirlgirlii =+=∑ vvv rrr mmm
Substitute to obtain: ( )( ) ( ) 0kg57ˆm/s5.2kg55 canoe =+ vi r
Solve for canoevr : ( ) iv ˆm/s83.1canoe −=
r
51 •
Picture the Problem If we include the earth in our system, then the net external force is zero and linear momentum is conserved as the spring delivers its energy to the two objects.
Apply conservation of momentum to the system:
0101055ii =+=∑ vvv rrr mmm
Substitute numerical values to obtain:
( )( ) ( ) 0kg10ˆm/s8kg5 10 =+− vi r
Solve for 10vr : ( ) iv ˆm/s410 =r
*52 • Picture the Problem This is an explosion-like event in which linear momentum is conserved. Thus we can equate the initial and final momenta in the x direction and the initial and final momenta in the y direction. Choose a coordinate system in the positive x direction is to the right and the positive y direction is upward. Equate the momenta in the y direction before and after the explosion:
( ) 022
2
11
12fy,iy,
=−=
−== ∑∑mvvm
mvmvpp
We can conclude that the momentum was
entirely in the x direction before the particle exploded.
Equate the momenta in the x direction before and after the explosion:
3i
fx,ix,
4 mvmvpp
=∴
=∑ ∑
Solve for v3: 34
1i vv = and correct. is )(c
Systems of Particles and Conservation of Momentum
539
53 • Picture the Problem Choose the direction the shell is moving just before the explosion to be the positive x direction and apply conservation of momentum. Use conservation of momentum to relate the masses of the fragments to their velocities:
fi pprr
=
or 'ˆˆ
21
21 vji
rmmvmv +=
Solve for 'vr : jiv ˆˆ2' vv −=r
*54 •• Picture the Problem Let the system include the earth and the platform, gun and block. Then extnet,F
r= 0 and momentum is conserved within the system.
(a) Apply conservation of momentum to the system just before and just after the bullet leaves the gun:
platformbullet
afterbefore
0or
pp
pp
rr
rr
+=
=
Substitute for platformbullet and pp
rrand
solve for platformvr : platformpbb
ˆ0 vi rmvm +=
and
iv ˆb
p
bplatform v
mm
−=r
(b) Apply conservation of momentum to the system just before the bullet leaves the gun and just after it comes to rest in the block:
afterbefore pp rr=
or platform0 pr= ⇒ 0platform =vr
(c) Express the distance ∆s traveled by the platform:
tvs ∆=∆ platform
Express the velocity of the bullet relative to the platform:
bp
bpb
p
b
bp
bbplatformbrel
1 vm
mmv
mm
vmmvvvv
+=⎟
⎟⎠
⎞⎜⎜⎝
⎛+=
+=−=
Relate the time of flight ∆t to L and vrel: relv
Lt =∆
Chapter 8
540
Substitute to find the distance ∆s moved by the platform in time ∆t:
Lmm
m
vm
mmLv
mm
vLv
mmtvs
bp
b
bp
bpb
p
b
relb
p
bplatform
+=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=∆=∆
55 ••
Picture the Problem The pictorial representation shows the wedge and small object, initially at rest, to the left, and, to the right, both in motion as the small object leaves the wedge. Choose the direction the small object is moving when it leaves the wedge be the positive x direction and the zero of potential energy to be at the surface of the table. Let the speed of the small object be v and that of the wedge V. We can use conservation of momentum to express v in terms of V and conservation of energy to express v in terms of h.
Apply conservation of momentum to the small object and the wedge:
Vi
pp
r
rr
mmv
xx
2ˆ0
orf,i,
+=
=
Solve for :V
r iV ˆ
21 v−=
r (1)
and vV 2
1=
Use conservation of energy to determine the speed of the small object when it exits the wedge: 0
or0
ifif =−+−
=∆+∆
UUKK
UK
Because Uf = Ki = 0: ( ) 02 2
212
21 =−+ mghVmmv
Systems of Particles and Conservation of Momentum
541
Substitute for V to obtain: ( )( ) 02 221
212
21 =−+ mghvmmv
Solve for v to obtain: 3
2 ghv =
Substitute in equation (1) to determine V
r: iiV ˆ
3ˆ
322
1 ghgh−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
r
i.e., the wedge moves in the direction
opposite to that of the small object with a
speed of3gh
.
*56 •• Picture the Problem Because no external forces act on either cart, the center of mass of the two-cart system can’t move. We can use the data concerning the masses and separation of the gliders initially to calculate its location and then apply the definition of the center of mass a second time to relate the positions X1 and X2 of the centers of the carts when they first touch. We can also use the separation of the centers of the gliders when they touch to obtain a second equation in X1 and X2 that we can solve simultaneously with the equation obtained from the location of the center of mass. (a) Apply its definition to find the center of mass of the 2-glider system:
( )( ) ( )( )
m10.1kg0.2kg0.1
m1.6kg0.2m0.1kg0.121
2211cm
=++
=
++
=mm
xmxmx
from the left end of the air track.
Use the definition of the center of mass to relate the coordinates of the centers of the two gliders when they first touch to the location of the center of mass:
( ) ( )
232
131
21
21
2211
kg0.2kg0.1kg0.2kg0.1
m10.1
XX
XXmm
XmXm
+=++
=
++
=
Also, when they first touch, their centers are separated by half their combined lengths:
( ) m0.15cm20cm1021
12 =+=− XX
Thus we have:
m10.1667.0333.0 21 =+ XX and
m0.1512 =− XX
Chapter 8
542
Solve these equations simultaneously to obtain:
m00.11 =X and m15.12 =X
(b)
collision. after thezero bemust it so zero, is system theof momentum initial The No.
Kinetic Energy of a System of Particles *57 • Picture the Problem Choose a coordinate system in which the positive x direction is to the right. Use the expression for the total momentum of a system to find the velocity of the center of mass and the definition of relative velocity to express the sum of the kinetic energies relative to the center of mass. (a) Find the sum of the kinetic energies:
( )( ) ( )( )J43.5
m/s2kg3m/s5kg3 2212
21
2222
12112
1
21
=
+=
+=
+=
vmvm
KKK
(b) Relate the velocity of the center of mass of the system to its total momentum:
2211cm vvv rrr mmM +=
Solve for :cmvr
21
2211cm mm
mm++
=vvvrr
r
Substitute numerical values and evaluate :cmvr
( )( ) ( )( )
( )i
iiv
ˆm/s50.1
kg3kg3
ˆm/s2kg3ˆm/s5kg3cm
=
+−
=r
(c) The velocity of an object relative to the center of mass is given by:
cmrel vvv rrr−=
Systems of Particles and Conservation of Momentum
543
Substitute numerical values to obtain:
( ) ( )( )
( ) ( )( )i
iiv
i
iiv
ˆm/s50.3
ˆm/s5.1ˆm/s2
ˆm/s50.3
ˆm/s5.1ˆm/s5
rel2,
rel1,
−=
−−=
=
−=
r
r
(d) Express the sum of the kinetic energies relative to the center of mass:
2rel,222
12rel,112
1rel,2rel,1rel vmvmKKK +=+=
Substitute numerical values and evaluate Krel:
( )( )( )( )
J75.63
m/s5.3kg3
m/s3.5kg32
21
221
rel
=
−+
=K
(e) Find Kcm: ( )( )
rel
2212
cmtot21
cm
J36.75J43.5J6.75
m/s1.5kg6
KK
vmK
−=
−==
==
58 •
Picture the Problem Choose a coordinate system in which the positive x direction is to the right. Use the expression for the total momentum of a system to find the velocity of the center of mass and the definition of relative velocity to express the sum of the kinetic energies relative to the center of mass. (a) Express the sum of the kinetic energies:
2222
12112
121 vmvmKKK +=+=
Substitute numerical values and evaluate K:
( )( ) ( )( )J0.06
m/s3kg5m/s5kg3 2212
21
=
+=K
(b) Relate the velocity of the center of mass of the system to its total momentum:
2211cm vvv rrr mmM +=
Solve for cmvr :
21
2211cm mm
mm++
=vvvrr
r
Chapter 8
544
Substitute numerical values and evaluate cmvr :
( )( ) ( )( )
( )i
iiv
ˆm/s75.3
kg5kg3
ˆm/s3kg5ˆm/s5kg3cm
=
++
=r
(c) The velocity of an object relative to the center of mass is given by:
cmrel vvv rrr−=
Substitute numerical values and evaluate the relative velocities:
( ) ( )( )i
iiv
ˆm/s25.1
ˆm/s75.3ˆm/s5rel1,
=
−=r
and ( ) ( )
( )iiiv
ˆm/s750.0
ˆm/s75.3ˆm/s3rel2,
−=
−=r
(d) Express the sum of the kinetic energies relative to the center of mass:
2rel,222
12rel,112
1
rel,2rel,1rel
vmvm
KKK
+=
+=
Substitute numerical values and evaluate Krel:
( )( )( )( )
J75.3
m/s75.0kg5
m/s25.1kg32
21
221
rel
=
−+
=K
(e) Find Kcm: ( )( )
rel
2212
cmtot21
cm
J3.65
m/s75.3kg8
KK
vmK
−==
==
Impulse and Average Force 59 • Picture the Problem The impulse imparted to the ball by the kicker equals the change in the ball’s momentum. The impulse is also the product of the average force exerted on the ball by the kicker and the time during which the average force acts. (a) Relate the impulse delivered to the ball to its change in momentum: 0since if
if
==−=∆=
vmvpppI
Substitute numerical values and evaluate I:
( )( ) sN10.8m/s25kg0.43 ⋅==I
Systems of Particles and Conservation of Momentum
545
(b) Express the impulse delivered to the ball as a function of the average force acting on it and solve for and evaluate avF :
tFI ∆= av
and
kN1.34s0.008sN10.8
av =⋅
=∆
=t
IF
60 • Picture the Problem The impulse exerted by the ground on the brick equals the change in momentum of the brick and is also the product of the average force exerted by the ground on the brick and the time during which the average force acts. (a) Express the impulse exerted by the ground on the brick:
bricki,brickf,brick pppI −=∆=
Because pf,brick = 0: vmpI brickbricki, == (1)
Use conservation of energy to determine the speed of the brick at impact: 0
or0
ifif =−+−
=∆+∆
UUKK
UK
Because Uf = Ki = 0:
0or
0
brick2
brick21
if
=−
=−
ghmvm
UK
Solve for v: ghv 2=
Substitute in equation (1) to obtain: ghmI 2brick=
Substitute numerical values and evaluate I:
( ) ( )( )sN76.3
m8m/s9.812kg0.3 2
⋅=
=I
(c) Express the impulse delivered to the brick as a function of the average force acting on it and solve for and evaluate avF :
tFI ∆= av
and
kN89.2s0.0013sN76.3
av =⋅
=∆
=t
IF
*61 • Picture the Problem The impulse exerted by the ground on the meteorite equals the change in momentum of the meteorite and is also the product of the average force exerted by the ground on the meteorite and the time during which the average force acts.
Chapter 8
546
Express the impulse exerted by the ground on the meteorite:
ifmeteorite pppI −=∆=
Relate the kinetic energy of the meteorite to its initial momentum and solve for its initial momentum:
ii
2i
i 22
mKpm
pK =⇒=
Express the ratio of the initial and final kinetic energies of the meteorite:
2
2m
22f
2i
2f
2i
f
i ===pp
pm
p
KK
Solve for pf:
2i
fp
p =
Substitute in our expression for I and simplify:
⎟⎠⎞
⎜⎝⎛ −=
⎟⎠⎞
⎜⎝⎛ −=−=
12
12
12
12
i
iii
mK
pppI
Because our interest is in its magnitude, evaluate I :
( )( ) sMN81.112
1J10617kg1030.82 63 ⋅=⎟⎠⎞
⎜⎝⎛ −××=I
Express the impulse delivered to the meteorite as a function of the average force acting on it and solve for and evaluate avF :
tFI ∆= av
and
MN602.0s3
sMN81.1av =
⋅=
∆=
tIF
62 •• Picture the Problem The impulse exerted by the bat on the ball equals the change in momentum of the ball and is also the product of the average force exerted by the bat on the ball and the time during which the bat and ball were in contact. (a) Express the impulse exerted by the bat on the ball in terms of the change in momentum of the ball:
( ) iii
pppIˆ2ˆˆ
if
ifball
mvmvmv =−−=
−=∆=rrrr
where v = vf = vi
Systems of Particles and Conservation of Momentum
547
Substitute for m and v and evaluate I:
( )( ) sN00.6m/s20kg15.02 ⋅==I
(b) Express the impulse delivered to the ball as a function of the average force acting on it and solve for and evaluate avF :
tFI ∆= av
and
kN62.4ms3.1
sN00.6av =
⋅=
∆=
tIF
*63 •• Picture the Problem The figure shows the handball just before and immediately after its collision with the wall. Choose a coordinate system in which the positive x direction is to the right. The wall changes the momentum of the ball by exerting a force on it during the ball’s collision with it. The reaction to this force is the force the ball exerts on the wall. Because these action and reaction forces are equal in magnitude, we can find the average force exerted on the ball by finding the change in momentum of the ball.
Using Newton’s 3rd law, relate the average force exerted by the ball on the wall to the average force exerted by the wall on the ball:
ballon avon wall av FFrr
−=
and ballon avon wall av FF = (1)
Relate the average force exerted by the wall on the ball to its change in momentum:
tm
t ∆∆
=∆∆
=vpFrrr
ballon av
Express xvr∆ for the ball: iiv ˆˆ,i,f xxx vv −=∆
r
or, because vi,x = vcosθ and vf,x = −vcosθ, iiiv ˆcos2ˆcosˆcos θθθ vvvx −=−−=∆
r
Substitute in our expression for
ballon avFr
: ivF ˆcos2
ballon av tmv
tm
∆−=
∆∆
=θ
rr
Chapter 8
548
Evaluate the magnitude of ballon avFr
:
( )( )
N230ms2
cos40m/s5kg0.062
cos2ballon av
=
°=
∆=
tmvF θ
Substitute in equation (1) to obtain: N230on wall av =F
64 •• Picture the Problem The pictorial representation shows the ball during the interval of time you are exerting a force on it to accelerate it upward. The average force you exert can be determined from the change in momentum of the ball. The change in the velocity of the ball can be found by applying conservation of mechanical energy to its rise in the air once it has left your hand. (a) Relate the average force exerted by your hand on the ball to the change in momentum of the ball:
tmv
tpp
tpF
∆=
∆−
=∆∆
= 212av
because v1 and, hence, p1 = 0.
Letting Ug = 0 at the initial elevation of your hand, use conservation of mechanical energy to relate the initial kinetic energy of the ball to its potential energy when it is at its highest point:
0since0
or0
if
fi
===+−
=∆+∆
UKUK
UK
Substitute for Kf and Ui and solve for v2:
ghv
mghmv
2
and0
2
222
1
=
=+−
Relate ∆t to the average speed of the ball while you are throwing it upward:
22av
2
2vd
vd
vdt ===∆
Systems of Particles and Conservation of Momentum
549
Substitute for ∆t and v2 in the expression for Fav to obtain: d
mghF =av
Substitute numerical values and evaluate Fav:
( )( )( )
N1.84
m0.7m40m/s9.81kg0.15 2
av
=
=F
(b) Express the ratio of the weight of the ball to the average force acting on it:
( )( ) %2N84.1
m/s9.81kg0.15 2
avav
<==Fmg
Fw
weight.its neglected have toreasonable isit ball, on theexerted force average theof 2% than less is ball theof weight theBecause
65 •• Picture the Problem Choose a coordinate system in which the direction the ball is moving after its collision with the wall is the positive x direction. The impulse delivered to the wall or received by the player equals the change in the momentum of the ball. We can find the average forces from the rate of change in the momentum of the ball. (a) Relate the impulse delivered to the wall to the change in momentum of the handball:
( )( )( )( )[ ]
( ) wall.into directed ˆsN08.1
ˆm/s01kg0.06
ˆm/s8kg0.06if
i
i
i
vvpI
⋅=
−−
=
−=∆=rrrr
mm
(b) Find Fav from the change in the ball’s momentum:
wall.into N,603
s0.003sN08.1
av
=
⋅=
∆∆
=tpF
(c) Find the impulse received by the player from the change in momentum of the ball:
( )( ) wall.fromaway s,N480.0
m/s8kg0.06ball
⋅=
=∆=∆= vmpI
(d) Relate Fav to the change in the ball’s momentum: t
pF
∆∆
= ballav
Express the stopping time in terms of the average speed vav of the ball avv
dt =∆
Chapter 8
550
and its stopping distance d:
Substitute to obtain: dpvF ballav
av∆
=
Substitute numerical values and evaluate Fav:
( )( )
wall.fromaway N,84.3
m0.5sN480.0m/s4
av
=
⋅=F
66 ••• Picture the Problem The average force exerted on the limestone by the droplets of water equals the rate at which momentum is being delivered to the floor. We’re given the number of droplets that arrive per minute and can use conservation of mechanical energy to determine their velocity as they reach the floor. (a) Letting N represent the rate at which droplets fall, relate Fav to the change in the droplet’s momentum:
tvmN
tp
F∆∆
=∆
∆= droplets
av
Find the mass of the droplets: ( )( )kg103
mL0.03kg/L15−×=
== Vm ρ
Letting Ug = 0 at the point of impact of the droplets, use conservation of mechanical energy to relate their speed at impact to their fall distance:
0or
0
ifif =−+−
=∆+∆
UUKK
UK
Because Ki = Uf = 0: 02f2
1 =− mghmv
Solve for and evaluate v = vf: ( )( )
m/s9.90m5m/s9.8122 2
=
== ghv
Substitute numerical values and evaluate Fav:
( )( )N1095.4
m/s9.90kg103
s60min1
mindroplets10
5
5
av
−
−
×=
××
⎟⎟⎠
⎞⎜⎜⎝
⎛×=
∆⎟⎠⎞
⎜⎝⎛
∆= vm
tNF
Systems of Particles and Conservation of Momentum
551
(b) Calculate the ratio of the weight of a droplet to Fav:
( )( ) 6N104.95m/s9.81kg103
5
25avav
≈×
×=
=
−
−
Fmg
Fw
Collisions in One Dimension *67 • Picture the Problem We can apply conservation of momentum to this perfectly inelastic collision to find the after-collision speed of the two cars. The ratio of the transformed kinetic energy to kinetic energy before the collision is the fraction of kinetic energy lost in the collision. (a) Letting V be the velocity of the two cars after their collision, apply conservation of momentum to their perfectly inelastic collision:
( )Vmmmvmv
pp
+=+
=
21
finalinitial
or
Solve for and evaluate V:
m/s0.202
m/s10m/s302
21
=
+=
+=
vvV
(b) Express the ratio of the kinetic energy that is lost to the kinetic energy of the two cars before the collision and simplify:
( )
12
12
1
22
21
2
222
1212
1
221
initial
final
initial
initialfinal
initial
−+
=
−+
=
−=
−=
∆
vvV
mvmvVm
KK
KKK
KK
Substitute numerical values to obtain: ( )
( ) ( )200.0
1m/s10m/s30
m/s20222
2
initial
−=
−+
=∆
KK
metal. ofn deformatio the and sound, heat, into ed transformisenergy kinetic initial theof 20%
Chapter 8
552
68 • Picture the Problem We can apply conservation of momentum to this perfectly inelastic collision to find the after-collision speed of the two players. Letting the subscript 1 refer to the running back and the subscript 2 refer to the linebacker, apply conservation of momentum to their perfectly inelastic collision:
( )Vmmvm
pp
2111
fi
or+=
=
Solve for V: 1
21
1 vmm
mV+
=
Substitute numerical values and evaluate V:
( ) m/s13.3m/s7kg051kg58
kg58=
+=V
69 • Picture the Problem We can apply conservation of momentum to this collision to find the after-collision speed of the 5-kg object. Let the direction the 5-kg object is moving before the collision be the positive direction. We can decide whether the collision was elastic by examining the initial and final kinetic energies of the system. (a) Letting the subscript 5 refer to the 5-kg object and the subscript 2 refer to the 10-kg object, apply conservation of momentum to obtain:
f,55i,10105i,5
fi
orvmvmvm
pp
=−
=
Solve for vf,5:
5
i,10105i,5f,5 m
vmvmv
−=
Substitute numerical values and evaluate vf,5:
( )( ) ( )( )
m/s00.2
kg5m/s3kg10m/s4kg5
f,5
−=
−=v
where the minus sign means that the 5-kg object is moving to the left after the collision.
Systems of Particles and Conservation of Momentum
553
(b) Evaluate ∆K for the collision:
( )( ) ( )( )[ ( )( ) ] J0.75m/s3kg10m/s4kg5m/s2kg5 2212
212
21
if −=+−=−=∆ KKK
inelastic. wascollision the0, K Because ≠∆
70 • Picture the Problem The pictorial representation shows the ball and bat just before and just after their collision. Take the direction the bat is moving to be the positive direction. Because the collision is elastic, we can equate the speeds of recession and approach, with the approximation that vi,bat ≈ vf,bat to find vf,ball.
Express the speed of approach of the bat and ball:
( )balli,bati,ballf,batf, vvvv −−=−
Because the mass of the bat is much greater than that of the ball:
batf,bati, vv ≈
Substitute to obtain:
( )balli,batf,ballf,batf, vvvv −−=−
Solve for and evaluate vf,ball: ( )
v
vvvvvvvv
3
22 batf,balli,
balli,batf,batf,ballf,
=
+=+−=
−+=
*71 •• Picture the Problem Let the direction the proton is moving before the collision be the positive x direction. We can use both conservation of momentum and conservation of mechanical energy to obtain an expression for velocity of the proton after the collision. (a) Use the expression for the total momentum of a system to find vcm:
( )
( )i
iv
v
vvP
ˆm/s1.23
ˆm/s30012
and
131ip,
cm
cm
=
=+
=
== ∑
mmm
Mmi
ii
rr
rrr
Chapter 8
554
(b) Use conservation of momentum to obtain one relation for the final velocities:
fnuc,nucfp,pip,p vmvmvm += (1)
Use conservation of mechanical energy to set the velocity of recession equal to the negative of the velocity of approach:
( ) ip,ip,inuc,fp,fnuc, vvvvv =−−=− (2)
To eliminate vnuc,f, solve equation (2) for vnuc,f, and substitute the result in equation (1):
fp,ip,fnuc, vvv +=
( )fp,ip,nucfp,pip,p vvmvmvm ++=
Solve for and evaluate vp,f:
( ) m/s254m/s30013
12
ip,nucp
nucpfp,
−=−
=
+−
=
mmm
vmmmm
v
72 •• Picture the Problem We can use conservation of momentum and the definition of an elastic collision to obtain two equations in v2f and v3f that we can solve simultaneously. Use conservation of momentum to obtain one relation for the final velocities:
2f23f33i3 vmvmvm += (1)
Use conservation of mechanical energy to set the velocity of recession equal to the negative of the velocity of approach:
( ) 3i3i2i3f2f vvvvv =−−=− (2)
Solve equation (2) for v3f , substitute in equation (1) to eliminate v3f, and solve for and evaluate v2f:
( )( )
m/s80.4
kg3kg2m/s4kg322
32
3i32f
=
+=
+=
mmvmv
Use equation (2) to find v3f:
m/s0.800
m/s4.00m/s4.803i2f3f
=
−=−= vvv
Evaluate Ki and Kf: ( )( )
J24.0m/s4kg3 2
212
3i321
3ii
==== vmKK
Systems of Particles and Conservation of Momentum
555
and
( )( )( )( )J0.24
m/s4.8kg2
m/s0.8kg32
21
221
22f22
123f32
12f3ff
=+
=
+=+= vmvmKKK
elastic.been havingcollision with theconsistent are and for obtained values that theconcludecan we, Because 3f2ffi vvKK =
73 •• Picture the Problem We can find the velocity of the center of mass from the definition of the total momentum of the system. We’ll use conservation of energy to find the maximum compression of the spring and express the initial (i.e., before collision) and final (i.e., at separation) velocities. Finally, we’ll transform the velocities from the center of mass frame of reference to the table frame of reference. (a) Use the definition of the total momentum of a system to relate the initial momenta to the velocity of the center of mass:
cmvvPrrr
Mmi
ii == ∑
or ( ) cm211i1 vmmvm +=
Solve for vcm:
21
2i21i1cm mm
vmvmv++
=
Substitute numerical values and evaluate vcm:
( )( ) ( )( )
m/s00.5
kg5kg2m/s3kg5m/s10kg2
cm
=
++
=v
(b) Find the kinetic energy of the system at maximum compression (u1 = u2 = 0):
( )( ) J87.5m/s5kg7 221
2cm2
1cm
==
== MvKK
Use conservation of energy to relate the kinetic energy of the system to the potential energy stored in the spring at maximum compression:
0s =∆+∆ UK
or 0sisfif =−+− UUKK
Because Kf = Kcm and Usi = 0: ( ) 0221
icm =∆+− xkKK
Chapter 8
556
Solve for ∆x: ( )
[ ]
kKvmvm
kKvmvm
kKKx
cm2i22
2i11
cm2i222
12i112
1
cmi
2
2
2
−+=
−+=
−=∆
Substitute numerical values and evaluate ∆x:
( )( ) ( )( ) ( ) m250.0N/m1120
J87.52N/m1120
m/s3kg5m/s10kg2 22
=⎥⎦
⎤−
+=∆x
(c) Find u1i, u2i, and u1f for this elastic collision:
m/s5m/s50and
m/s,2m/s5m/s3m/s,5m/s5m/s10
cm1f1f
cm2i2i
cm1i1i
−=−=−=
−=−=−==−=−=
vvu
vvuvvu
Use conservation of mechanical energy to set the velocity of recession equal to the negative of the velocity of approach and solve for u2f:
( )1i2i1f2f uuuu −−=−
and
( )m/s2
m/s5m/s5m/s21f1i2i2f
=−+−−=
++−= uuuu
Transform u1f and u2f to the table frame of reference:
0m/s5m/s5cm1f1f =+−=+= vuv
and
m/s00.7m/s5m/s2cm2f2f
=+=
+= vuv
*74 •• Picture the Problem Let the system include the earth, the bullet, and the sheet of plywood. Then Wext = 0. Choose the zero of gravitational potential energy to be where the bullet enters the plywood. We can apply both conservation of energy and conservation of momentum to obtain the various physical quantities called for in this problem. (a) Use conservation of mechanical energy after the bullet exits the sheet of plywood to relate its exit speed to the height to which it rises:
0=∆+∆ UK or, because Kf = Ui = 0,
0221 =+− mghmvm
Systems of Particles and Conservation of Momentum
557
Solve for vm: ghvm 2=
Proceed similarly to relate the initial velocity of the plywood to the height to which it rises:
gHvM 2=
(b) Apply conservation of momentum to the collision of the bullet and the sheet of plywood:
fi pp rr=
or Mmm Mvmvmv +=i
Substitute for vm and vM and solve for vmi:
gHmMghvm 22i +=
(c) Express the initial mechanical energy of the system (i.e., just before the collision):
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛++=
=
HmMhH
mMhmg
mvE m
2
2i2
1i
2
Express the final mechanical energy of the system (i.e., when the bullet and block have reached their maximum heights):
( )MHmhgMgHmghE +=+=f
(d) Use the work-energy theorem with Wext = 0 to find the energy dissipated by friction in the inelastic collision:
0frictionif =+− WEE
and
⎥⎦
⎤⎢⎣
⎡−+=
−=
12
fifriction
mM
HhgMH
EEW
75 •• Picture the Problem We can find the velocity of the center of mass from the definition of the total momentum of the system. We’ll use conservation of energy to find the speeds of the particles when their separation is least and when they are far apart. (a) Noting that when the distance between the two particles is least, both move at the same speed, namely vcm, use the definition of the total momentum of a system to relate the initial momenta to the velocity of
cmvvPrrr
Mmi
ii == ∑
or ( ) cmppip vmmvm α+= .
Chapter 8
558
the center of mass: Solve for and evaluate vcm:
0
0
21
ipipcm
200.0
40'
v
mmmv
mmvmvm
vv
=
++
=++
== αα
(b) Use conservation of momentum to obtain one relation for the final velocities:
fpfp0p ααvmvmvm += (1)
Use conservation of mechanical energy to set the velocity of recession equal to the negative of the velocity of approach:
( ) piipifpf vvvvv −=−−=− αα (2)
Solve equation (2) for vpf , substitute in equation (1) to eliminate vpf, and solve for vαf:
00
p
0pf 400.0
422
vmm
mvmmvm
v =+
=+
=α
α
76 • Picture the Problem Let the numeral 1 denote the electron and the numeral 2 the hydrogen atom. We can find the final velocity of the electron and, hence, the fraction of its initial kinetic energy that is transferred to the atom, by transforming to the center-of-mass reference frame, calculating the post-collision velocity of the electron, and then transforming back to the laboratory frame of reference. Express f, the fraction of the electron’s initial kinetic energy that is transferred to the atom: 2
1i
f12i112
1
2f112
1
i
f
i
fi
11
1
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−=
−=−
=
vv
vmvm
KK
KKKf
(1)
Find the velocity of the center of mass:
21
i11cm mm
vmv+
=
or, because m2 = 1840m1,
1i11
i11cm 1841
11840
vmm
vmv =+
=
Find the initial velocity of the electron in the center-of-mass reference frame:
1i
1i1icm1i1i
184111
18411
v
vvvvu
⎟⎠⎞
⎜⎝⎛ −=
−=−=
Systems of Particles and Conservation of Momentum
559
Find the post-collision velocity of the electron in the center-of-mass reference frame by reversing its velocity:
1i1i1f 11841
1 vuu ⎟⎠⎞
⎜⎝⎛ −=−=
To find the final velocity of the electron in the original frame, add vcm to its final velocity in the center-of-mass reference frame:
1icm1f1f 11841
2 vvuv ⎟⎠⎞
⎜⎝⎛ −=+=
Substitute in equation (1) to obtain:
%217.01017.2
11841
211
18412
1
3
2
2
1i
1i
=×=
⎟⎠⎞
⎜⎝⎛ −−=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛ −
−=
−
v
vf
77 •• Picture the Problem The pictorial representation shows the bullet about to imbed itself in the bob of the ballistic pendulum and then, later, when the bob plus bullet have risen to their maximum height. We can use conservation of momentum during the collision to relate the speed of the bullet to the initial speed of the bob plus bullet (V). The initial kinetic energy of the bob plus bullet is transformed into gravitational potential energy when they reach their maximum height. Hence we apply conservation of mechanical energy to relate V to the angle through which the bullet plus bob swings and then solve the momentum and energy equations simultaneously for the speed of the bullet.
Use conservation of momentum to relate the speed of the bullet just before impact to the initial speed of the bob plus bullet:
( )VMmmv +=b
Solve for the speed of the bullet:
VmMv ⎟
⎠⎞
⎜⎝⎛ += 1b (1)
Use conservation of energy to relate 0=∆+∆ UK
Chapter 8
560
the initial kinetic energy of the bullet to the final potential energy of the system:
or, because Kf = Ui = 0, 0fi =+− UK
Substitute for Ki and Uf and solve for V:
( )( ) ( ) 0cos1
221
=−+++−
θgLMmVMm
and ( )θcos12 −= gLV
Substitute for V in equation (1) to obtain:
( )θcos121b −⎟⎠⎞
⎜⎝⎛ += gL
mMv
Substitute numerical values and evaluate vb:
( )( )( ) m/s450cos601m2.3m/s9.812kg0.016
kg1.51 2b =°−⎟⎟
⎠
⎞⎜⎜⎝
⎛+=v
*78 •• Picture the Problem We can apply conservation of momentum and the definition of an elastic collision to obtain equations relating the initial and final velocities of the colliding objects that we can solve for v1f and v2f. Apply conservation of momentum to the elastic collision of the particles to obtain:
2i21i1f22f11 vmvmvmvm +=+ (1)
Relate the initial and final kinetic energies of the particles in an elastic collision:
2i222
12i112
12f222
12f112
1 vmvmvmvm +=+
Rearrange this equation and factor to obtain:
( ) ( )2f1
2i11
2i2
2f22 vvmvvm −=−
or ( )( )
( )( )1fi11fi11
2if22if22
vvvvmvvvvm
+−=+−
(2)
Rearrange equation (1) to obtain:
( ) ( )1f1i12i2f2 vvmvvm −=− (3)
Divide equation (2) by equation (3) to obtain:
1fi12if2 vvvv +=+
Rearrange this equation to obtain equation (4):
1ii2f2f1 vvvv −=− (4)
Multiply equation (4) by m2 and add it to equation (1) to obtain:
( ) ( ) 2i21i211f21 2 vmvmmvmm +−=+
Systems of Particles and Conservation of Momentum
561
Solve for v1f to obtain: iif v
mmmv
mmmmv 2
21
21
21
211
2+
++−
=
Multiply equation (4) by m1 and subtract it from equation (1) to obtain:
( ) ( ) 1i1i212f221 2 vmvmmvmm +−=+
Solve for v2f to obtain: i2
21
12i1
21
1f2
2 vmmmmv
mmmv
+−
++
=
Remarks: Note that the velocities satisfy the condition that ( )1i2i1f2f vvvv −−=− . This verifies that the speed of recession equals the speed of approach. 79 •• Picture the Problem As in this problem, Problem 78 involves an elastic, one-dimensional collision between two objects. Both solutions involve using the conservation of momentum equation 2i21i1f22f11 vmvmvmvm +=+ and the elastic collision equation 1ii2f2f1 vvvv −=− . In part (a) we can simply set the masses equal to each other and substitute in the equations in Problem 78 to show that the particles "swap" velocities. In part (b) we can divide the numerator and denominator of the equations in Problem 78 by m2 and use the condition that m2 >> m1 to show that v1f ≈ −v1i+2v2i and v2f ≈ v2i. (a) From Problem 78 we have: 2i
21
2i1
21
21f1
2 vmm
mvmmmmv
++
+−
= (1)
and
2i21
121i
21
12f
2 vmmmmv
mmmv
+−
++
= (2)
Set m1 = m2 = m to obtain:
i2i2f12 vv
mmmv =+
=
and
1i1if22 vv
mmmv =+
=
(b) Divide the numerator and denominator of both terms in equation (1) by m2 to obtain:
2i
2
1i1
2
1
2
1
f1
1
2
1
1v
mmv
mmmm
v+
++
−=
If m2 >> m1:
2ii1f1 2vvv +−≈
Chapter 8
562
Divide the numerator and denominator of both terms in equation (2) by m2 to obtain:
2i
2
1
2
1
1i
2
1
2
1
2f
1
1
1
2v
mm
mm
v
mm
mm
v+
−+
+=
If m2 >> m1:
2i2f vv ≈
Remarks: Note that, in both parts of this problem, the velocities satisfy the condition that ( )1i2i1f2f vvvv −−=− . This verifies that the speed of recession equals the speed of approach. Perfectly Inelastic Collisions and the Ballistic Pendulum 80 •• Picture the Problem Choose Ug = 0 at the bob’s equilibrium position. Momentum is conserved in the collision of the bullet with bob and the initial kinetic energy of the bob plus bullet is transformed into gravitational potential energy as it swings up to the top of the circle. If the bullet plus bob just makes it to the top of the circle with zero speed, it will swing through a complete circle. Use conservation of momentum to relate the speed of the bullet just before impact to the initial speed of the bob plus bullet:
( )Vmmvm 211 +=
Solve for the speed of the bullet: V
mmv ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
1
21 (1)
Use conservation of energy to relate the initial kinetic energy of the bob plus bullet to their potential energy at the top of the circle:
0=∆+∆ UK or, because Kf = Ui = 0,
0fi =+− UK
Substitute for Ki and Uf: ( ) ( ) ( ) 02212
2121 =+++− LgmmVmm
Solve for V:
gLV =
Substitute for V in equation (1) and simplify to obtain: gL
mmv ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
1
21
Systems of Particles and Conservation of Momentum
563
*81 •• Picture the Problem Choose Ug = 0 at the equilibrium position of the ballistic pendulum. Momentum is conserved in the collision of the bullet with the bob and kinetic energy is transformed into gravitational potential energy as the bob swings up to its maximum height. Letting V represent the initial speed of the bob as it begins its upward swing, use conservation of momentum to relate this speed to the speeds of the bullet just before and after its collision with the bob:
( ) Vmvmvm 221
11 +=
Solve for the speed of the bob: vmmV
2
1
2= (1)
Use conservation of energy to relate the initial kinetic energy of the bob to its potential energy at its maximum height:
0=∆+∆ UK or, because Kf = Ui = 0,
0fi =+− UK
Substitute for Ki and Uf: 022
221 =+− ghmVm
Solve for h: g
Vh2
2
= (2)
Substitute V from equation (1) in equation (2) and simplify to obtain: 2
2
12
2
2
1
822
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛
=mm
gv
g
vmm
h
82 • Picture the Problem Let the mass of the bullet be m, that of the wooden block M, the pre-collision velocity of the bullet v, and the post-collision velocity of the block+bullet be V. We can use conservation of momentum to find the velocity of the block with the bullet imbedded in it just after their perfectly inelastic collision. We can use Newton’s 2nd law to find the acceleration of the sliding block and a constant-acceleration equation to find the distance the block slides.
Chapter 8
564
Using a constant-acceleration equation, relate the velocity of the block+bullet just after their collision to their acceleration and displacement before stopping:
xaV ∆+= 20 2 because the final velocity of the block+bullet is zero.
Solve for the distance the block slides before coming to rest: a
Vx2
2
−=∆ (1)
Use conservation of momentum to relate the pre-collision velocity of the bullet to the post-collision velocity of the block+bullet:
( )VMmmv +=
Solve for V: v
MmmV+
=
Substitute in equation (1) to obtain: 2
21
⎟⎠⎞
⎜⎝⎛
+−=∆ v
Mmm
ax (2)
Apply aF rr
m=∑ to the block+bullet (see the FBD in the diagram):
( )aMmfFx +=−=∑ k (3) and
( ) 0n =+−=∑ gMmFFy (4)
Use the definition of the coefficient of kinetic friction and equation (4) to obtain:
( )gMmFf +== knkk µµ
Substitute in equation (3): ( ) ( )aMmgMm +=+− kµ
Solve for a to obtain: ga kµ−=
Substitute in equation (2) to obtain: 2
k21
⎟⎠⎞
⎜⎝⎛
+=∆ v
Mmm
gx
µ
Systems of Particles and Conservation of Momentum
565
Substitute numerical values and evaluate ∆x:
( )( ) ( ) m130.0m/s750kg10.5kg0.0105
kg0.0105m/s9.810.222
12
2 =⎟⎟⎠
⎞⎜⎜⎝
⎛+
=∆x
83 •• Picture the Problem The collision of the ball with the box is perfectly inelastic and we can find the speed of the box-and-ball immediately after their collision by applying conservation of momentum. If we assume that the kinetic friction force is constant, we can use a constant-acceleration equation to find the acceleration of the box and ball combination and the definition of µk to find its value. Using its definition, express the coefficient of kinetic friction of the table:
( )( ) g
agmMamM
Ff
=++
==n
kkµ (1)
Use conservation of momentum to relate the speed of the ball just before the collision to the speed of the ball+box immediately after the collision:
( )vMmMV +=
Solve for v:
MmMVv+
= (2)
Use a constant-acceleration equation to relate the sliding distance of the ball+box to its initial and final velocities and its acceleration:
xavv ∆+= 22i
2f
or, because vf = 0 and vi = v, xav ∆+= 20 2
Solve for a:
xva∆
−=2
2
Substitute in equation (1) to obtain:
xgv∆
=2
2
kµ
Use equation (2) to eliminate v:
2
2
k
121
21
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+∆=
⎟⎠⎞
⎜⎝⎛
+∆=
Mm
Vxg
MmMV
xgµ
Chapter 8
566
Substitute numerical values and evaluate µk:
( )( ) 0529.01
kg0.425kg0.327m/s1.3
m0.52m/s9.8121
2
2k =
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
+=µ
*84 •• Picture the Problem Jane’s collision with Tarzan is a perfectly inelastic collision. We can find her speed v1 just before she grabs Tarzan from conservation of energy and their speed V just after she grabs him from conservation of momentum. Their kinetic energy just after their collision will be transformed into gravitational potential energy when they have reached their greatest height h.
Use conservation of energy to relate the potential energy of Jane and Tarzan at their highest point (2) to their kinetic energy immediately after Jane grabbed Tarzan:
12 KU = or
2TJ2
1TJ Vmghm ++ =
Solve for h to obtain: g
Vh2
2
= (1)
Use conservation of momentum to relate Jane’s velocity just before she collides with Tarzan to their velocity just after their perfectly inelastic collision:
Vmvm TJ1J +=
Solve for V: 1
TJ
J vmmV
+
= (2)
Apply conservation of energy to relate Jane’s kinetic energy at 1 to her potential energy at 0:
01 UK = or
gLmvm J21J2
1 =
Systems of Particles and Conservation of Momentum
567
Solve for v1: gLv 21 =
Substitute in equation (2) to obtain: gL
mmV 2
TJ
J
+
=
Substitute in equation (1) and simplify:
LmmgL
mm
gh
2
TJ
J
2
TJ
J 221
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
++
Substitute numerical values and evaluate h: ( ) m3.94m25
kg82kg45kg45
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
=h
Exploding Objects and Radioactive Decay 85 •• Picture the Problem This nuclear reaction is 4Be → 2α + 1.5×10−14 J. In order to conserve momentum, the alpha particles will have move in opposite directions with the same velocities. We’ll use conservation of energy to find their speeds. Letting E represent the energy released in the reaction, express conservation of energy for this process:
( ) EvmK == 22122 ααα
Solve for vα:
αα m
Ev =
Substitute numerical values and evaluate vα: m/s1050.1
kg106.68J101.5 6
27
14
×=××
= −
−
αv
86 •• Picture the Problem This nuclear reaction is 5Li → α + p + 3.15 × 10−13 J. To conserve momentum, the alpha particle and proton must move in opposite directions. We’ll apply both conservation of energy and conservation of momentum to find the speeds of the proton and alpha particle. Use conservation of momentum in this process to express the alpha particle’s velocity in terms of the proton’s:
0fi == pp
and αα vmvm −= pp0
Chapter 8
568
Solve for vα and substitute for mα to obtain: p4
1p
p
pp
p
4vv
mm
vmm
v ===α
α
Letting E represent the energy released in the reaction, apply conservation of energy to the process:
EKK =+ αp
or Evmvm =+ 2
212
pp21
αα
Substitute for vα: ( ) Evmvm =+ 2
p41
212
pp21
α
Solve for vp and substitute for mα to obtain:
pppp 416
3216
32mm
Emm
Ev+
=+
=α
Substitute numerical values and evaluate vp:
( )( )
m/s1074.1
kg101.6720J103.1532
7
27
13
p
×=
××
= −
−
v
Use the relationship between vp and vα to obtain vα:
( )m/s104.34
m/s101.746
741
p41
×=
×== vvα
87 ••• Picture the Problem The pictorial representation shows the projectile at its maximum elevation and is moving horizontally. It also shows the two fragments resulting from the explosion. We chose the system to include the projectile and the earth so that no external forces act to change the momentum of the system during the explosion. With this choice of system we can also use conservation of energy to determine the elevation of the projectile when it explodes. We’ll also find it useful to use constant-acceleration equations in our description of the motion of the projectile and its fragments.
Systems of Particles and Conservation of Momentum
569
(a) Use conservation of momentum to relate the velocity of the projectile before its explosion to the velocities of its two parts after the explosion:
jjii
vvvpp
ˆˆˆˆ22111133
221133
fi
yyx vmvmvmvm
mmm
−+=
+==
rrr
rr
The only way this equality can hold is if:
2211
1133
and
yy
x
vmvm
vmvm
=
=
Express v3 in terms of v0 and substitute for the masses to obtain: ( ) m/s312cos30m/s1203
cos33 031
=°=== θvvvx
and 21 2 yy vv = (1)
Using a constant-acceleration equation with the downward direction positive, relate vy2 to the time it takes the 2-kg fragment to hit the ground:
( )221
2 tgtvy y ∆+∆=∆
( )t
tgyvy ∆∆−∆
=2
21
2 (2)
With Ug = 0 at the launch site, apply conservation of energy to the climb of the projectile to its maximum elevation:
0=∆+∆ UK Because Kf = Ui = 0, 0fi =+− UK
or 03
2032
1 =∆+− ygmvm y
Solve for ∆y: ( )
gv
gv
y y
230sin
2
20
20 °
==∆
Substitute numerical values and evaluate ∆y:
( )[ ]( ) m183.5
m/s9.812sin30m/s120
2
2
=°
=∆y
Substitute in equation (2) and evaluate vy2:
( )( )
m/s33.3s3.6
s3.6m/s9.81m183.5 2221
2
=
−=yv
Substitute in equation (1) and evaluate vy1:
( ) m/s66.6m/s33.321 ==yv
Chapter 8
570
Express 1vr in vector form:
( ) ( ) ji
jiv
ˆm/s6.66ˆm/s312
ˆˆ111
+=
+= yx vvr
(b) Express the total distance d traveled by the 1-kg fragment:
'xxd ∆+∆= (3)
Relate ∆x to v0 and the time-to-explosion:
( )( )exp0 cos tvx ∆=∆ θ (4)
Using a constant-acceleration equation, express ∆texp: g
vg
vt y θsin00exp ==∆
Substitute numerical values and evaluate ∆texp:
( ) s6.12m/s9.81
sin30m/s1202exp =
°=∆t
Substitute in equation (4) and evaluate ∆x:
( )( )( )m636.5
s6.12cos30m/s120=
°=∆x
Relate the distance traveled by the 1-kg fragment after the explosion to the time it takes it to reach the ground:
t'vx' x ∆=∆ 1
Using a constant-acceleration equation, relate the time ∆t′ for the 1-kg fragment to reach the ground to its initial speed in the y direction and the distance to the ground:
( )221
1 t'gt'vy y ∆−∆=∆
Substitute to obtain the quadratic equation:
( ) ( ) 0s4.37s6.13 22 =−∆−∆ t't'
Solve the quadratic equation to find ∆t′:
∆t′ = 15.9 s
Substitute in equation (3) and evaluate d: ( )( )
km5.61
s15.9m/s312m636.51
=
+=∆+∆=∆+∆= t'vxx'xd x
Systems of Particles and Conservation of Momentum
571
(c) Express the energy released in the explosion:
ifexp KKKE −=∆= (5)
Find the kinetic energy of the fragments after the explosion: ( ) ( ) ( )[ ]
( )( )kJ0.52
m/s33.3kg2
m/s66.6m/s312kg12
21
2221
2222
12112
121f
=+
+=
+=+= vmvmKKK
Find the kinetic energy of the projectile before the explosion:
( )( ) ( )[ ]
kJ2.1630cosm/s201kg3
cos2
21
2032
12332
1i
=
°=
== θvmvmK
Substitute in equation (5) to determine the energy released in the explosion:
kJ35.8
kJ16.2kJ0.52ifexp
=
−=−= KKE
*88 ••• Picture the Problem This nuclear reaction is 9B → 2α + p + 4.4×10−14 J. Assume that the proton moves in the –x direction as shown in the figure. The sum of the kinetic energies of the decay products equals the energy released in the decay. We’ll use conservation of momentum to find the angle between the velocities of the proton and the alpha particles. Note that 'αα vv = .
Express the energy released to the kinetic energies of the decay products:
relp 2 EKK =+ α
or ( ) rel
2212
pp21 2 Evmvm =+ αα
Solve for vα:
αα m
vmEv
2pp2
1rel −
=
Chapter 8
572
Substitute numerical values and evaluate vα:
( )( ) m/s1044.1kg106.68
m/s106kg101.67kg106.68J104.4 6
27
262721
27
14
×=×
××−
××
= −
−
−
−
αv
Given that the boron isotope was at rest prior to the decay, use conservation of momentum to relate the momenta of the decay products:
0if == pp rr ⇒ 0f =xp
( ) 0cos2 pp =−∴ vmvm θαα
or ( ) 0cos42 ppp =− vmvm θα
Solve for θ :
( ) °±=⎥⎦
⎤⎢⎣
⎡×
×=
⎥⎦
⎤⎢⎣
⎡=
−
−
7.58m/s101.448
m/s106cos
8cos
6
61
p1
α
θvv
Let θ′ equal the angle the velocities of the alpha particles make with that of the proton:
( )°±=
°−°±=
121
7.58180'θ
Coefficient of Restitution 89 • Picture the Problem The coefficient of restitution is defined as the ratio of the velocity of recession to the velocity of approach. These velocities can be determined from the heights from which the ball was dropped and the height to which it rebounded by using conservation of mechanical energy. Use its definition to relate the coefficient of restitution to the velocities of approach and recession:
app
rec
vve =
Letting Ug = 0 at the surface of the steel plate, apply conservation of energy to express the velocity of approach:
0=∆+∆ UK Because Ki = Uf = 0,
0or
0
app2app2
1
if
=−
=−
mghmv
UK
Solve for vapp: appapp 2ghv =
Systems of Particles and Conservation of Momentum
573
In like manner, show that: recrec 2ghv =
Substitute in the equation for e to obtain:
app
rec
app
rec
22
hh
ghgh
e ==
Substitute numerical values and evaluate e:
913.0m3m2.5
==e
*90 • Picture the Problem The coefficient of restitution is defined as the ratio of the velocity of recession to the velocity of approach. These velocities can be determined from the heights from which an object was dropped and the height to which it rebounded by using conservation of mechanical energy. Use its definition to relate the coefficient of restitution to the velocities of approach and recession:
app
rec
vve =
Letting Ug = 0 at the surface of the steel plate, apply conservation of energy to express the velocity of approach:
0=∆+∆ UK Because Ki = Uf = 0,
0or
0
app2app2
1
if
=−
=−
mghmv
UK
Solve for vapp: appapp 2ghv =
In like manner, show that: recrec 2ghv =
Substitute in the equation for e to obtain:
app
rec
app
rec
22
hh
ghgh
e ==
Find emin: 825.0
cm254cm173
min ==e
Find emax: 849.0
cm254cm183
max ==e
Chapter 8
574
and 849.0825.0 ≤≤ e
91 • Picture the Problem Because the rebound kinetic energy is proportional to the rebound height, the percentage of mechanical energy lost in one bounce can be inferred from knowledge of the rebound height. The coefficient of restitution is defined as the ratio of the velocity of recession to the velocity of approach. These velocities can be determined from the heights from which an object was dropped and the height to which it rebounded by using conservation of mechanical energy. (a) We know, from conservation of energy, that the kinetic energy of an object dropped from a given height h is proportional to h:
K α h.
If, for each bounce of the ball, hrec = 0.8happ:
lost. isenergy mechanical its of %20
(b) Use its definition to relate the coefficient of restitution to the velocities of approach and recession:
app
rec
vve =
Letting Ug = 0 at the surface from which the ball is rebounding, apply conservation of energy to express the velocity of approach:
0=∆+∆ UK Because Ki = Uf = 0,
0or
0
app2app2
1
if
=−
=−
mghmv
UK
Solve for vapp: appapp 2ghv =
In like manner, show that: recrec 2ghv =
Substitute in the equation for e to obtain:
app
rec
app
rec
22
hh
ghgh
e ==
Substitute for app
rec
hh
to obtain: 894.08.0 ==e
Systems of Particles and Conservation of Momentum
575
92 •• Picture the Problem Let the numeral 2 refer to the 2-kg object and the numeral 4 to the 4-kg object. Choose a coordinate system in which the direction the 2-kg object is moving before the collision is the positive x direction and let the system consist of the earth, the surface on which the objects slide, and the objects. Then we can use conservation of momentum to find the velocity of the recoiling 4-kg object. We can find the energy transformed in the collision by calculating the difference between the kinetic energies before and after the collision and the coefficient of restitution from its definition. (a) Use conservation of momentum in one dimension to relate the initial and final momenta of the participants in the collision:
f22f44i22
fi
orvmvmvm −=
= pp rr
Solve for and evaluate the final velocity of the 4-kg object:
( )( ) m/s3.50kg4
m/s1m/s6kg24
f22i22f4
=+
=
+=
mvmvmv
(b) Express the energy lost in terms of the kinetic energies before and after the collision:
( )( )[ ]2
f442f2
2i222
1
2f442
12f222
12i222
1
filost
vmvvm
vmvmvm
KKE
−−=
+−=
−=
Substitute numerical values and evaluate Elost:
( ) ( ) ( ){ }( ) ( )( )[ ] J5.10m/s3.5kg4m/s1m/s6kg2 22221
lost =−−=E
(c) Use the definition of the coefficient of restitution:
( ) 0.750m/s6
m/s1m/s3.5
i2
f2f4
app
rec =−−
=−
==v
vvvve
93 •• Picture the Problem Let the numeral 2 refer to the 2-kg block and the numeral 3 to the 3-kg block. Choose a coordinate system in which the direction the blocks are moving before the collision is the positive x direction and let the system consist of the earth, the surface on which the blocks move, and the blocks. Then we can use conservation of momentum find the velocity of the 2-kg block after the collision. We can find the coefficient of restitution from its definition.
Chapter 8
576
(a) Use conservation of momentum in one dimension to relate the initial and final momenta of the participants in the collision:
f33f223i3i22
fi
orvmvmvmvm +=+
= pp rr
Solve for the final velocity of the 2-kg object: 2
f33i33i22f2 m
vmvmvmv −+=
Substitute numerical values and evaluate v2f:
( )( ) ( )( ) m/s70.1kg2
m/s4.2m/s2kg3m/s5kg2f2 =
−+=v
(b) Use the definition of the coefficient of restitution:
0.833
m/s2m/s5m/s7.1m/s2.4
i3i2
f2f3
app
rec
=
−−
=−−
==vvvv
vve
Collisions in Three Dimensions *94 •• Picture the Problem We can use the definition of the magnitude of a vector and the definition of the dot product to establish the result called for in (a). In part (b) we can use the result of part (a), the conservation of momentum, and the definition of an elastic collision (kinetic energy is conserved) to show that the particles separate at right angles. (a) Find the dot product of CB
rr+
with itself: ( ) ( )
CB
CBCBrr
rrrr
⋅++=
+⋅+
222 CB
Because CBA
rrr+= : ( ) ( )CBCBCB
rrrrrr+⋅+=+=
22A
Substitute to obtain: CB
rr⋅++= 2222 CBA
(b) Apply conservation of momentum to the collision of the particles:
Ppprrr
=+ 21
Form the dot product of each side of this equation with itself to obtain:
( ) ( ) PPpppprrrrrr
⋅=+⋅+ 2121 or
221
22
21 2 Ppp =⋅++ pp
rr (1)
Apply the definition of an elastic collision to obtain: m
Pm
pm
p222
222
21 =+
Systems of Particles and Conservation of Momentum
577
or 22
221 Ppp =+ (2)
Subtract equation (1) from equation (2) to obtain:
02 21 =⋅ pp rror 021 =⋅ pp rr
i.e., the particles move apart along paths that are at right angles to each other.
95 •
Picture the Problem Let the initial direction of motion of the cue ball be the positive x direction. We can apply conservation of energy to determine the angle the cue ball makes with the positive x direction and the conservation of momentum to find the final velocities of the cue ball and the eight ball. (a) Use conservation of energy to relate the velocities of the collision participants before and after the collision:
28
2cf
2ci
282
12cf2
12ci2
1
orvvv
mvmvmv
+=
+=
This Pythagorean relationship tells us that 8cfci and,, vvv rrr
form a right
triangle. Hence: °=
°=+
60
and90
cf
8cf
θ
θθ
(b) Use conservation of momentum in the x direction to relate the velocities of the collision participants before and after the collision:
88cfcfci
xfxi
coscosor
θθ mvmvmv +=
= pp rr
Use conservation of momentum in the y direction to obtain a second equation relating the velocities of the collision participants before and after the collision:
88cfcf
yfyi
sinsin0or
θθ mvmv +=
= pprr
Solve these equations simultaneously to obtain:
m/s33.4
and
m/s50.2
8
cf
=
=
v
v
Chapter 8
578
96 •• Picture the Problem We can find the final velocity of the object whose mass is M1 by using the conservation of momentum. Whether the collision was elastic can be decided by examining the difference between the initial and final kinetic energy of the interacting objects. (a) Use conservation of momentum to relate the initial and final velocities of the two objects:
fi pp rr=
or ( ) ( ) 1f04
102
10
ˆ2ˆ2ˆ viji rmvmvmmv +=+
Simplify to obtain:
1f021
00ˆˆˆ viji r
+=+ vvv
Solve for 1fvr : jiv ˆˆ002
11f vv +=r
(b) Express the difference between the kinetic energy of the system before the collision and its kinetic energy after the collision:
( ) [ ][ ] [ ]
( ) ( )[ ] 2016
12016
1204
5204
1202
1
2f2
2f1
2i2
2i12
12f2
2f1
2i2
2i12
1
2f22
2f11
2i22
2i112
12f1f2i1ifi
22
2222
mvvvvvm
vvvvmmvmvmvmv
vMvMvMvMKKKKKKE
=−−+=
−−+=−−+=
−−+=+−+=−=∆
. iscollision the0, Because inelasticE ≠∆
*97 •• Picture the Problem Let the direction of motion of the puck that is moving before the collision be the positive x direction. Applying conservation of momentum to the collision in both the x and y directions will lead us to two equations in the unknowns v1 and v2 that we can solve simultaneously. We can decide whether the collision was elastic by either calculating the system’s kinetic energy before and after the collision or by determining whether the angle between the final velocities is 90°. (a) Use conservation of momentum in the x direction to relate the velocities of the collision participants before and after the collision: °+°=
°+°=
=
60cos30cosor
60cos30cosor
21
21
xfxi
vvv
mvmvmv
pp
Systems of Particles and Conservation of Momentum
579
Use conservation of momentum in the y direction to obtain a second equation relating the velocities of the collision participants before and after the collision: °−°=
°−°=
=
60sin30sin0or
60sin30sin0or
21
21
yfyi
vv
mvmv
pp
Solve these equations simultaneously to obtain:
m/s00.1andm/s73.1 21 == vv
(b) . wascollision the,90 is and between angle theBecause 21 elastic°vv rr
98 •• Picture the Problem Let the direction of motion of the object that is moving before the collision be the positive x direction. Applying conservation of momentum to the motion in both the x and y directions will lead us to two equations in the unknowns v2 and θ2 that we can solve simultaneously. We can show that the collision was elastic by showing that the system’s kinetic energy before and after the collision is the same. (a) Use conservation of momentum in the x direction to relate the velocities of the collision participants before and after the collision:
22100
22100
xfxi
cos2cos53
orcos2cos53
or
θθ
θθ
vvv
mvmvmv
pp
+=
+=
=
Use conservation of momentum in the y direction to obtain a second equation relating the velocities of the collision participants before and after the collision: 2210
2210
yfyi
sin2sin50
orsin2sin50
or
θθ
θθ
vv
mvmv
pp
−=
−=
=
Note that if tanθ1 = 2, then:
52sinand
51cos 11 == θθ
Substitute in the momentum equations to obtain:
220
2200
cosor
cos25
153
θ
θ
vv
vvv
=
+=
and
Chapter 8
580
220
220
sin0or
sin25
250
θ
θ
vv
vv
−=
−=
Solve these equations simultaneously for θ2 :
°== − 0.451tan 12θ
Substitute to find v2:
00
2
02 2
45coscosvvvv =
°==
θ
(b) To show that the collision was elastic, find the before-collision and after-collision kinetic energies:
( )
( ) ( )( )20
2
021
2
021
f
20
202
1i
5.4
225
and5.43
mv
vmvmK
mvvmK
=
+=
==
elastic. iscollision the, Because fi KK =
*99 •• Picture the Problem Let the direction of motion of the ball that is moving before the collision be the positive x direction. Let v represent the velocity of the ball that is moving before the collision, v1 its velocity after the collision and v2 the velocity of the initially-at-rest ball after the collision. We know that because the collision is elastic and the balls have the same mass, v1 and v2 are 90° apart. Applying conservation of momentum to the collision in both the x and y directions will lead us to two equations in the unknowns v1 and v2 that we can solve simultaneously. Noting that the angle of deflection for the recoiling ball is 60°, use conservation of momentum in the x direction to relate the velocities of the collision participants before and after the collision:
°+°=
°+°=
=
60cos30cosor
60cos30cosor
21
21
xfxi
vvv
mvmvmv
pp
Use conservation of momentum in the y direction to obtain a second equation relating the velocities of the collision participants before and after the collision: °−°=
°−°=
=
60sin30sin0or
60sin30sin0or
21
21
yfyi
vv
mvmv
pp
Systems of Particles and Conservation of Momentum
581
Solve these equations simultaneously to obtain:
m/s00.5andm/s66.8 21 == vv
100 •• Picture the Problem Choose the coordinate system shown in the diagram below with the x-axis the axis of initial approach of the first particle. Call V the speed of the target particle after the collision. In part (a) we can apply conservation of momentum in the x and y directions to obtain two equations that we can solve simultaneously for tanθ. In part (b) we can use conservation of momentum in vector form and the elastic-collision equation to show that v = v0cosφ.
(a) Apply conservation of momentum in the x direction to obtain:
θφ coscos0 Vvv += (1)
Apply conservation of momentum in the y direction to obtain:
θφ sinsin Vv = (2)
Solve equation (1) for Vcosθ :
φθ coscos 0 vvV −= (3)
Divide equation (2) by equation (3) to obtain: φ
φθθ
cossin
cossin
0 vvv
VV
−=
or
φφθ
cossintan
0 vvv−
=
(b) Apply conservation of momentum to obtain:
Vvvrrr
+=0
Draw the vector diagram representing this equation:
Use the definition of an elastic 222
0 Vvv +=
Chapter 8
582
collision to obtain: If this Pythagorean condition is to hold, the third angle of the triangle must be a right angle and, using the definition of the cosine function:
φcos0vv =
Center-of-Mass Frame 101 •• Picture the Problem The total kinetic energy of a system of particles is the sum of the kinetic energy of the center of mass and the kinetic energy relative to the center of mass. The kinetic energy of a particle of mass m is related to momentum according to mpK 22= .
Express the total kinetic energy of the system:
cmrel KKK += (1)
Relate the kinetic energy relative to the center of mass to the momenta of the two particles:
( )21
2121
2
21
1
21
rel 222 mmmmp
mp
mpK +
=+=
Express the kinetic energy of the center of mass of the two particles:
( )( ) 21
21
21
21
cm2
22
mmp
mmpK
+=
+=
Substitute in equation (1) and simplify to obtain:
( )
⎥⎦
⎤⎢⎣
⎡+
++=
++
+=
2212
21
2221
21
21
21
21
21
2121
62
22
mmmmmmmmp
mmp
mmmmpK
In an elastic collision:
⎥⎦
⎤⎢⎣
⎡+
++=
⎥⎦
⎤⎢⎣
⎡+
++=
=
2212
21
2221
21
21
2212
21
2221
21
21
fi
62
62
mmmmmmmmp'
mmmmmmmmp
KK
Simplify to obtain: ( ) ( ) 11
21
21 pppp '' ±=⇒=
and collide.not do particles the, If 11 pp' +=
Systems of Particles and Conservation of Momentum
583
*102 •• Picture the Problem Let the numerals 3 and 1 denote the blocks whose masses are 3 kg and 1 kg respectively. We can use cmvv rr Mm
iii =∑ to find the velocity of the center-of-
mass of the system and simply follow the directions in the problem step by step. (a) Express the total momentum of this two-particle system in terms of the velocity of its center of mass:
( ) cm31cm
3311
vv
vvvPrr
rrrr
mmM
mmmi
ii
+==
+== ∑
Solve for cmvr :
13
1133cm mm
mm++
=vvvrr
r
Substitute numerical values and evaluate cmvr :
( )( ) ( )( )
( )i
iiv
ˆm/s3.00
kg1kg3
ˆm/s3kg1ˆm/s5kg3cm
−=
++−
=r
(b) Find the velocity of the 3-kg block in the center of mass reference frame:
( ) ( )( )i
iivvuˆm/s2.00
ˆm/s3ˆm/s5cm33
−=
−−−=−=rrr
Find the velocity of the 1-kg block in the center of mass reference frame:
( ) ( )( )i
iivvuˆm/s00.6
ˆm/s3ˆm/s3cm11
=
−−=−=rrr
(c) Express the after-collision velocities of both blocks in the center of mass reference frame:
( )iu ˆm/s00.23 ='r
and
( )iu ˆm/s00.61 −='r
(d) Transform the after-collision velocity of the 3-kg block from the center of mass reference frame to the original reference frame:
( ) ( )( )i
iivuvˆm/s00.1
ˆm/s3ˆm/s2cm33
−=
−+=+=rrr ''
Transform the after-collision velocity of the 1-kg block from the center of mass reference frame to the original reference frame:
( ) ( )( )i
iivuvˆm/s00.9
ˆm/s3ˆm/s6cm11
−=
−+−=+=rrr ''
(e) Express Ki in the original frame of 2112
12332
1i vmvmK +=
Chapter 8
584
reference:
Substitute numerical values and evaluate Ki:
( )( ) ( )( )[ ]J42.0
m/s3kg1m/s5kg3 2221
i
=
+=K
Express Kf in the original frame of reference:
2112
12332
1f v'mv'mK +=
Substitute numerical values and evaluate Kf:
( )( ) ( )( )[ ]J42.0
m/s9kg1m/s1kg3 2221
f
=
+=K
103 •• Picture the Problem Let the numerals 3 and 1 denote the blocks whose masses are 3 kg and 1 kg respectively. We can use cmvv rr Mm
iii =∑ to find the velocity of the center-of-
mass of the system and simply follow the directions in the problem step by step. (a) Express the total momentum of this two-particle system in terms of the velocity of its center of mass:
( ) cm53cm
5533
vv
vvvPrr
rrrr
mmM
mmmi
ii
+==
+== ∑
Solve for cmvr :
53
5533cm mm
mm++
=vvvrr
r
Substitute numerical values and evaluate cmvr :
( )( ) ( )( )
0
kg5kg3
ˆm/s3kg5ˆm/s5kg3cm
=
++−
=iivr
(b) Find the velocity of the 3-kg block in the center of mass reference frame:
( )( ) i
ivvuˆm/s5
0ˆm/s5cm33
−=
−−=−=rrr
Find the velocity of the 5-kg block in the center of mass reference frame:
( )( )i
ivvuˆm/s3
0ˆm/s3cm55
=
−=−=rrr
(c) Express the after-collision velocities of both blocks in the center of mass reference frame:
( )iu ˆm/s53 ='r
and
Systems of Particles and Conservation of Momentum
585
m/s75.05 ='u
(d) Transform the after-collision velocity of the 3-kg block from the center of mass reference frame to the original reference frame:
( )( )i
ivuvˆm/s5
0ˆm/s5cm33
=
+=+=rrr ''
Transform the after-collision velocity of the 5-kg block from the center of mass reference frame to the original reference frame:
( )( )i
ivuvˆm/s3
0ˆm/s3cm55
−=
+−=+=rrr ''
(e) Express Ki in the original frame of reference:
2552
12332
1i vmvmK +=
Substitute numerical values and evaluate Ki:
( )( ) ( )( )[ ]J60.0
m/s3kg5m/s5kg3 2221
i
=
+=K
Express Kf in the original frame of reference:
2552
12332
1f v'mv'mK +=
Substitute numerical values and evaluate Kf:
( )( )[ ( )( ) ] J60.0m/s3kg5m/s5kg3 2221
f =+=K
Systems With Continuously Varying Mass: Rocket Propulsion 104 •• Picture the Problem The thrust of a rocket Fth depends on the burn rate of its fuel dm/dt and the relative speed of its exhaust gases uex according to exth udtdmF = .
Using its definition, relate the rocket’s thrust to the relative speed of its exhaust gases:
exth udtdmF =
Substitute numerical values and evaluate Fth:
( )( ) MN20.1km/s6kg/s200th ==F
Chapter 8
586
105 •• Picture the Problem The thrust of a rocket Fth depends on the burn rate of its fuel dm/dt and the relative speed of its exhaust gases uex according to exth udtdmF = . The final
velocity vf of a rocket depends on the relative speed of its exhaust gases uex, its payload to initial mass ratio mf/m0 and its burn time according to ( ) b0fexf ln gtmmuv −−= .
(a) Using its definition, relate the rocket’s thrust to the relative speed of its exhaust gases:
exth udtdmF =
Substitute numerical values and evaluate Fth:
( )( ) kN360km/s8.1kg/s200th ==F
(b) Relate the time to burnout to the mass of the fuel and its burn rate:
dtdmm
dtdmmt
/8.0
/0fuel
b ==
Substitute numerical values and evaluate tb:
( ) s120kg/s200
kg30,0000.8b ==t
(c) Relate the final velocity of a rocket to its initial mass, exhaust velocity, and burn time:
b0
fexf ln gt
mmuv −⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
Substitute numerical values and evaluate vf:
( ) ( )( ) km/s72.1s120m/s9.8151lnkm/s1.8 2
f =−⎟⎠⎞
⎜⎝⎛−=v
*106 •• Picture the Problem We can use the dimensions of thrust, burn rate, and acceleration to show that the dimension of specific impulse is time. Combining the definitions of rocket thrust and specific impulse will lead us to spex gIu = . (a) Express the dimension of specific impulse in terms of the dimensions of Fth, R, and g:
[ ] [ ][ ][ ] T
TL
TM
TLM
2
2th
sp =⋅
⋅
==gR
FI
(b) From the definition of rocket thrust we have:
exth RuF =
Solve for uex:
RFu th
ex =
Systems of Particles and Conservation of Momentum
587
Substitute for Fth to obtain: sp
spex gI
RRgI
u == (1)
(c) Solve equation (1) for Isp and substitute for uex to obtain: Rg
FI thsp =
From Example 8-21 we have: R = 1.384×104 kg/s and Fth = 3.4×106 N
Substitute numerical values and evaluate Isp: ( )( )
s25.0
m/s81.9kg/s101.384N103.4
24
6
sp
=
××
=I
*107 ••• Picture the Problem We can use the rocket equation and the definition of rocket thrust to show that ga00 1+=τ . In part (b) we can express the burn time tb in terms of the initial and final masses of the rocket and the rate at which the fuel burns, and then use this equation to express the rocket’s final velocity in terms of Isp, τ0, and the mass ratio m0/mf. In part (d) we’ll need to use trial-and-error methods or a graphing calculator to solve the transcendental equation giving vf as a function of m0/mf. (a) Express the rocket equation:
maRumg =+− ex
From the definition of rocket thrust we have:
exth RuF =
Substitute to obtain:
maFmg =+− th
Solve for Fth at takeoff: 000th amgmF +=
Divide both sides of this equation by m0g to obtain: g
agm
F 0
0
th 1+=
Because )/( 0th0 gmF=τ :
ga0
0 1+=τ
(b) Use equation 8-42 to express the final speed of a rocket that starts from rest with mass m0:
bf
0exf ln gt
mmuv −= , (1)
where tb is the burn time.
Express the burn time in terms of the burn rate R (assumed constant):
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
−=
0
f0f0b 1
mm
Rm
Rmmt
Chapter 8
588
Multiply tb by one in the form gT/gT and simplify to obtain:
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
0
f
0
sp
0
fth
th
0
0
f0
th
thb
1
1
1
mmI
mm
gRF
Fgm
mm
Rm
gFgFt
τ
Substitute in equation (1):
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
0
f
0
sp
f
0exf 1ln
mmgI
mmuv
τ
From Problem 32 we have:
spex gIu = , where uex is the exhaust velocity of the propellant.
Substitute and factor to obtain:
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
0
f
0f
0sp
0
f
0
sp
f
0spf
11ln
1ln
mm
mmgI
mmgI
mmgIv
τ
τ
(c) A spreadsheet program to calculate the final velocity of the rocket as a function of the mass ratio m0/mf is shown below. The constants used in the velocity function and the formulas used to calculate the final velocity are as follows:
Cell Content/Formula Algebraic Form B1 250 Isp B2 9.81 g B3 2 τ D9 D8 + 0.25 m0/mf E8 $B$2*$B$1*(LOG(D8) −
(1/$B$3)*(1/D8)) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛
0
f
0f
0sp 11ln
mm
mmgI
τ
A B C D E 1 Isp = 250 s 2 g = 9.81 m/s^2 3 tau = 2 4 5 6 7 mass ratio vf 8 2.00 1.252E+029 2.25 3.187E+02
Systems of Particles and Conservation of Momentum
589
10 2.50 4.854E+0211 2.75 6.316E+0212 3.00 7.614E+02
36 9.00 2.204E+0337 9.25 2.237E+0338 9.50 2.269E+0339 9.75 2.300E+0340 10.00 2.330E+0341 725.00 7.013E+03
A graph of final velocity as a function of mass ratio is shown below.
0
1
2
2 4 6 8 10
m 0/m f
v f (k
m/s
)
(d) Substitute the data given in part (c) in the equation derived in part (b) to obtain:
( )( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
0
f
f
02 121lns250m/s9.81km/s7
mm
mm
or
xx 5.05.0ln854.2 +−= where x = m0/mf.
Use trial-and-error methods or a graphing calculator to solve this transcendental equation for the root greater than 1:
1.28=x ,
a value considerably larger than the practical limit of 10 for single-stage rockets.
108 •• Picture the Problem We can use the velocity-at-burnout equation from Problem 106 to find vf and constant-acceleration equations to approximate the maximum height the rocket will reach and its total flight time. (a) Assuming constant acceleration, relate the maximum height reached
2top2
1 gth = (1)
Chapter 8
590
by the model rocket to its time-to-top-of-trajectory: From Problem 106 we have: ⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
0
f
f
0spf 11ln
mm
mmgIv
τ
Evaluate the velocity at burnout vf for Isp = 100 s, m0/mf = 1.2, and τ = 5:
( )( )
( )
m/s1462.1
11512.1ln
s100m/s9.81 2f
=
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −−×
=v
Assuming that the time for the fuel to burn up is short compared to the total flight time, find the time to the top of the trajectory:
s14.9m/s9.81m/s146
2f
top ===gvt
Substitute in equation (1) and evaluate h:
( )( ) km1.09s14.9m/s9.81 2221 ==h
(b) Find the total flight time from the time it took the rocket to reach its maximum height:
( ) s29.8s14.922 topflight === tt
(c) Express and evaluate the fuel burn time tb:
s3.331.211
5s1001
0
=
⎟⎠⎞
⎜⎝⎛ −=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
mmI
t fspb τ
burnout. untilon acceleraticonstant assuming m, 243 is timein this movepossibly couldrocket model the
distance maximum theas accuracy, 30%about togood be however, should,It accurate. very be to)(Part in obtained answer we expect the
tcan' wee,flight tim total theof 1/5ely approximat is burn time thisBecause
b21 =vt
b
General Problems 109 • Picture the Problem Let the direction of motion of the 250-g car before the collision be the positive x direction. Let the numeral 1 refer to the 250-kg car, the numeral 2 refer to the 400-kg car, and V represent the velocity of the linked cars. Let the system include the earth and the cars. We can use conservation of momentum to find their speed after they have linked together and the definition of kinetic energy to find their initial and final kinetic energies.
Systems of Particles and Conservation of Momentum
591
Use conservation of momentum to relate the speeds of the cars immediately before and immediately after their collision:
( )Vmmvm
pp
2111
fxix
or+=
=
Solve for V:
21
11
mmvmV
+=
Substitute numerical values and evaluate V:
( )( ) m/s192.0kg0.400kg0.250
m/s0.50kg0.250=
+=V
Find the initial kinetic energy of the cars:
( )( )mJ3.31
m/s0.50kg0.250 2212
1121
i
=
== vmK
Find the final kinetic energy of the coupled cars:
( )( )( )
mJ0.12
m/s0.192kg0.400kg0.250 221
2212
1f
=
+=
+= VmmK
110 • Picture the Problem Let the direction of motion of the 250-g car before the collision be the positive x direction. Let the numeral 1 refer to the 250-kg car and the numeral 2 refer to the 400-g car and the system include the earth and the cars. We can use conservation of momentum to find their speed after they have linked together and the definition of kinetic energy to find their initial and final kinetic energies. (a) Express and evaluate the initial kinetic energy of the cars:
( )( )mJ3.31
m/s0.50kg0.250 2212
1121
i
=
== vmK
(b) Relate the velocity of the center of mass to the total momentum of the system:
cmi
ii vvPrrr
mm == ∑
Solve for vcm:
21
2211cm mm
vmvmv++
=
Substitute numerical values and evaluate vcm:
( )( ) m/s192.0kg0.400kg0.250
m/s0.50kg0.250cm =
+=v
Chapter 8
592
Find the initial velocity of the 250-g car relative to the velocity of the center of mass:
m/s0.308
m/s0.192m/s.500cm11
=
−=−= vvu
Find the initial velocity of the 400-g car relative to the velocity of the center of mass:
m/s192.0
m/s0.192m/s0cm22
−=
−=−= vvu
Express the initial kinetic energy of the system relative to the center of mass:
2222
12112
1reli, umumK +=
Substitute numerical values and evaluate Ki,rel:
( )( )( )( )
mJ19.2
m/s0.192kg0.400
m/s0.308kg0.2502
21
221
reli,
=
−+
=K
(c) Express the kinetic energy of the center of mass:
2cm2
1cm MvK =
Substitute numerical values and evaluate Kcm:
( )( )mJ12.0
m/s0.192kg0.650 221
cm
=
=K
(d) Relate the initial kinetic energy of the system to its initial kinetic energy relative to the center of mass and the kinetic energy of the center of mass:
mJ31.2mJ12.0mJ9.21
cmreli,i
=+=
+= KKK
cmreli,i KKK +=∴
*111 • Picture the Problem Let the direction the 4-kg fish is swimming be the positive x direction and the system include the fish, the water, and the earth. The velocity of the larger fish immediately after its lunch is the velocity of the center of mass in this perfectly inelastic collision. Relate the velocity of the center of mass to the total momentum of the system:
cmi
ii vvPrrr
mm == ∑
Systems of Particles and Conservation of Momentum
593
Solve for vcm:
2.14
2.12.144cm mm
vmvmv+−
=
Substitute numerical values and evaluate vcm:
( )( )
m/s462.0
kg2.1kg4)m/s(3kg)2.1(m/s5.1kg4
cm
=
+−
=v
112 • Picture the Problem Let the direction the 3-kg block is moving be the positive x direction and include both blocks and the earth in the system. The total kinetic energy of the two-block system is the sum of the kinetic energies of the blocks. We can relate the momentum of the system to the velocity of its center of mass and use this relationship to find vcm. Finally, we can use the definition of kinetic energy to find the kinetic energy relative to the center of mass. (a) Express the total kinetic energy of the system in terms of the kinetic energy of the blocks:
2662
12332
1tot vmvmK +=
Substitute numerical values and evaluate Ktot:
( )( ) ( )( )J81.0
m/s3kg6m/s6kg3 2212
21
tot
=
+=K
(b) Relate the velocity of the center of mass to the total momentum of the system:
cmi
ii vvPrrr
mm == ∑
Solve for vcm:
21
6633cm mm
vmvmv++
=
Substitute numerical values and evaluate vcm:
( )( ) ( )( )
m/s00.4
kg6kg3m/s3kg6m/s6kg3
cm
=
++
=v
(c) Find the center of mass kinetic energy from the velocity of the center of mass:
( )( )J72.0
m/s4kg9 2212
cm21
cm
=
== MvK
Chapter 8
594
(d) Relate the initial kinetic energy of the system to its initial kinetic energy relative to the center of mass and the kinetic energy of the center of mass:
J00.9
J0.27J0.81cmtotrel
=
−=−= KKK
113 • Picture the Problem Let east be the positive x direction and north the positive y direction. Include both cars and the earth in the system and let the numeral 1 denote the 1500-kg car and the numeral 2 the 2000-kg car. Because the net external force acting on the system is zero, momentum is conserved in this perfectly inelastic collision. (a) Express the total momentum of the system: ij
vvpppˆˆ
2211
221121
vmvm
mm
−=
+=+=rrrrr
Substitute numerical values and evaluate p
r:
( )( ) ( )( )
( ) ( )ji
ijpˆkm/hkg1005.1ˆkm/hkg1010.1
ˆkm/h55kg2000ˆkm/h70kg150055 ⋅×+⋅×−=
−=r
(b) Express the velocity of the wreckage in terms of the total momentum of the system:
Mpvvr
rr== cmf
Substitute numerical values and evaluate fvr :
( ) ( )
( ) ( ) ji
jiv
ˆkm/h0.30ˆkm/h4.31
kg2000kg1500
ˆkm/hkg101.05kg2000kg1500
ˆkm/hkg101.10 55
f
+−=
+⋅×
++
⋅×−=
r
Find the magnitude of the velocity of the wreckage:
( ) ( )km/h43.4
km/h30.0km/h31.4 22f
=
+=v
Find the direction of the velocity of the wreckage: °−=⎥
⎦
⎤⎢⎣
⎡−
= − 7.43km/h31.4
km/h30.0tan 1θ
north. of west 46.3is wreckage theofdirection The
°
Systems of Particles and Conservation of Momentum
595
*114 •• Picture the Problem Take the origin to be at the initial position of the right-hand end of raft and let the positive x direction be to the left. Let ″w″ denote the woman and ″r″ the raft, d be the distance of the end of the raft from the pier after the woman has walked to its front. The raft moves to the left as the woman moves to the right; with the center of mass of the woman-raft system remaining fixed (because Fext,net = 0). The diagram shows the initial (xw,i) and final (xw,f) positions of the woman as well as the initial (xr_cm,i) and final (xr_cm,f) positions of the center of mass of the raft both before and after the woman has walked to the front of the raft.
x
x
xw,
f
xr_cm,i
xr_cm,f
xw,
i =6 m
xr_cm,i
0
0×
×
CM
CM
xCM
d
0.5 mP I E R
(a) Express the distance of the raft from the pier after the woman has walked to the front of the raft:
wf,m5.0 xd += (1)
Express xcm before the woman has walked to the front of the raft:
rw
i r_cm,riw,wcm mm
xmxmx
+
+=
Express xcm after the woman has walked to the front of the raft:
rw
fr_cm,rfw,wcm mm
xmxmx
+
+=
Because Fext,net = 0, the center of mass remains fixed and we can equate these two expressions for xcm to obtain:
fr_cm,rfw,wir_cm,ri,ww xmxmxmxm +=+
Solve for xw,f: ( )ir_cm,fr_cm,w
riw,fw, xx
mmxx −−=
From the figure it can be seen that xr_cm,f – xr_cm,i = xw,f. Substitute xw,f rw
iw,wfw, mm
xmx
+=
Chapter 8
596
for xr_cm,f – xr_cm,i and to obtain: Substitute numerical values and evaluate xw,f:
( )( ) m00.2kg120kg60
m6kg60fw, =
+=x
Substitute in equation (1) to obtain: m50.2m5.0m00.2 =+=d
(b) Express the total kinetic energy of the system:
2rr2
12ww2
1tot vmvmK +=
Noting that the elapsed time is 2 s, find vw and vr:
m/s2s2
m6m2iw,fw,w −=
−=
∆−
=txx
v
relative to the dock, and
m/s1s2
m0.5m50.2ir,fr,r =
−=
∆−
=txx
v ,
also relative to the dock.
Substitute numerical values and evaluate Ktot:
( )( )( )( )J180
m/s1kg120
m/s2kg602
21
221
tot
=
+
−=K
Evaluate K with the raft tied to the pier:
( )( )J270
m/s3kg60 2212
ww21
tot
=
== vmK
(c) energy. internalher
into ed transformisenergy kinetic thefriction, static viastops she assuming and, woman theofenergy chemical thefrom derivesenergy kinetic theAll
(d)
raft. theoffront at the lands and m 6 of range a has alsoshot theframe,woman -raft in the Thus, land. the
of frame reference in the m 6 of range a hadshot that thedid asvelocity initial same thehasshot theframeIn that frame. reference inertialan
sconstitute systemwoman -raft thehand, s woman' theleavesshot After the
Systems of Particles and Conservation of Momentum
597
115 •• Picture the Problem Let the zero of gravitational potential energy be at the elevation of the 1-kg block. We can use conservation of energy to find the speed of the bob just before its perfectly elastic collision with the block and conservation of momentum to find the speed of the block immediately after the collision. We’ll apply Newton’s 2nd law to find the acceleration of the sliding block and use a constant-acceleration equation to find how far it slides before coming to rest. (a) Use conservation of energy to find the speed of the bob just before its collision with the block: 0
or0
ifif =−+−
=∆+∆
UUKK
UK
Because Ki = Uf = 0:
hgv
hgmvm
∆=
=∆+
2
and0
ball
ball2ballball2
1
Substitute numerical values and evaluate vball:
( )( ) m/s6.26m2m/s9.812 2ball ==v
Because the collision is perfectly elastic and the ball and block have the same mass:
m/s26.6ballblock == vv
(b) Using a constant-acceleration equation, relate the displacement of the block to its acceleration and initial speed and solve for its displacement: block
2block
block
2i
f
block2i
2f
22
0, Since2
av
avx
vxavv
−=
−=∆
=∆+=
Apply ∑ = aF rr
m to the sliding
block:
∑
∑
=−=
=−=
0and
blockn
blockk
gmFF
mafF
y
x
Using the definition of fk (µkFn) eliminate fk and Fn between the two equations and solve for ablock:
ga kblock µ−=
Substitute for ablock to obtain: g
vg
vxk
2block
k
2block
22 µµ=
−−
=∆
Chapter 8
598
Substitute numerical values and evaluate ∆x:
( )( )( ) m0.20
m/s9.810.12m/s6.26
2
2
==∆x
*116 •• Picture the Problem We can use conservation of momentum in the horizontal direction to find the recoil velocity of the car along the track after the firing. Because the shell will neither rise as high nor be moving as fast at the top of its trajectory as it would be in the absence of air friction, we can apply the work-energy theorem to find the amount of thermal energy produced by the air friction.
(a) conserved. benot willsystem theof momentum the
so and force externalan is rails theof forcereaction verticalThe No.
(b) Use conservation of momentum in the horizontal (x) direction to obtain:
0=∆ xp
or 030cos recoil =−° Mvmv
Solve for and evaluate vrecoil:
Mmvv °
=30cos
recoil
Substitute numerical values and evaluate vrecoil:
( )( )
m/s33.4
kg5000cos30m/s125kg200
recoil
=
°=v
(c) Using the work-energy theorem, relate the thermal energy produced by air friction to the change in the energy of the system:
KUEWW ∆+∆=∆== sysfext
Substitute for ∆U and ∆K to obtain:
( ) ( )22f2
1if
2212
f21
ifext
i
i
vvmyymg
mvmvmgymgyW
−+−=
−+−=
Substitute numerical values and evaluate Wext:
( )( )( ) ( ) ( ) ( )[ ] kJ569m/s125m/s80kg200m180m/s81.9kg200 22212
ext −=−+=W
Systems of Particles and Conservation of Momentum
599
117 •• Picture the Problem Because this is a perfectly inelastic collision, the velocity of the block after the collision is the same as the velocity of the center of mass before the collision. The distance the block travels before hitting the floor is the product of its velocity and the time required to fall 0.8 m; which we can find using a constant-acceleration equation. Relate the distance D to the velocity of the center of mass and the time for the block to fall to the floor:
tvD ∆= cm
Relate the velocity of the center of mass to the total momentum of the system and solve for vcm:
cmi
ii vvPrrr
Mm == ∑
and
blockbullet
blockblockbulletbulletcm mm
vmvmv++
=
Substitute numerical values and evaluate vcm:
( )( ) m/s9.20kg0.8kg0.015m/s500kg0.015
cm =+
=v
Using a constant-acceleration equation, find the time for the block to fall to the floor:
( )
gytv
tatvy
∆=∆=
∆+∆=∆
2 0, Because 0
221
0
Substitute to obtain:
gyvD ∆
=2
cm
Substitute numerical values and evaluate D: ( ) ( ) m72.3
m/s9.81m0.82m/s20.9 2 ==D
118 •• Picture the Problem Let the direction the particle whose mass is m is moving initially be the positive x direction and the direction the particle whose mass is 4m is moving initially be the negative y direction. We can determine the impulse delivered by F
r and,
hence, the change in the momentum of the system from the change in the momentum of the particle whose mass is m. Knowing p
r∆ , we can express the final momentum of the
particle whose mass is 4m and solve for its final velocity. Express the impulse delivered by the force F
r: ( ) iii
pppFIˆ3ˆˆ4
if
mvmvvm
T
=−=
−=∆==rrrrr
Chapter 8
600
Express m4'pr : ( )ij
ppvpˆ3ˆ4
04 m44m
mvmv
'm'
+−=
∆+==rrrr
Solve for 'vr : jiv ˆˆ
43 vv' −=
r
119 •• Picture the Problem Let the numeral 1 refer to the basketball and the numeral 2 to the baseball. The left-hand side of the diagram shows the balls after the basketball’s elastic collision with the floor and just before they collide. The right-hand side of the diagram shows the balls just after their collision. We can apply conservation of momentum and the definition of an elastic collision to obtain equations relating the initial and final velocities of the masses of the colliding objects that we can solve for v1f and v2f.
(a) Because both balls are in free-fall, and both are in the air for the same amount of time, they have the same velocity just before the basketball rebounds. After the basketball rebounds elastically, its velocity will have the same magnitude, but the opposite direction than just before it hit the ground.
baseball. theof velocity thetodirection in
oppositebut magnitudein equal be willbasketball theof velocity The
(b) Apply conservation of momentum to the collision of the balls to obtain:
2i21i1f22f11 vmvmvmvm +=+ (1)
Relate the initial and final kinetic energies of the balls in their elastic collision:
2i222
12i112
12f222
12f112
1 vmvmvmvm +=+
Rearrange this equation and factor to obtain:
( ) ( )2f1
2i11
2i2
2f22 vvmvvm −=−
or ( )( )
( )( )1fi11fi11
2if22if22
vvvvmvvvvm
+−=+−
(2)
Rearrange equation (1) to obtain:
( ) ( )1f1i12i2f2 vvmvvm −=− (3)
Divide equation (2) by equation (3) to obtain:
1fi12if2 vvvv +=+
Systems of Particles and Conservation of Momentum
601
Rearrange this equation to obtain equation (4):
1ii2f2f1 vvvv −=− (4)
Multiply equation (4) by m2 and add it to equation (1) to obtain:
( ) ( ) 2i21i211f21 2 vmvmmvmm +−=+
Solve for v1f to obtain: i2
21
2i1
21
21f1
2 vmm
mvmmmmv
++
+−
=
or, because v2i = −v1i,
i121
21
i121
2i1
21
21f1
3
2
vmmmm
vmm
mvmmmmv
+−
=
+−
+−
=
For m1 = 3m2 and v1i = v: 0
333
22
22f1 =
+−
= vmmmmv
(c) Multiply equation (4) by m1 and subtract it from equation (1) to obtain:
( ) ( ) 1i1i212f221 2 vmvmmvmm +−=+
Solve for v2f to obtain: i2
21
12i1
21
1f2
2 vmmmmv
mmmv
+−
++
=
or, because v2i = −v1i,
i121
21
i121
12i1
21
1f2
3
2
vmmmm
vmmmmv
mmmv
+−
=
+−
−+
=
For m1 = 3m2 and v1i = v:
( ) vvmm
mmv 2333
22
22f2 =
+−
=
Chapter 8
602
120 ••• Picture the Problem In Problem 119 only two balls are dropped. They collide head on, each moving at speed v, and the collision is elastic. In this problem, as it did in Problem 119, the solution involves using the conservation of momentum equation
2i21i1f22f11 vmvmvmvm +=+ and the elastic collision equation
,1ii2f2f1 vvvv −=− where the numeral 1 refers to the baseball, and the numeral 2 to the top ball. The diagram shows the balls just before and just after their collision. From Problem 119 we know that that v1i = 2v and v2i = −v.
(a) Express the final speed v1f of the baseball as a function of its initial speed v1i and the initial speed of the top ball v2i (see Problem 78):
i221
2i1
21
21f1
2 vmm
mvmmmmv
++
+−
=
Substitute for v1i and , v2i to obtain: ( ) ( )v
mmmv
mmmmv −
++
+−
=21
2
21
21f1
22
Divide the numerator and denominator of each term by m2 to introduce the mass ratio of the upper ball to the lower ball:
( ) ( )v
mmv
mmmm
v −+
++
−=
1
221
1
2
1
2
1
2
1
f1
Set the final speed of the baseball v1f equal to zero, let x represent the mass ratio m1/m2, and solve for x:
( ) ( )vx
vxx
−+
++−
=1
22110
and
21
2
1 ==mmx
(b) Apply the second of the two equations in Problem 78 to the collision between the top ball and the baseball:
2i21
12i1
21
1f2
2 vmmmmv
mmmv
+−
++
=
Substitute v1i = 2v and are given that v2i = −v to obtain: ( ) ( )v
mmmmv
mmmv −
+−
++
=21
12
21
1f2 22
Systems of Particles and Conservation of Momentum
603
In part (a) we showed that m2 = 2m1. Substitute and simplify:
( ) ( )
( )
v
vvvmmv
mm
vmmmmv
mmmv
37
31
38
1
1
1
1
11
11
11
13f
32
34
222
222
=
−=−=
+−
−+
=
*121 •• Picture the Problem Let the direction the probe is moving after its elastic collision with Saturn be the positive direction. The probe gains kinetic energy at the expense of the kinetic energy of Saturn. We’ll relate the velocity of approach relative to the center of mass to urec and then to v. (a) Relate the velocity of recession to the velocity of recession relative to the center of mass:
cmrec vuv +=
Find the velocity of approach: km/s0.20
km/s0.41km/s9.6app
−=
−−=u
Relate the relative velocity of approach to the relative velocity of recession for an elastic collision:
km/s0.20apprec =−= uu
Because Saturn is so much more massive than the space probe:
km/s6.9Saturncm == vv
Substitute and evaluate v:
km/s29.6
km/s9.6km/s02cmrec
=
+=+= vuv
(b) Express the ratio of the final kinetic energy to the initial kinetic energy:
10.8km/s10.4km/s29.6
2
2
i
rec2i2
1
2rec2
1
i
f
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛==
vv
MvMv
KK
Saturn. of slowing smallly immeasuraban from comesenergy The
Chapter 8
604
*122 •• Picture the Problem We can use the relationships mcP ∆= and 2mcE ∆=∆ to show that .cEP ∆= We can then equate this expression with the change in momentum of the flashlight to find the latter’s final velocity. (a) Express the momentum of the mass lost (i.e., carried away by the light) by the flashlight:
mcP ∆=
Relate the energy carried away by the light to the mass lost by the flashlight:
2cEm ∆
=∆
Substitute to obtain: c
EcEcP ∆
=∆
= 2
(b) Relate the final momentum of the flashlight to ∆E:
mvpcE
=∆=∆
because the flashlight is initially at rest.
Solve for v:
mcEv ∆
=
Substitute numerical values and evaluate v: ( )( )
m/s33.3
m/s1033.3m/s102.998kg1.5
J101.5
6
8
3
µ=
×=
××
=
−
v
123 • Picture the Problem We can equate the change in momentum of the block to the momentum of the beam of light and relate the momentum of the beam of light to the mass converted to produce the beam. Combining these expressions will allow us to find the speed attained by the block. Relate the change in momentum of the block to the momentum of the beam:
( ) beamPvmM =− because the block is initially at rest.
Express the momentum of the mass converted into a well-collimated beam of light:
mcP =beam
Substitute to obtain:
( ) mcvmM =−
Solve for v:
mMmcv−
=
Systems of Particles and Conservation of Momentum
605
Substitute numerical values and evaluate v:
( )( )
m/s1000.3
kg0.001kg1m/s102.998kg0.001
5
8
×=
−×
=v
124 •• Picture the Problem Let the origin of the coordinate system be at the end of the boat at which your friend is sitting prior to changing places. If we let the system include you and your friend, the boat, the water and the earth, then Fext,net = 0 and the center of mass is at the same location after you change places as it was before you shifted. Express the center of mass of the system prior to changing places:
( )mmmmxmmx
mmmmxxmxm
x
++++
=
++++
=
youboat
friendyouboatyou
friendyouboat
friendyouyouboatboatcm
Substitute numerical values and simplify to obtain an expression for xcm in terms of m:
( )( ) ( )
m
mmx
+⋅
=
++++
=
kg140mkg280
kg80kg600kg80kg60m2
cm
Find the center of mass of the system after changing places:
( )( ) ( )mmm
mmmm
mmmmm
mxxmxmx'
++±
+++±+
=++
++=
youboat
you
youboat
boat
friendyouboat
friendyouyouboatboatcm
m0.2m0.2m2
Substitute numerical values and simplify to obtain:
( )( ) ( )( )
( )m
mmmmm
mx'
+⋅±±
+
+⋅±⋅
=++
±+
++±+
=
kg140mkg16m2.0m2
kg140mkg12mkg120
kg80kg60m0.2kg80
kg80kg60m0.2m2kg60
cm
Because Fext,net = 0, cmcm xx' = .
Equate the two expressions and solve for m to obtain:
( )( ) kg
0.2228160
±±
=m
Calculate the largest possible mass for your friend:
( )( ) kg104kg
0.2228160
=−
+=m
Chapter 8
606
Calculate the smallest possible mass for your friend:
( )( ) kg0.60kg
0.2228160
=+
−=m
125 •• Picture the Problem Let the system include the woman, both vehicles, and the earth. Then Fext,net = 0 and acm = 0. Include the mass of the man in the mass of the truck. We can use Newton’s 2nd and 3rd laws to find the acceleration of the truck and net force acting on both the car and the truck. (a) Relate the action and reaction forces acting on the car and truck:
truckcar FF =
or truckwomantruckcarcar amam +=
Solve for the acceleration of the truck:
womantruck
carcartruck
+
=m
ama
Substitute numerical values and evaluate atruck:
( )( ) 22
truck m/s600.0kg1600
m/s1.2kg800==a
(b) Apply Newton’s 2nd law to either vehicle to obtain:
carcarnet amF =
Substitute numerical values and evaluate Fnet:
( )( ) N960m/s1.2kg800 2net ==F
126 •• Picture the Problem Let the system include the block, the putty, and the earth. Then Fext,net = 0 and momentum is conserved in this perfectly inelastic collision. We’ll use conservation of momentum to relate the after-collision velocity of the block plus blob and conservation of energy to find their after-collision velocity. Noting that, because this is a perfectly elastic collision, the final velocity of the block plus blob is the velocity of the center of mass, use conservation of momentum to relate the velocity of the center of mass to the velocity of the glob before the collision:
cmglgl
fi
orMvvm
pp
=
=
where M = mgl + mbl.
Systems of Particles and Conservation of Momentum
607
Solve for vgl to obtain: cm
glgl v
mMv = (1)
Use conservation of energy to find the initial energy of the block plus glob:
0f =+∆+∆ WUK
Because ∆U = Kf = 0, 0k
2cm2
1 =∆+− xfMv
Use fk = µkMg to eliminate fk and solve for vcm:
xgv ∆= kcm 2µ
Substitute numerical values and evaluate vcm:
( )( )( )m/s1.08
m0.15m/s9.810.42 2cm
=
=v
Substitute numerical values in equation (1) and evaluate vgl:
( )
m/s36.2
m/s1.08kg0.4
kg0.4kg13gl
=
+=v
*127 •• Picture the Problem Let the direction the moving car was traveling before the collision be the positive x direction. Let the numeral 1 denote this car and the numeral 2 the car that is stopped at the stop sign and the system include both cars and the earth. We can use conservation of momentum to relate the speed of the initially-moving car to the speed of the meshed cars immediately after their perfectly inelastic collision and conservation of energy to find the initial speed of the meshed cars. Using conservation of momentum, relate the before-collision velocity to the after-collision velocity of the meshed cars:
( )Vmmvm
pp
2111
fi
or+=
=
Solve for v1: VmmV
mmmv ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
+=
1
2
1
211 1
Using conservation of energy, relate the initial kinetic energy of the meshed cars to the work done by friction in bringing them to a stop:
0thermal =∆+∆ EK
or, because Kf = 0 and ∆Ethermal = f∆s, 0ki =∆+− sfK
Substitute for Ki and, using fk = µkFn = µkMg, eliminate fk to
0k2
21 =∆+− xMgMV µ
Chapter 8
608
obtain:
Solve for V: xgV ∆= k2µ
Substitute to obtain:
xgmmv ∆⎟⎟
⎠
⎞⎜⎜⎝
⎛+= k
1
21 21 µ
Substitute numerical values and evaluate v1:
( )( )( ) km/h23.3m/s48.6m0.76m/s9.810.922kg1200kg9001 2
1 ==⎟⎟⎠
⎞⎜⎜⎝
⎛+=v
km/h. 23.3at traveling wasHe truth. thegnot tellin driver was The
128 •• Picture the Problem Let the zero of gravitational potential energy be at the lowest point of the bob’s swing and note that the bob can swing either forward or backward after the collision. We’ll use both conservation of momentum and conservation of energy to relate the velocities of the bob and the block before and after their collision. Express the kinetic energy of the block in terms of its after-collision momentum:
mpK m
2
2
m =
Solve for m to obtain:
m
m
Kpm
2
2
= (1)
Use conservation of energy to relate Km to the change in the potential energy of the bob:
0=∆+∆ UK or, because Ki = 0,
0if =−+ UUKm
Solve for Km:
( ) ( )[ ][ ]ifbob
fibob
if
coscoscos1cos1
θθθθ
−=−−−=
+−=
gLmLLgm
UUKm
Substitute numerical values and evaluate Km:
( )( )( )[ ] J2.47cos53cos5.73m1.6m/s9.81kg0.4 2 =°−°=mK
Systems of Particles and Conservation of Momentum
609
Use conservation of energy to find the velocity of the bob just before its collision with the block:
0=∆+∆ UK or, because Ki = Uf = 0,
0if =−UK
( )
( )i
ibob2
bob21
cos12
or0cos1
θ
θ
−=
=−−∴
gLv
gLmvm
Substitute numerical values and evaluate v:
( )( )( )m/s3.544
cos531m1.6m/s9.812 2
=
°−=v
Use conservation of energy to find the velocity of the bob just after its collision with the block:
0=∆+∆ UK or, because Kf = Ui = 0,
0fi =+− UK
Substitute for Ki and Uf to obtain: ( ) 0cos1' fbob
2bob2
1 =−+− θgLmvm
Solve for v′: ( )fcos12' θ−= gLv
Substitute numerical values and evaluate v′:
( )( )( )m/s396.0
cos5.731m1.6m/s9.812' 2
=
°−=v
Use conservation of momentum to relate pm after the collision to the momentum of the bob just before and just after the collision:
mpvmvm
pp
±=
=
'or
bobbob
fi
Solve for and evaluate pm:
( )( )m/skg0.158m/skg.4181m/s0.396m/s3.544kg0.4
'bobbob
⋅±⋅=±=
±= vmvmpm
Find the larger value for pm:
m/skg1.576m/skg0.158m/skg.4181
⋅=⋅+⋅=mp
Find the smaller value for pm:
m/skg1.260m/skg0.158m/skg.4181
⋅=⋅−⋅=mp
Substitute in equation (1) to determine the two values for m:
( )( ) kg503.0
J47.22m/skg576.1 2
=⋅
=m
Chapter 8
610
or ( )
( ) kg321.0J47.22m/skg260.1 2
=⋅
=m
129 •• Picture the Problem Choose the zero of gravitational potential energy at the location of the spring’s maximum compression. Let the system include the spring, the blocks, and the earth. Then the net external force is zero as is work done against friction. We can use conservation of energy to relate the energy transformations taking place during the evolution of this system. Apply conservation of energy: 0sg =∆+∆+∆ UUK
Because ∆K = 0:
0sg =∆+∆ UU
Express the change in the gravitational potential energy:
θsing MgxhmgU −∆−=∆
Express the change in the potential energy of the spring:
221
s kxU =∆
Substitute to obtain:
0sin 221 =+−∆− kxMgxhmg θ
Solve for M: x
hmgkx
gxhmgkxM ∆
−=°∆−
=2
30sin
221
Relate ∆h to the initial and rebound positions of the block whose mass is m:
( ) m720.030sinm56.2m4 =°−=∆h
Substitute numerical values and evaluate M:
( ) ( ) ( )( ) kg8.85m0.04
m0.72kg12m/s9.81
m0.04N/m10112
3
=−×
=M
*130 •• Picture the Problem By symmetry, xcm = 0. Let σ be the mass per unit area of the disk. The mass of the modified disk is the difference between the mass of the whole disk and the mass that has been removed.
Systems of Particles and Conservation of Momentum
611
Start with the definition of ycm:
hole
holeholediskdisk
hole
iii
cm
mMymym
mM
ymy
−−
=
−=
∑
Express the mass of the complete disk: 2rAM σπσ ==
Express the mass of the material removed:
Mrrm 412
41
2
hole 2==⎟
⎠⎞
⎜⎝⎛= σπσπ
Substitute and simplify to obtain: ( ) ( )( ) r
MMrMMy 6
1
41
21
41
cm0
=−
−−=
131 •• Picture the Problem Let the horizontal axis by the y axis and the vertical axis the z axis. By symmetry, xcm = ycm = 0. Let ρ be the mass per unit volume of the sphere. The mass of the modified sphere is the difference between the mass of the whole sphere and the mass that has been removed. Start with the definition of ycm:
hole
holeholespheresphere
hole
iii
cm
mMymym
mM
ymz
−
−=
−=
∑
Express the mass of the complete sphere:
334 rVM ρπρ ==
Express the mass of the material removed: ( ) Mrrm 8
1334
81
3
34
hole 2==⎟
⎠⎞
⎜⎝⎛= ρπρπ
Substitute and simplify to obtain: ( ) ( )( ) r
MMrMMz 14
1
81
21
81
cm0
=−
−−=
*132 •• Picture the Problem In this elastic head-on collision, the kinetic energy of recoiling nucleus is the difference between the initial and final kinetic energies of the neutron. We can derive the indicated results by using both conservation of energy and conservation of momentum and writing the kinetic energies in terms of the momenta of the particles before and after the collision.
Chapter 8
612
(a) Use conservation of energy to relate the kinetic energies of the particles before and after the collision:
Mp
mp
mp
222
2nucleus
2nf
2ni += (1)
Apply conservation of momentum to obtain a second relationship between the initial and final momenta:
nucleusnfni ppp += (2)
Eliminate pnf in equation (1) using equation (2):
022
ninucleusnucleus =−+mp
mp
Mp
(3)
Use equation (3) to write mp 22
ni in
terms of pnucleus:
( )mM
mMpKm
p2
22nucleus
n
2ni
82+
== (4)
Use equation (4) to express MpK 22
nucleusnucleus = in terms of
Kn:
( ) ⎥⎦
⎤⎢⎣
⎡
+= 2nnucleus
4mM
MmKK (5)
(b) Relate the change in the kinetic energy of the neutron to the after-collision kinetic energy of the nucleus:
nucleusn KK −=∆
Using equation (5), express the fraction of the energy lost in the collision: ( ) 22
n
n
1
44
⎟⎠⎞
⎜⎝⎛ +
=+
=∆−
MmMm
mMMm
KK
133 •• Picture the Problem Problem 132 (b) provides an expression for the fractional loss of energy per collision. (a) Using the result of Problem 132 (b), express the fractional loss of energy per collision:
( )( )2
2
0
nni
ni
nf
mMmM
EKK
KK
+−
=∆−
=
Evaluate this fraction to obtain: ( )
( )716.0
1212
2
2
0
nf =+−
=mmmm
EK
Express the kinetic energy of one
0nf 716.0 EK N=
Systems of Particles and Conservation of Momentum
613
neutron after N collisions: (b) Substitute for Knf and E0 to obtain:
810716.0 −=N
Take the logarithm of both sides of the equation and solve for N:
550.716log
8N ≈−
=
134 •• Picture the Problem We can relate the number of collisions needed to reduce the energy of a neutron from 2 MeV to 0.02 eV to the fractional energy loss per collision and solve the resulting exponential equation for N. (a) Using the result of Problem 132 (b), express the fractional loss of energy per collision: 37.0
63.0
ni
nini
0
nni
ni
nf
=
−=
∆−=
KKK
EKK
KK
Express the kinetic energy of one neutron after N collisions:
0nf 37.0 EK N=
Substitute for Knf and E0 to obtain:
81037.0 −=N
Take the logarithm of both sides of the equation and solve for N:
190.37log8
≈−
=N
(b) Proceed as in (a) to obtain:
89.0
11.0
ni
nini
0
nni
ni
nf
=
−=
∆−=
KKK
EKK
KK
Express the kinetic energy of one neutron after N collisions:
0nf 89.0 EK N=
Substitute for Knf and E0 to obtain:
81089.0 −=N
Take the logarithm of both sides of the equation and solve for N:
1580.89log8
≈−
=N
Chapter 8
614
135 •• Picture the Problem Let λ = M/L be the mass per unit length of the rope, the subscript 1 refer to the portion of the rope that is being supported by the force F at any given time, and the subscript 2 refer to the rope that is still on the table at any given time. We can find the height hcm of the center of mass as a function of time and then differentiate this expression twice to find the acceleration of the center of mass. (a) Apply the definition of the center of mass to obtain:
Mhmhm
h cm2,2cm1,1cm
+= (1)
From the definition of λ we have:
vtm
LM 1= ⇒ vt
LMm =1
h1,cm and h2,cm are given by : vth
21
cm1, = and h2,cm = 0
Substitute for m1, h1,cm, and h2,cm in equation (1) and simplify to obtain:
( )2
22cm1,
cm 2
0t
Lv
M
mhvtLM
h =+⎟
⎠⎞
⎜⎝⎛
=
(b) Differentiate hcm twice to obtain acm:
Lva
dthd
tLvt
Lv
dtdh
2
cm2cm
2
22cm
and2
2
==
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
(c) Letting N represent the normal force that the table exerts on the rope, apply ∑ = cmmaFy to the
rope to obtain:
cmMaMgNF =−+
Solve for F, substitute for acm and N to obtain: gm
LvMMg
NMaMgF
2
2cm
−+=
−+=
Use the definition of λ again to obtain:
LM
vtLm
=−
2 ⇒ ⎟⎠⎞
⎜⎝⎛ −=
LvtMm 12
Systems of Particles and Conservation of Momentum
615
Substitute for m2 and simplify:
Mggtv
Lvt
Lvt
gLvMgg
Lvtg
LvgMg
LvtM
LvMMgF
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=⎟⎟
⎠
⎞⎜⎜⎝
⎛+−+=⎟
⎠⎞
⎜⎝⎛ −−+=
1
1222
136 •• Picture the Problem The free-body diagram shows the forces acting on the platform when the spring is partially compressed. The scale reading is the force the scale exerts on the platform and is represented on the FBD by Fn. We can use Newton’s 2nd law to determine the scale reading in part (a). We’ll use both conservation of energy and momentum to obtain the scale reading when the ball collides inelastically with the cup.
(a) Apply ∑ = yy maF to the
spring when it is compressed a distance d:
0springonballpn =−− FgmF
Solve for Fn:
( )gmmgmgm
kgm
kgm
kdgm
FgmF
bpbp
bp
p
springonballpn
+=+=
⎟⎠⎞
⎜⎝⎛+=
+=
+=
(b) Letting the zero of gravitational energy be at the initial elevation of the cup and vbi represent the velocity of the ball just before it hits the cup, use conservation of energy to find this velocity:
0 where0 gfig ===∆+∆ UKUK
ghv
mghvm
2
and0
bi
2bib2
1
=
=−∴
Use conservation of momentum to fi pp rr=
Chapter 8
616
find the velocity of the center of mass: ⎥
⎦
⎤⎢⎣
⎡+
=+
=∴cb
b
cb
bibcm 2
mmmgh
mmvmv
Apply conservation of energy to the collision to obtain:
0scm =∆+∆ UK
or, with Kf = Usi = 0, ( ) 02
212
cmcb21 =++− kxvmm
Substitute for vcm and solve for kx2: ( )
( )
cb
2b
2
cb
bcb
2cmcb
2
2
2
mmghm
mmmmmgh
vmmkx
+=
⎥⎦
⎤⎢⎣
⎡+
+=
+=
Solve for x:
( )cbb
2mmk
ghmx+
=
From part (a):
( )
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛
++=
++=
+=
cbbp
cbbp
pn
2
2
mmgkhmmg
mmkghkmgm
kxgmF
(c) height. original its toreturnsnever ball theinelastic, iscollision theBecause
137 •• Picture the Problem Let the direction that astronaut 1 first throws the ball be the positive direction and let vb be the initial speed of the ball in the laboratory frame. Note that each collision is perfectly inelastic. We can apply conservation of momentum and the definition of the speed of the ball relative to the thrower to each of the perfectly inelastic collisions to express the final speeds of each astronaut after one throw and one catch. Use conservation of momentum to relate the speeds of astronaut 1 and the ball after the first throw:
0bb11 =+ vmvm (1)
Relate the speed of the ball in the laboratory frame to its speed relative
1b vvv −= (2)
Systems of Particles and Conservation of Momentum
617
to astronaut 1: Eliminate vb between equations (1) and (2) and solve for v1:
vmm
mv
b1
b1 +
−= (3)
Substitute equation (3) in equation (2) and solve for vb:
vmm
mvb1
1b +
= (4)
Apply conservation of momentum to express the speed of astronaut 2 and the ball after the first catch:
( ) 2b2bb0 vmmvm +== (5)
Solve for v2: b
b2
b2 v
mmm
v+
= (6)
Express v2 in terms of v by substituting equation (4) in equation (6):
( )( ) vmmmm
mm
vmm
mmm
mv
⎥⎦
⎤⎢⎣
⎡++
=
++=
b1b2
1b
b1
1
b2
b2
(7)
Use conservation of momentum to express the speed of astronaut 2 and the ball after she throws the ball:
( ) 2f2bfb2b2 vmvmvmm +=+ (8)
Relate the speed of the ball in the laboratory frame to its speed relative to astronaut 2:
bf2f vvv −= (9)
Eliminate vbf between equations (8) and (9) and solve for v2f: v
mmm
mmmv ⎥
⎦
⎤⎢⎣
⎡+
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
=b1
1
b2
bf2 1 (10)
Substitute equation (10) in equation (9) and solve for vbf:
vmm
m
mmmv
⎥⎦
⎤⎢⎣
⎡+
+×
⎥⎦
⎤⎢⎣
⎡+
−−=
b1
1
b2
bbf
1
1 (11)
Apply conservation of momentum to express the speed of astronaut 1 and the ball after she catches the ball:
( ) 11bfb1fb1 vmvmvmm +=+ (12)
Chapter 8
618
Using equations (3) and (11), eliminate vbf and v1 in equation (12) and solve for v1f:
( )( ) ( )
vmmmm
mmmmv
b22
b1
b1b21f
2++
+−=
*138 •• Picture the Problem We can use the definition of the center of mass of a system containing multiple objects to locate the center of mass of the earth−moon system. Any object external to the system will exert accelerating forces on the system. (a) Express the center of mass of the earth−moon system relative to the center of the earth:
∑=i
iicm rrrr
mM
or ( )
1
0
m
e
em
me
emm
me
emmecm
+=
+=
++
=
mM
rmM
rmmM
rmMr
Substitute numerical values and evaluate rcm:
km467013.81km1084.3 5
cm =+
×=r
earth. theof surface thebelow is systemmoon earth theof mass ofcenter theofposition theearth, theof radius than theless is distance thisBecause
−
(b) planets.other andsun thee.g.,
system, on the forces exerts systemmoon earth in thenot object Any −
(c) sun. the towardis system theofon accelerati thesystem,
moon earth on the force externaldominant theexertssun theBecause −
(d) Because the center of mass is at a fixed distance from the sun, the distance d moved by the earth in this time interval is:
( ) km9340km467022 em === rd
139 •• Picture the Problem Let the numeral 2 refer to you and the numeral 1 to the water leaving the hose. Apply conservation of momentum to the system consisting of yourself, the water, and the earth and then differentiate this expression to relate your recoil acceleration to your mass, the speed of the water, and the rate at which the water is
Systems of Particles and Conservation of Momentum
619
leaving the hose. Use conservation of momentum to relate your recoil velocity to the velocity of the water leaving the hose:
0or
0
2211
21
=+
=+
vmvm
pp rr
Differentiate this expression with respect to t:
022
22
11
11 =+++
dtdmv
dtdvm
dtdmv
dtdvm
or
0222
1111 =+++
dtdmvma
dtdmvam
Because the acceleration of the water leaving the hose, a1, is zero …
as is dt
dm2 , the rate at which you are
losing mass: dtdm
mv
a
amdt
dmv
1
2
12
221
1
and
0
−=
=+
Substitute numerical values and evaluate a2:
2
2
m/s960.0
)kg/s(2.4kg75m/s30
−=
−=a
*140 ••• Picture the Problem Take the zero of gravitational potential energy to be at the elevation of the pan and let the system include the balance, the beads, and the earth. We can use conservation of energy to find the vertical component of the velocity of the beads as they hit the pan and then calculate the net downward force on the pan from Newton’s 2nd law. Use conservation of energy to relate the y component of the bead’s velocity as it hits the pan to its height of fall:
0=∆+∆ UK or, because Ki = Uf = 0,
0221 =− mghmvy
Solve for vy: ghvy 2=
Substitute numerical values and evaluate vy:
( )( ) m/s3.13m0.5m/s9.812 2 ==yv
Express the change in momentum in the y direction per bead:
( ) yyyyyy mvmvmvppp 2if =−−=−=∆
Chapter 8
620
Use Newton’s 2nd law to express the net force in the y direction exerted on the pan by the beads:
tp
NF yy ∆
∆=net,
Letting M represent the mass to be placed on the other pan, equate its weight to the net force exerted by the beads, substitute for ∆py, and solve for M:
tp
NMg y
∆
∆=
and
⎟⎟⎠
⎞⎜⎜⎝
⎛∆
=g
mvt
NM y2
Substitute numerical values and evaluate M:
( ) ( )( )[ ]
g9.31
m/s9.81m/s3.13kg0.00052s/100 2
=
=M
141 ••• Picture the Problem Assume that the connecting rod goes halfway through both balls, i.e., the centers of mass of the balls are separated by L. Let the system include the dumbbell, the wall and floor, and the earth. Let the zero of gravitational potential be at the center of mass of the lower ball and use conservation of energy to relate the speeds of the balls to the potential energy of the system. By symmetry, the speeds will be equal when the angle with the vertical is 45°. Use conservation of energy to express the relationship between the initial and final energies of the system:
fi EE =
Express the initial energy of the system:
mgLE =i
Express the energy of the system when the angle with the vertical is 45°:
( ) 221
f 245sin vmmgLE +°=
Substitute to obtain: 2
21 vgLgL +⎟
⎠
⎞⎜⎝
⎛=
Solve for v:
⎟⎠⎞
⎜⎝⎛ −=
211gLv
Systems of Particles and Conservation of Momentum
621
Substitute numerical values and evaluate v: ( )
( ) L
Lv
/sm70.1
211m/s81.9
21
2
=
⎟⎠⎞
⎜⎝⎛ −=
Chapter 8
622
623
Chapter 9 Rotation Conceptual Problems *1 • Determine the Concept Because r is greater for the point on the rim, it moves the greater distance. Both turn through the same angle. Because r is greater for the point on the rim, it has the greater speed. Both have the same angular velocity. Both have zero tangential acceleration. Both have zero angular acceleration. Because r is greater for the point on the rim, it has the greater centripetal acceleration. 2 •
(a) False. Angular velocity has the dimensions ⎥⎦⎤
⎢⎣⎡T1
whereas linear velocity has
dimensions ⎥⎦⎤
⎢⎣⎡TL
.
(b) True. The angular velocity of all points on the wheel is dθ/dt. (c) True. The angular acceleration of all points on the wheel is dω/dt. 3 •• Picture the Problem The constant-acceleration equation that relates the given variables is θαωω ∆+= 22
02 . We can set up a proportion to determine the number of revolutions
required to double ω and then subtract to find the number of additional revolutions to accelerate the disk to an angular speed of 2ω. Using a constant-acceleration equation, relate the initial and final angular velocities to the angular acceleration:
θαωω ∆+= 220
2
or, because 20ω = 0,
θαω ∆= 22
Let ∆θ10 represent the number of revolutions required to reach an angular velocity ω:
102 2 θαω ∆= (1)
Let ∆θ2ω represent the number of revolutions required to reach an angular velocity ω:
( ) ωθαω 22 22 ∆= (2)
Divide equation (2) by equation (1) and solve for ∆θ2ω:
( )10102
2
2 42 θθωωθ ω ∆=∆=∆
Chapter 9
624
The number of additional revolutions is: ( ) rev30rev10334 101010 ==∆=∆−∆ θθθ
and correct. is )(c
*4 •
Determine the Concept Torque has the dimension ⎥⎦
⎤⎢⎣
⎡2
2
TML
.
(a) Impulse has the dimension ⎥⎦⎤
⎢⎣⎡
TML
.
(b) Energy has the dimension ⎥⎦
⎤⎢⎣
⎡2
2
TML
. correct. is )(b
(c) Momentum has the dimension ⎥⎦⎤
⎢⎣⎡
TML
.
5 • Determine the Concept The moment of inertia of an object is the product of a constant that is characteristic of the object’s distribution of matter, the mass of the object, and the square of the distance from the object’s center of mass to the axis about which the object is rotating. Because both (b) and (c) are correct correct. is )(d
*6 • Determine the Concept Yes. A net torque is required to change the rotational state of an object. In the absence of a net torque an object continues in whatever state of rotational motion it was at the instant the net torque became zero. 7 • Determine the Concept No. A net torque is required to change the rotational state of an object. A net torque may decrease the angular speed of an object. All we can say for sure is that a net torque will change the angular speed of an object. 8 • (a) False. The net torque acting on an object determines the angular acceleration of the object. At any given instant, the angular velocity may have any value including zero. (b) True. The moment of inertia of a body is always dependent on one’s choice of an axis of rotation. (c) False. The moment of inertia of an object is the product of a constant that is characteristic of the object’s distribution of matter, the mass of the object, and the square of the distance from the object’s center of mass to the axis about which the object is
Rotation
625
rotating. 9 • Determine the Concept The angular acceleration of a rotating object is proportional to the net torque acting on it. The net torque is the product of the tangential force and its lever arm. Express the angular acceleration of the disk as a function of the net torque acting on it:
dIF
IFd
I=== netτα
i.e., d∝α
Because d∝α , doubling d will double the angular acceleration.
correct. is )(b
*10 • Determine the Concept From the parallel-axis theorem we know that
,2cm MhII += where Icm is the moment of inertia of the object with respect to an axis
through its center of mass, M is the mass of the object, and h is the distance between the parallel axes. Therefore, I is always greater than Icm by Mh2. correct. is )( d
11 • Determine the Concept The power delivered by the constant torque is the product of the torque and the angular velocity of the merry-go-round. Because the constant torque causes the merry-go-round to accelerate, neither the power input nor the angular velocity of the merry-go-round is constant. correct. is )(b
12 • Determine the Concept Let’s make the simplifying assumption that the object and the surface do not deform when they come into contact, i.e., we’ll assume that the system is rigid. A force does no work if and only if it is perpendicular to the velocity of an object, and exerts no torque on an extended object if and only if it’s directed toward the center of the object. Because neither of these conditions is satisfied, the statement is false. 13 • Determine the Concept For a given applied force, this increases the torque about the hinges of the door, which increases the door’s angular acceleration, leading to the door being opened more quickly. It is clear that putting the knob far from the hinges means that the door can be opened with less effort (force). However, it also means that the hand on the knob must move through the greatest distance to open the door, so it may not be the quickest way to open the door. Also, if the knob were at the center of the door, you would have to walk around the door after opening it, assuming the door is opening toward you.
Chapter 9
626
*14 • Determine the Concept If the wheel is rolling without slipping, a point at the top of the wheel moves with a speed twice that of the center of mass of the wheel, but the bottom of the wheel is momentarily at rest. correct. is )(c
15 •• Picture the Problem The kinetic energies of both objects is the sum of their translational and rotational kinetic energies. Their speed dependence will differ due to the differences in their moments of inertia. We can express the total kinetic of both objects and equate them to decide which of their translational speeds is greater. Express the kinetic energy of the cylinder:
( )2cyl4
3
2cyl2
12
2cyl2
21
21
2cyl2
12cylcyl2
1cyl
mv
mvrv
mr
mvIK
=
+=
+= ω
Express the kinetic energy of the sphere:
( )2sph10
7
2sph2
12
2sph2
52
21
2sph2
12sphlsph2
1sph
mv
mvr
vmr
mvIK
=
+=
+= ω
Equate the kinetic energies and simplify to obtain:
sphsph1514
cyl vvv <=
and correct. is )(b
*16 • Determine the Concept You could spin the pipes about their center. The one which is easier to spin has its mass concentrated closer to the center of mass and, hence, has a smaller moment of inertia. 17 •• Picture the Problem Because the coin and the ring begin from the same elevation, they will have the same kinetic energy at the bottom of the incline. The kinetic energies of both objects is the sum of their translational and rotational kinetic energies. Their speed dependence will differ due to the differences in their moments of inertia. We can express the total kinetic of both objects and equate them to their common potential energy loss to decide which of their translational speeds is greater at the bottom of the incline.
Rotation
627
Express the kinetic energy of the coin at the bottom of the incline:
( )2coincoin4
3
2coincoin2
12
2coin2
coin21
21
2coincoin2
12coincyl2
1coin
vm
vmr
vrm
vmIK
=
+=
+= ω
Express the kinetic energy of the ring at the bottom of the incline:
( )2ringring
2ringring2
12
2ring2
ring21
2ringring2
12ringring2
1ring
vm
vmr
vrm
vmIK
=
+=
+= ω
Equate the kinetic of the coin to its change in potential energy as it rolled down the incline and solve for vcoin:
ghv
ghmvm
342
coin
coin2coincoin4
3
and=
=
Equate the kinetic of the ring to its change in potential energy as it rolled down the incline and solve for vring:
ghv
ghmvm
=
=
2ring
ring2ringring
and
correct. is )( and Therefore, ringcoin bvv >
18 •• Picture the Problem We can use the definitions of the translational and rotational kinetic energies of the hoop and the moment of inertia of a hoop (ring) to express and compare the kinetic energies. Express the translational kinetic energy of the hoop:
221
trans mvK =
Express the rotational kinetic energy of the hoop:
( ) 221
2
22
212
hoop21
rot mvrvmrIK === ω
Therefore, the translational and rotational
kinetic energies are the same and correct. is )( c
Chapter 9
628
19 •• Picture the Problem We can use the definitions of the translational and rotational kinetic energies of the disk and the moment of inertia of a disk (cylinder) to express and compare the kinetic energies. Express the translational kinetic energy of the disk:
221
trans mvK =
Express the rotational kinetic energy of the disk:
( ) 241
2
22
21
212
hoop21
rot mvrvmrIK === ω
Therefore, the translational kinetic energy is
greater and correct. is )( a
20 •• Picture the Problem Let us assume that f ≠ 0 and acts along the direction of motion. Now consider the acceleration of the center of mass and the angular acceleration about the point of contact with the plane. Because Fnet ≠ 0, acm ≠ 0. However, τ = 0 because l = 0, so α = 0. But α = 0 is not consistent with acm ≠ 0. Consequently, f = 0. 21 • Determine the Concept True. If the sphere is slipping, then there is kinetic friction which dissipates the mechanical energy of the sphere. 22 • Determine the Concept Because the ball is struck high enough to have topspin, the frictional force is forward; reducing ω until the nonslip condition is satisfied.
correct. is )(a
Estimation and Approximation 23 •• Picture the Problem Assume the wheels are hoops, i.e., neglect the mass of the spokes, and express the total kinetic energy of the bicycle and rider. Let M represent the mass of the rider, m the mass of the bicycle, mw the mass of each bicycle wheel, and r the radius of the wheels. Express the ratio of the kinetic energy associated with the rotation of the wheels to that associated with the total kinetic energy of the bicycle and rider:
rottrans
rot
tot
rot
KKK
KK
+= (1)
Rotation
629
Express the translational kinetic energy of the bicycle and rider: 2
212
21
riderbicycletrans
Mvmv
KKK
+=
+=
Express the rotational kinetic energy of the bicycle wheels:
( )( ) 2
w2
22
w
2w2
1wheel1rot,rot 22
vmrvrm
IKK
==
== ω
Substitute in equation (1) to obtain:
w
w21
21
w2
w2
212
21
2w
tot
rot
2
2
mMmmMm
mvmMvmv
vmKK
++
=++
=++
=
Substitute numerical values and evaluate Krot/Ktot:
%3.10
kg3kg38kg142
2
tot
rot =+
+=
KK
24 •• Picture the Problem We can apply the definition of angular velocity to find the angular orientation of the slice of toast when it has fallen a distance of 0.5 m from the edge of the table. We can then interpret the orientation of the toast to decide whether it lands jelly-side up or down. Relate the angular orientation θ of the toast to its initial angular orientation, its angular velocity ω, and time of fall ∆t:
t∆+= ωθθ 0 (1)
Use the equation given in the problem statement to find the angular velocity corresponding to this length of toast:
rad/s9.470.1m
m/s9.81956.02
==ω
Using a constant-acceleration equation, relate the distance the toast falls ∆y to its time of fall ∆t:
( )221
0 tatvy yy ∆+∆=∆ or, because v0y = 0 and ay = g,
( )221 tgy ∆=∆
Solve for ∆t: g
yt ∆=∆
2
Substitute numerical values and evaluate ∆t:
( ) s0.319m/s9.81
m0.522 ==∆t
Chapter 9
630
( )0
2f cos
2' θ+
gLv
Substitute in equation (1) to
find θ :
( )( )
°=°
×=
+=
203rad
180rad54.3
s0.319rad/s9.476
π
πθ
down. side-jelly with thei.e. ground, therespect towith 203 of anglean at be thereforel toast wilof slice theofn orientatio The °
*25 •• Picture the Problem Assume that the mass of an average adult male is about 80 kg, and that we can model his body when he is standing straight up with his arms at his sides as a cylinder. From experience in men’s clothing stores, a man’s average waist circumference seems to be about 34 inches, and the average chest circumference about 42 inches. We’ll also assume that about 20% of the body’s mass is in the two arms, and each has a length L = 1 m, so that each arm has a mass of about m = 8 kg. Letting Iout represent his moment of inertia with his arms straight out and Iin his moment of inertia with his arms at his side, the ratio of these two moments of inertia is:
in
armsbody
in
out
III
II +
= (1)
Express the moment of inertia of the ″man as a cylinder″:
221
in MRI =
Express the moment of inertia of his arms:
( ) 231
arms 2 mLI =
Express the moment of inertia of his body-less-arms:
( ) 221
body RmMI −=
Substitute in equation (1) to obtain:
( ) ( )2
21
2312
21
in
out 2MR
mLRmMII +−
=
Assume the circumference of the cylinder to be the average of the average waist circumference and the average chest circumference:
in382
in42in34av =
+=c
Find the radius of a circle whose circumference is 38 in:
m154.02π
cm100m1
incm2.54in38
2av
=
××==
πcR
Substitute numerical values and evaluate Iout/ Iin:
Rotation
631
( )( ) ( )( )( )( )
42.6m0.154kg80
m1kg8m0.154kg16kg802
21
2322
21
in
out =+−
=II
Angular Velocity and Angular Acceleration 26 • Picture the Problem The tangential and angular velocities of a particle moving in a circle are directly proportional. The number of revolutions made by the particle in a given time interval is proportional to both the time interval and its angular speed. (a) Relate the angular velocity of the particle to its speed along the circumference of the circle:
ωrv =
Solve for and evaluate ω: rad/s0.278m90
m/s25===
rvω
(b) Using a constant-acceleration equation, relate the number of revolutions made by the particle in a given time interval to its angular velocity:
( )
rev33.1
rad2rev1s30
srad278.0
=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=∆=∆
πωθ t
27 • Picture the Problem Because the angular acceleration is constant; we can find the various physical quantities called for in this problem by using constant-acceleration equations. (a) Using a constant-acceleration equation, relate the angular velocity of the wheel to its angular acceleration and the time it has been accelerating:
t∆+= αωω 0
or, when ω0 = 0, t∆= αω
Evaluate ω when ∆t = 6 s: ( ) rad/s15.6s62rad/s2.6 =⎟⎠⎞⎜
⎝⎛=ω
(b) Using another constant-acceleration equation, relate the angular displacement to the wheel’s angular acceleration and the time it
( )221
0 tt ∆+∆=∆ αωθ
or, when ω0 = 0, ( )2
21 t∆=∆ αθ
Chapter 9
632
has been accelerating: Evaluate θ∆ when ∆t = 6 s: ( ) ( )( ) rad8.46s6rad/s2.6s6 22
21 ==∆θ
(c) Convert ( )s6θ∆ from rad to
revolutions: ( ) rev45.7
rad2rev1rad8.46s6 =×=∆
πθ
(d) Relate the angular velocity of the particle to its tangential speed and evaluate the latter when ∆t = 6 s:
( )( ) m/s4.68rad/s15.6m0.3 === ωrv
Relate the resultant acceleration of the point to its tangential and centripetal accelerations when ∆t = 6 s:
( ) ( )42
2222c
2t
ωα
ωα
+=
+=+=
r
rraaa
Substitute numerical values and evaluate a:
( ) ( ) ( )2
422
m/s73.0
rad/s15.6rad/s2.6m0.3
=
+=a
*28 • Picture the Problem Because we’re assuming constant angular acceleration; we can find the various physical quantities called for in this problem by using constant-acceleration equations. (a) Using its definition, express the angular acceleration of the turntable:
tt ∆−
=∆∆
= 0ωωωα
Substitute numerical values and evaluate α:
2
31
rad/s0.134
s26s60
min1rev
rad2πminrev330
=
××−=α
Rotation
633
(b) Because the angular acceleration is constant, the average angular velocity is the average of its initial and final values:
rad/s75.12
s60min1
revrad2
minrev33
2
31
0av
=
××=
+=
π
ωωω
(c) Using the definition of ωav, find the number or revolutions the turntable makes before stopping:
( )( )
rev24.7rad2
rev1rad5.45
s26rad/s1.75av
=×=
=∆=∆
π
ωθ t
29 • Picture the Problem Because the angular acceleration of the disk is constant, we can use a constant-acceleration equation to relate its angular velocity to its acceleration and the time it has been accelerating. We can find the tangential and centripetal accelerations from their relationships to the angular velocity and angular acceleration of the disk. (a) Using a constant-acceleration equation, relate the angular velocity of the disk to its angular acceleration and time during which it has been accelerating:
t∆+= αωω 0
or, because ω0 = 0, t∆= αω
Evaluate ω when t = 5 s: ( ) ( )( ) rad/s40.0s5rad/s8s5 2 ==ω
(b) Express at in terms of α:
αra =t
Evaluate at when t = 5 s: ( ) ( )( )2
2t
m/s960.0
rad/s8m0.12s5
=
=a
Express ac in terms of ω:
2c ωra =
Evaluate ac when t = 5 s: ( ) ( )( )2
2c
m/s192
rad/s40.0m0.12s5
=
=a
30 • Picture the Problem We can find the angular velocity of the Ferris wheel from its definition and the linear speed and centripetal acceleration of the passenger from the relationships between those quantities and the angular velocity of the Ferris wheel.
Chapter 9
634
(a) Find ω from its definition: rad/s233.0s27
rad2==
∆∆
=πθω
t
(b) Find the linear speed of the passenger from his/her angular speed:
( )( )m/s79.2
rad/s0.233m12
=
== ωrv
Find the passenger’s centripetal acceleration from his/her angular velocity:
( )( )2
22c
m/s651.0
rad/s0.233m12
=
== ωra
31 • Picture the Problem Because the angular acceleration of the wheels is constant, we can use constant-acceleration equations in rotational form to find their angular acceleration and their angular velocity at any given time. (a) Using a constant-acceleration equation, relate the angular displacement of the wheel to its angular acceleration and the time it has been accelerating:
( )221
0 tt ∆+∆=∆ αωθ
or, because ω0 = 0, ( )2
21 t∆=∆ αθ
Solve for α: ( )22
t∆∆
=θα
Substitute numerical values and evaluate α:
( )
( )2
2 rad/s589.0s8
revrad2rev32
=⎟⎠⎞
⎜⎝⎛
=
π
α
(b) Using a constant-acceleration equation, relate the angular velocity of the wheel to its angular acceleration and the time it has been accelerating:
t∆+= αωω 0
or, when ω0 = 0, t∆= αω
Evaluate ω when ∆t = 8 s: ( ) ( )( ) rad/s71.4s8rad/s589.0s8 2 ==ω
Rotation
635
32 • Picture the Problem The earth rotates through 2π radians every 24 hours. Find ω using its definition:
rad/s1027.7h
s3600h24
rad2
5−×=
×=
∆∆
≡πθω
t
33 • Picture the Problem When the angular acceleration of a wheel is constant, its average angular velocity is the average of its initial and final angular velocities. We can combine this relationship with the always applicable definition of angular velocity to find the initial angular velocity of the wheel. Express the average angular velocity of the wheel in terms of its initial and final angular speeds:
20
avωω
ω+
=
or, because ω = 0, 02
1av ωω =
Express the definition of the average angular velocity of the wheel:
t∆∆
≡θω av
Equate these two expressions and solve for ω0:
( ) s57.3s2.8
rad5220 ==
∆∆
=tθω and
correct. is )(d
34 • Picture the Problem The tangential and angular accelerations of the wheel are directly proportional to each other with the radius of the wheel as the proportionality constant. Provided there is no slippage, the acceleration of a point on the rim of the wheel is the same as the acceleration of the bicycle. We can use its defining equation to determine the acceleration of the bicycle. Relate the tangential acceleration of a point on the wheel (equal to the acceleration of the bicycle) to the wheel’s angular acceleration and solve for its angular acceleration:
αraa == t
and
ra
=α
Chapter 9
636
Use its definition to express the acceleration of the wheel: t
vvtva
∆−
=∆∆
= 0
or, because v0 = 0,
tva∆
=
Substitute in the expression for α to obtain: tr
v∆
=α
Substitute numerical values and evaluate α:
( )( )2rad/s794.0
s14.0m0.6km
m1000s3600
h1h
km24
=
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
=α
*35 •• Picture the Problem The two tapes will have the same tangential and angular velocities when the two reels are the same size, i.e., have the same area. We can calculate the tangential speed of the tape from its length and running time and relate the angular velocity to the constant tangential speed and the radius of the reels when they are turning with the same angular velocity. Relate the angular velocity of the tape to its tangential speed:
rv
=ω (1)
Letting Rf represent the outer radius of the reel when the reels have the same area, express the condition that they have the same speed:
( )222122
f rRrR ππππ −=−
Solve for Rf:
2
22
frRR +
=
Substitute numerical values and evaluate Rf:
( ) ( ) mm32.92
mm12mm45 22
f =+
=R
Find the tangential speed of the tape from its length and running time: cm/s42.3
hs3600h2
mcm100m246
∆=
×
×==
tLv
Rotation
637
Substitute in equation (1) and evaluate ω:
rad/s1.04
mm10cm1mm32.9
cm/s3.42
f
=
×==
Rvω
Convert 1.04 rad/s to rev/min:
rev/min93.9
mins60
rad2rev1
srad04.1rad/s04.1
=
××=π
Torque, Moment of Inertia, and Newton’s Second Law for Rotation 36 • Picture the Problem The force that the woman exerts through her axe, because it does not act at the axis of rotation, produces a net torque that changes (decreases) the angular velocity of the grindstone. (a) From the definition of angular acceleration we have: tt ∆
−=
∆∆
= 0ωωωα
or, because ω = 0,
t∆−
= 0ωα
Substitute numerical values and evaluate α:
2rad/s49.8
s9s60
min1rev
rad2minrev730
−=
××−=
π
α
where the minus sign means that the grindstone is slowing down.
(b) Use Newton’s 2nd law in rotational form to relate the angular acceleration of the grindstone to the net torque slowing it:
ατ I=net
Express the moment of inertia of disk with respect to its axis of rotation:
221 MRI =
Chapter 9
638
Substitute to obtain: ατ MR21
net =
Substitute numerical values and evaluate τnet:
( )( ) ( )mN0462.0
rad/s8.49m0.08kg1.7 2221
net
⋅=
=τ
*37 • Picture the Problem We can find the torque exerted by the 17-N force from the definition of torque. The angular acceleration resulting from this torque is related to the torque through Newton’s 2nd law in rotational form. Once we know the angular acceleration, we can find the angular velocity of the cylinder as a function of time. (a) Calculate the torque from its definition:
( )( ) mN1.87m0.11N17 ⋅=== lFτ
(b) Use Newton’s 2nd law in rotational form to relate the acceleration resulting from this torque to the torque:
Iτα =
Express the moment of inertia of the cylinder with respect to its axis of rotation:
221 MRI =
Substitute to obtain: 2
2MR
τα =
Substitute numerical values and evaluate α:
( )( )( )
22 rad/s124
m0.11kg2.5mN1.872
=⋅
=α
(c) Using a constant-acceleration equation, express the angular velocity of the cylinder as a function of time:
tαωω += 0
or, because ω0 = 0, tαω =
Evaluate ω (5 s): ( ) ( )( ) rad/s620s5rad/s124s5 2 ==ω
38 •• Picture the Problem We can find the angular acceleration of the wheel from its definition and the moment of inertia of the wheel from Newton’s 2nd law.
Rotation
639
(a) Express the moment of inertia of the wheel in terms of the angular acceleration produced by the applied torque:
ατ
=I
Find the angular acceleration of the wheel:
2rad/s14.3s20
s60min1
revrad2
minrev600
=
××=
∆∆
=
πωαt
Substitute and evaluate I: 2
2 mkg9.15rad/s3.14
mN50⋅=
⋅=I
(b) Because the wheel takes 120 s to slow to a stop (it took 20 s to acquire an angular velocity of 600 rev/min) and its angular acceleration is directly proportional to the accelerating torque:
( ) mN33.8mN5061
61
fr ⋅=⋅== ττ
39 •• Picture the Problem The pendulum and the forces acting on it are shown in the free-body diagram. Note that the tension in the string is radial, and so exerts no tangential force on the ball. We can use Newton’s 2nd law in both translational and rotational form to find the tangential component of the acceleration of the bob. (a) Referring to the FBD, express the component of g
rm that is tangent
to the circular path of the bob:
θsint mgF =
Use Newton’s 2nd law to express the tangential acceleration of the bob:
θsintt g
mFa ==
(b) Noting that, because the line-of-action of the tension passes through the pendulum’s pivot point, its lever arm is zero and the net torque is due
∑ = θτ sinpointpivot mgL
Chapter 9
640
to the weight of the bob, sum the torques about the pivot point to obtain: (c) Use Newton’s 2nd law in rotational form to relate the angular acceleration of the pendulum to the net torque acting on it:
αθτ ImgL == sinnet
Solve for α to obtain: I
mgL θα sin=
Express the moment of inertia of the bob with respect to the pivot point:
2mLI =
Substitute to obtain: L
gmL
mgL θθα sinsin2 ==
Relate α to at: θθα sinsin
t gL
gLra =⎟⎠⎞
⎜⎝⎛==
*40 ••• Picture the Problem We can express the velocity of the center of mass of the rod in terms of its distance from the pivot point and the angular velocity of the rod. We can find the angular velocity of the rod by using Newton’s 2nd law to find its angular acceleration and then a constant-acceleration equation that relates ω to α. We’ll use the impulse-momentum relationship to derive the expression for the force delivered to the rod by the pivot. Finally, the location of the center of percussion of the rod will be verified by setting the force exerted by the pivot to zero. (a) Relate the velocity of the center of mass to its distance from the pivot point:
ω2cmLv = (1)
Express the torque due to F0:
ατ pivot0 IxF ==
Solve for α: pivot
0
IxF
=α
Express the moment of inertia of the rod with respect to an axis through
231
pivot MLI =
Rotation
641
its pivot point: Substitute to obtain:
203
MLxF
=α
Express the angular velocity of the rod in terms of its angular acceleration:
203
MLtxFt ∆
=∆= αω
Substitute in equation (1) to obtain:
MLtxF
v2
3 0cm
∆=
(b) Let IP be the impulse exerted by the pivot on the rod. Then the total impulse (equal to the change in momentum of the rod) exerted on the rod is:
cm0P MvtFI =∆+
and tFMvI ∆−= 0cmP
Substitute our result from (a) to obtain:
⎟⎠⎞
⎜⎝⎛ −∆=∆−
∆= 1
23
23
000
P LxtFtF
LtxFI
Because tFI ∆= PP :
⎟⎠⎞
⎜⎝⎛ −= 1
23
0P LxFF
In order for FP to be zero:
0123
=−Lx
⇒3
2Lx =
41 ••• Picture the Problem We’ll first express the torque exerted by the force of friction on the elemental disk and then integrate this expression to find the torque on the entire disk. We’ll use Newton’s 2nd law to relate this torque to the angular acceleration of the disk and then to the stopping time for the disk. (a) Express the torque exerted on the elemental disk in terms of the friction force and the distance to the elemental disk:
kf rdfd =τ (1)
Using the definition of the coefficient of friction, relate the
gdmdf kk µ= (2)
Chapter 9
642
force of friction to µk and the weight of the circular element: Letting σ represent the mass per unit area of the disk, express the mass of the circular element:
drrdm σπ2= (3)
Substitute equations (2) and (3) in (1) to obtain:
drrgd 2kf 2 σµπτ = (4)
Because 2RM
πσ = : drr
RgMd 2
2k
f2µτ =
(b) Integrate fτd to obtain the total
torque on the elemental disk: gMRdrr
RgM R
k32
0
22
kf
2 µµτ == ∫
(c) Relate the disk’s stopping time to its angular velocity and acceleration:
αω
=∆t
Using Newton’s 2nd law, express α in terms of the net torque acting on the disk:
Ifτ
α =
The moment of inertia of the disk, with respect to its axis of rotation, is:
221 MRI =
Substitute and simplify to obtain: g
Rtk4
3µ
ω=∆
Calculating the Moment of Inertia 42 • Picture the Problem One can find the formula for the moment of inertia of a thin spherical shell in Table 9-1. The moment of inertia of a thin spherical shell about its diameter is:
232 MRI =
Rotation
643
Substitute numerical values and evaluate I:
( )( )25
232
mkg104.66
m0.035kg0.057
⋅×=
=−
I
*43 • Picture the Problem The moment of inertia of a system of particles with respect to a given axis is the sum of the products of the mass of each particle and the square of its distance from the given axis. Use the definition of the moment of inertia of a system of particles to obtain:
244
233
222
211
i
2ii
rmrmrmrm
rmI
+++=
= ∑
Substitute numerical values and evaluate I:
( )( ) ( )( )( )( ) ( )( )
2
22
22
mkg0.56
m2kg30kg4
m22kg4m2kg3
⋅=
++
+=I
44 • Picture the Problem Note, from symmetry considerations, that the center of mass of the system is at the intersection of the diagonals connecting the four masses. Thus the distance of each particle from the axis through the center of mass is m2 . According to the parallel-axis theorem, 2
cm MhII += , where Icm is the moment of inertia of the
object with respect to an axis through its center of mass, M is the mass of the object, and h is the distance between the parallel axes. Express the parallel axis theorem:
2cm MhII +=
Solve for Icm and substitute from Problem 44: ( )( )
2
22
2cm
mkg28.0
m2kg14mkg6.05
⋅=
−⋅=
−= MhII
Use the definition of the moment of inertia of a system of particles to express Icm:
244
233
222
211
i
2iicm
rmrmrmrm
rmI
+++=
= ∑
Substitute numerical values and evaluate Icm:
( )( ) ( )( )( )( ) ( )( )
2
22
22
cm
mkg0.28
m2kg3m2kg4
m2kg4m2kg3
⋅=
++
+=I
Chapter 9
644
45 • Picture the Problem The moment of inertia of a system of particles with respect to a given axis is the sum of the products of the mass of each particle and the square of its distance from the given axis. (a) Apply the definition of the moment of inertia of a system of particles to express Ix:
244
233
222
211
i
2ii
rmrmrmrm
rmI x
+++=
= ∑
Substitute numerical values and evaluate Ix:
( )( ) ( )( )( )( ) ( )( )
2
22
mkg0.28
0kg30kg4m2kg4m2kg3
⋅=
+++=xI
(b) Apply the definition of the moment of inertia of a system of particles to express Iy:
244
233
222
211
i
2ii
rmrmrmrm
rmI y
+++=
= ∑
Substitute numerical values and evaluate Iy:
( )( ) ( )( )( )( ) ( )( )
2
2
2
mkg0.28
m2kg30kg4
m2kg40kg3
⋅=
++
+=yI
Remarks: We could also use a symmetry argument to conclude that Iy = Ix . 46 • Picture the Problem According to the parallel-axis theorem, ,2
cm MhII += where Icm
is the moment of inertia of the object with respect to an axis through its center of mass, M is the mass of the object, and h is the distance between the parallel axes. Use Table 9-1 to find the moment of inertia of a sphere with respect to an axis through its center of mass:
252
cm MRI =
Express the parallel axis theorem:
2cm MhII +=
Substitute for Icm and simplify to obtain:
25722
52 MRMRMRI =+=
Rotation
645
47 •• Picture the Problem The moment of inertia of the wagon wheel is the sum of the moments of inertia of the rim and the six spokes. Express the moment of inertia of the wagon wheel as the sum of the moments of inertia of the rim and the spokes:
spokesrimwheel III +=
Using Table 9-1, find formulas for the moments of inertia of the rim and spokes: 2
spoke31
spoke
2rimrim
andLMI
RMI
=
=
Substitute to obtain: ( )
2spoke
2rim
2spoke3
12rimwheel
2
6
LMRM
LMRMI
+=
+=
Substitute numerical values and evaluate Iwheel:
( )( ) ( )( )2
22wheel
mkg60.2
m0.5kg1.22m5.0kg8
⋅=
+=I
*48 •• Picture the Problem The moment of inertia of a system of particles depends on the axis with respect to which it is calculated. Once this choice is made, the moment of inertia is the sum of the products of the mass of each particle and the square of its distance from the chosen axis. (a) Apply the definition of the moment of inertia of a system of particles:
( )22
21
i
2ii xLmxmrmI −+== ∑
(b) Set the derivative of I with respect to x equal to zero in order to identify values for x that correspond to either maxima or minima:
( )( )
( )extremafor 0
2
122
221
21
=−+=
−−+=
Lmxmxm
xLmxmdxdI
If 0=dxdI
, then:
0221 =−+ Lmxmxm
Chapter 9
646
Solve for x:
21
2
mmLmx
+=
Convince yourself that you’ve found
a minimum by showing that 2
2
dxId
is
positive at this point. . from mass ofcenter theof distance
the,definitionby is, 21
2
mmmLmx
+=
49 •• Picture the Problem Let σ be the mass per unit area of the uniform rectangular plate. Then the elemental unit has mass dm = σ dxdy. Let the corner of the plate through which the axis runs be the origin. The distance of the element whose mass is dm from the corner r is related to the coordinates of dm through the Pythagorean relationship r2 = x2 + y2. (a) Express the moment of inertia of the element whose mass is dm with respect to an axis perpendicular to it and passing through one of the corners of the uniform rectangular plate:
( )dxdyyxdI 22 += σ
Integrate this expression to find I: ( )
( ) ( )323133
31
0 0
22
bamabba
dxdyyxIa b
+=+=
+= ∫ ∫σ
σ
(b) Letting d represent the distance from the origin to the center of mass of the plate, use the parallel axis theorem to relate the moment of inertia found in (a) to the moment of inertia with respect to an axis through the center of mass:
( ) 222312
cm
2cm
ormdbammdII
mdII
−+=−=
+=
Using the Pythagorean theorem, relate the distance d to the center of
( ) ( ) ( )22412
212
212 babad +=+=
Rotation
647
mass to the lengths of the sides of the plate: Substitute for d2 in the expression for Icm and simplify to obtain:
( ) ( )( )22
121
2224122
31
cm
bam
bambamI
+=
+−+=
*50 •• Picture the Problem Corey will use the point-particle relationship
222
211
i
2iiapp rmrmrmI +== ∑ for his calculation whereas Tracey’s calculation will take
into account not only the rod but also the fact that the spheres are not point particles. (a) Using the point-mass approximation and the definition of the moment of inertia of a system of particles, express Iapp:
222
211
i
2iiapp rmrmrmI +== ∑
Substitute numerical values and evaluate Iapp:
( )( ) ( )( )2
22app
mkg0.0400
m0.2kg0.5m0.2kg0.5
⋅=
+=I
Express the moment of inertia of the two spheres and connecting rod system:
rodspheres III +=
Use Table 9-1 to find the moments of inertia of a sphere (with respect to its center of mass) and a rod (with respect to an axis through its center of mass):
2rod12
1rod
2sphere5
2sphere
andLMI
RMI
=
=
Because the spheres are not on the axis of rotation, use the parallel axis theorem to express their moment of inertia with respect to the axis of rotation:
rotation. of axis the tosphere a of mass of
center thefrom distance theish where
2sphere
2sphere5
2sphere hMRMI +=
Substitute to obtain: { } 2rod12
12sphere
2sphere5
22 LMhMRMI ++=
Substitute numerical values and evaluate I:
Chapter 9
648
( )( ) ( )( ){ } ( )( )2
212122
52
mkg0415.0
m0.3kg0.06m0.2kg0.5m0.05kg0.52
⋅=
++=I
Compare I and Iapp by taking their ratio:
964.0mkg0.0415mkg0.0400
2
2app =
⋅⋅
=I
I
(b) sphere. solid a of an greater th
is sphere hollow a of because increase wouldinertia rotational The
cm
cm
II
51 •• Picture the Problem The axis of rotation passes through the center of the base of the tetrahedron. The carbon atom and the hydrogen atom at the apex of the tetrahedron do not contribute to I because the distance of their nuclei from the axis of rotation is zero. From the geometry, the distance of the three H nuclei from the rotation axis is 3/a , where a is the length of a side of the tetrahedron. Apply the definition of the moment of inertia for a system of particles to obtain: 2
H
2
H
23H
22H
21H
i
2ii
33 amam
rmrmrmrmI
=⎟⎠
⎞⎜⎝
⎛=
++== ∑
Substitute numerical values and evaluate I:
( )( )247
2927
mkg1041.5
m1018.0kg101.67
⋅×=
××=−
−−I
52 •• Picture the Problem Let the mass of the element of volume dV be dm = ρdV = 2πρhrdr where h is the height of the cylinder. We’ll begin by expressing the moment of inertia dI for the element of volume and then integrating it between R1 and R2.
Rotation
649
Express the moment of inertia of the element of mass dm:
drhrdmrdI 32 2πρ==
Integrate dI from R1 to R2 to obtain: ( )
( )( )21
22
21
222
1
41
422
132
1
2
RRRRh
RRhdrrhIR
R
+−=
−== ∫πρ
πρπρ
The mass of the hollow cylinder is ( )2
122 RRhm −= ρπ , so:
( )21
22 RRhm
−=
πρ
Substitute for ρ and simplify to obtain:
( ) ( )( ) ( )21
222
121
22
21
222
122
21 RRmRRRRh
RRhmI +=+−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
=π
π
53 ••• Picture the Problem We can derive the given expression for the moment of inertia of a spherical shell by following the procedure outlined in the problem statement. Find the moment of inertia of a sphere, with respect to an axis through a diameter, in Table 9-1:
252 mRI =
Express the mass of the sphere as a function of its density and radius:
334 Rm ρπ=
Substitute to obtain:
5158 RI ρπ=
Express the differential of this expression:
dRRdI 438 ρπ= (1)
Express the increase in mass dm as the radius of the sphere increases by dR:
dRRdm 24 ρπ= (2)
Eliminate dR between equations (1) and (2) to obtain:
dmRdI 232=
. is mass of shell sphericalthe of inertia ofmoment theTherefore,2
32 mRm
Chapter 9
650
*54 ••• Picture the Problem We can find C in terms of M and R by integrating a spherical shell of mass dm with the given density function to find the mass of the earth as a function of M and then solving for C. In part (b), we’ll start with the moment of inertia of the same spherical shell, substitute the earth’s density function, and integrate from 0 to R. (a) Express the mass of the earth using the given density function:
33
0
3
0
2
0
2
22.13
4
422.14
4
CRCR
drrR
CdrrC
drrdmM
RR
R
ππ
ππ
ρπ
−=
−=
==
∫∫
∫∫
Solve for C as a function of M and R to obtain:
3508.0RMC =
(b) From Problem 9-40 we have: drrdI 438 ρπ=
Integrate to obtain:
( )
2
553
0 0
543
0
438
329.0
61
522.126.4
122.13508.08
MR
RRR
M
drrR
drrR
M
drrI
R R
R
=
⎥⎦⎤
⎢⎣⎡ −=
⎥⎦
⎤⎢⎣
⎡−=
=
∫ ∫
∫π
ρπ
Rotation
651
55 ••• Picture the Problem Let the origin be at the apex of the cone, with the z axis along the cone’s symmetry axis. Then the radius of the elemental ring, at a distance z from the apex, can be obtained from the
proportionHR
zr
= . The mass dm of the
elemental disk is ρdV = ρπr2dz. We’ll integrate r2dm to find the moment of inertia of the disk in terms of R and H and then integrate dm to obtain a second equation in R and H that we can use to eliminate H in our expression for I.
Express the moment of inertia of the cone in terms of the moment of inertia of the elemental disk:
102
4
0
44
4
22
22
02
2
21
221
HRdzzHR
dzzHRz
HR
dmrI
H
H
πρπρ
ρπ
==
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=
∫
∫
∫
Express the total mass of the cone in terms of the mass of the elemental disk: HR
dzzHRdzrM
HH
231
0
22
2
0
2
πρ
πρπρ
=
== ∫∫
Divide I by M, simplify, and solve for I to obtain:
2103 MRI =
56 ••• Picture the Problem Let the axis of rotation be the x axis. The radius r of the
elemental area is 22 zR − and its mass,
dm, is dzzRdA 222 −= σσ . We’ll
integrate z2 dm to determine I in terms of σ and then divide this result by M in order to eliminate σ and express I in terms of M and R.
Chapter 9
652
Express the moment of inertia about the x axis:
( )4
41
222
22
2
R
dzzRz
dAzdmzIR
R
σπ
σ
σ
=
−=
==
∫
∫∫
−
The mass of the thin uniform disk is:
2RM σπ=
Divide I by M, simplify, and solve for I to obtain:
241 MRI = , a result in agreement with
the expression given in Table 9-1 for a cylinder of length L = 0.
57 ••• Picture the Problem Let the origin be at the apex of the cone, with the z axis along the cone’s symmetry axis, and the axis of rotation be the x rotation. Then the radius of the elemental disk, at a distance z from the apex, can be obtained from the
proportionHR
zr
= . The mass dm of the
elemental disk is ρdV = ρπr2dz. Each elemental disk rotates about an axis that is parallel to its diameter but removed from it by a distance z. We can use the result from Problem 9-57 for the moment of inertia of the elemental disk with respect to a diameter and then use the parallel axis theorem to express the moment of inertia of the cone with respect to the x axis.
Using the parallel axis theorem, express the moment of inertia of the elemental disk with respect to the x axis:
2disk zdmdIdI x += (1)
where dzrdVdm 2ρπρ ==
In Problem 9-57 it was established that the moment of inertia of a thin uniform disk of mass M and radius R rotating about a diameter is 2
41 MR . Express this result in
( )
dzzHR
rdzrdI2
22
2
41
2241
disk
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=
ρπ
ρπ
Rotation
653
terms of our elemental disk: Substitute in equation (1) to obtain:
22
22
2
2
41
zdzzHR
dzzHRdI x
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛=
πρ
πρ
Integrate from 0 to H to obtain:
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛= ∫
520
41
324
0
42
222
2
2
HRHR
dzzHRz
HRI
H
x
πρ
πρ
Express the total mass of the cone in terms of the mass of the elemental disk: HR
dzzHRdzrM
HH
231
0
22
2
0
2
πρ
πρπρ
=
== ∫∫
Divide Ix by M, simplify, and solve for Ix to obtain: ⎟⎟
⎠
⎞⎜⎜⎝
⎛+=
2053
22 RHMI x
Remarks: Because both H and R appear in the numerator, the larger the cones are,
the greater their moment of inertia and the greater the energy consumption required to set them into motion. Rotational Kinetic Energy 58 • Picture the Problem The kinetic energy of this rotating system of particles can be calculated either by finding the tangential velocities of the particles and using these values to find the kinetic energy or by finding the moment of inertia of the system and using the expression for the rotational kinetic energy of a system. (a) Use the relationship between v and ω to find the speed of each particle:
( )( )
( )( ) m/s0.8rad/s2m0.4and
m/s0.4rad/s2m0.2
11
33
===
===
ω
ω
rv
rv
Chapter 9
654
Find the kinetic energy of the system: ( )( ) ( )( )
J1.12
m/s0.8kg1m/s0.4kg3
2222
211
23313
=
+=
+=+= vmvmKKK
(b) Use the definition of the moment of inertia of a system of particles to obtain:
244
233
222
211
2
rmrmrmrm
rmIi
ii
+++=
= ∑
Substitute numerical values and evaluate I:
( )( ) ( )( )( )( ) ( )( )
2
22
22
mkg560.0m0.2kg3m0.4kg1
m0.2kg3m0.4kg1
⋅=
++
+=I
Calculate the kinetic energy of the system of particles:
( )( )J1.12
rad/s2mkg0.560 22212
21
=
⋅== ωIK
*59 • Picture the Problem We can find the kinetic energy of this rotating ball from its angular speed and its moment of inertia. We can use the same relationship to find the new angular speed of the ball when it is supplied with additional energy. (a) Express the kinetic energy of the ball:
221 ωIK =
Express the moment of inertia of ball with respect to its diameter:
252 MRI =
Substitute for I: 2251 ωMRK =
Substitute numerical values and evaluate K:
( )( )
Jm6.84
s60min1
revrad2
minrev70
m0.075kg1.42
251
=
⎟⎟⎠
⎞⎜⎜⎝
⎛×××
=
π
K
(b) Express the new kinetic energy with K′ = 2.0846 J:
221 '' ωIK =
Express the ratio of K to K′: 2
221
221
'⎟⎠⎞
⎜⎝⎛==
ωω
ωω '
I'I
KK'
Rotation
655
Solve for ω′:
KK'' ωω =
Substitute numerical values and evaluate ω′: ( )
rev/min347
J0.0846J2.0846rev/min70
=
='ω
60 • Picture the Problem The power delivered by an engine is the product of the torque it develops and the angular speed at which it delivers the torque. Express the power delivered by the engine as a function of the torque it develops and the angular speed at which it delivers this torque:
ωτ=P
Substitute numerical values and evaluate P:
( ) kW155s60
min1rev
rad2minrev3700mN400 =⎟⎟
⎠
⎞⎜⎜⎝
⎛××⋅=
πP
61 •• Picture the Problem Let r1 and r2 be the distances of m1 and m2 from the center of mass. We can use the definition of rotational kinetic energy and the definition of the center of mass of the two point masses to show that K1/K2 = m2/m1. Use the definition of rotational kinetic energy to express the ratio of the rotational kinetic energies:
222
211
2222
2211
222
1
212
1
2
1
rmrm
rmrm
II
KK
===ωω
ωω
Use the definition of the center of mass to relate m1, m2, r1, and r2:
2211 mrmr =
Solve for 2
1
rr
, substitute and
simplify to obtain: 1
2
2
1
2
2
1
2
1
mm
mm
mm
KK
=⎟⎟⎠
⎞⎜⎜⎝
⎛=
62 •• Picture the Problem The earth’s rotational kinetic energy is given by
221
rot ωIK = where I is its moment of inertia with respect to its axis of rotation. The
Chapter 9
656
center of mass of the earth-sun system is so close to the center of the sun and the earth-sun distance so large that we can use the earth-sun distance as the separation of their centers of mass and assume each to be point mass. Express the rotational kinetic energy of the earth:
221
rot ωIK = (1)
Find the angular speed of the earth’s rotation using the definition of ω:
rad/s1027.7h
s3600h24
rad2
5−×=
×=
∆∆
=πθω
t
From Table 9-1, for the moment of inertia of a homogeneous sphere, we find:
( )( )237
262452
252
mkg109.83
m106.4kg106.0
⋅×=
××=
= MRI
Substitute numerical values in equation (1) to obtain:
( )( )
J102.60
rad/s107.27
mkg109.83
29
25
23721
rot
×=
××
⋅×=−
K
Express the earth’s orbital kinetic energy:
2orb2
1orb ωIK = (2)
Find the angular speed of the center of mass of the earth-sun system:
rad/s1099.1h
s3600dayh24days365.25
rad2
7−×=
××=
∆∆
=
π
θωt
Express and evaluate the orbital moment of inertia of the earth: ( )( )
247
21124
2orbE
mkg101.35m101.50kg106.0
⋅×=
××=
= RMI
Substitute in equation (2) to obtain: ( )
( )J102.67
rad/s101.99
mkg101.35
33
27
24721
orb
×=
××
⋅×=−
K
Rotation
657
Evaluate the ratio rot
orb
KK
: 429
33
rot
orb 10J102.60J102.67
≈××
=KK
*63 •• Picture the Problem Because the load is not being accelerated, the tension in the cable equals the weight of the load. The role of the massless pulley is to change the direction the force (tension) in the cable acts. (a) Because the block is lifted at constant speed:
( )( )kN19.6
m/s9.81kg2000 2
=
== mgT
(b) Apply the definition of torque at the winch drum:
( )( )mkN5.89
m0.30kN19.6
⋅=
== Trτ
(c) Relate the angular speed of the winch drum to the rate at which the load is being lifted (the tangential speed of the cable on the drum):
rad/s0.267m0.30
m/s0.08===
rvω
(d) Express the power developed by the motor in terms of the tension in the cable and the speed with which the load is being lifted:
( )( )kW1.57
m/s0.08kN19.6
=
== TvP
64 •• Picture the Problem Let the zero of gravitational potential energy be at the lowest point of the small particle. We can use conservation of energy to find the angular velocity of the disk when the particle is at its lowest point and Newton’s 2nd law to find the force the disk will have to exert on the particle to keep it from falling off. (a) Use conservation of energy to relate the initial potential energy of the system to its rotational kinetic energy when the small particle is at its lowest point:
0=∆+∆ UK or, because Uf = Ki = 0,
( ) 02fparticledisk2
1 =∆−+ hmgII ω
Solve for ωf:
particlediskf
2II
hmg+
∆=ω
Chapter 9
658
Substitute for Idisk, Iparticle, and ∆h and simplify to obtain:
( )( )MmR
mgmRMRRmg
+=
+=
2822
2221fω
(b) The mass is in uniform circular motion at the bottom of the disk, so the sum of the force F exerted by the disk and the gravitational force must be the centripetal force:
2fωmRmgF =−
Solve for F and simplify to obtain:
( )
⎟⎠⎞
⎜⎝⎛
++=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
+=
Mmmmg
mMRmgmRmg
mRmgF
281
28
2fω
65 •• Picture the Problem Let the zero of gravitational potential energy be at the center of mass of the ring when it is directly below the point of support. We’ll use conservation of energy to relate the maximum angular velocity and the initial angular velocity required for a complete revolution to the changes in the potential energy of the ring. (a) Use conservation of energy to relate the initial potential energy of the ring to its rotational kinetic energy when its center of mass is directly below the point of support:
0=∆+∆ UK or, because Uf = Ki = 0,
02max2
1 =∆− hmgIPω (1)
Use the parallel axis theorem and Table 9-1 to express the moment of inertia of the ring with respect to its pivot point P:
2cm mRII P +=
Substitute in equation (1) to obtain: ( ) 02max
2221 =−+ mgRmRmR ω
Solve for ωmax:
Rg
=maxω
Substitute numerical values and evaluate ωmax:
rad/s3.62m0.75
m/s9.81 2
max ==ω
Rotation
659
(b) Use conservation of energy to relate the final potential energy of the ring to its initial rotational kinetic energy:
0=∆+∆ UK or, because Ui = Kf = 0,
02i2
1 =∆+− hmgI Pω
Noting that the center of mass must rise a distance R if the ring is to make a complete revolution, substitute for IP and ∆h to obtain:
( ) 02i
2221 =++− mgRmRmR ω
Solve for ωi:
Rg
i =ω
Substitute numerical values and evaluate ωi: rad/s3.62
m0.75m/s9.81 2
==iω
66 •• Picture the Problem We can find the energy that must be stored in the flywheel and relate this energy to the radius of the wheel and use the definition of rotational kinetic energy to find the wheel’s radius. Relate the kinetic energy of the flywheel to the energy it must deliver:
( )( )MJ600
km300MJ/km22cyl2
1rot
=
== ωIK
Express the moment of inertia of the flywheel:
221
cyl MRI =
Substitute for Icyl and solve for ω:
MKR rot2
ω=
Substitute numerical values and evaluate R:
m95.1
kg100MJ
J10MJ600
revrad2
srev400
26
=
×
×= πR
67 •• Picture the Problem We’ll solve this problem for the general case of a ladder of length L, mass M, and person of mass m. Let the zero of gravitational potential energy be at floor level and include you, the ladder, and the earth in the system. We’ll use
Chapter 9
660
conservation of energy to relate your impact speed falling freely to your impact speed riding the ladder to the ground. Use conservation of energy to relate the speed with which a person will strike the ground to the fall distance L:
0=∆+∆ UK or, because Ki = Uf = 0,
02f2
1 =− mgLmv
Solve for 2fv : gLv 22
f =
Letting ωr represent the angular velocity of the ladder+person system as it strikes the ground, use conservation of energy to relate the initial and final momenta of the system:
0=∆+∆ UK or, because Ki = Uf = 0,
( ) 02
2rladderperson2
1 =⎟⎠⎞
⎜⎝⎛ +−+
LMgmgLII ω
Substitute for the moments of inertia to obtain:
023
1 2f
221 =⎟
⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ +
LMgmgLLMm ω
Substitute vr for Lωf and solve for 2rv :
3
22
2r Mm
MmgLv
+
⎟⎠⎞
⎜⎝⎛ +
=
Express the ratio 2f
2r
vv
:
3
22f
2r
Mm
Mm
vv
+
+=
Solve for vr to obtain: Mm
Mmvv2636
fr ++
=
ground. the tofall and golet better to isIt . zero, is ladder, theof mass the, Unless fr vvM >
Rotation
661
Pulleys, Yo-Yos, and Hanging Things *68 •• Picture the Problem We’ll solve this problem for the general case in which the mass of the block on the ledge is M, the mass of the hanging block is m, and the mass of the pulley is Mp, and R is the radius of the pulley. Let the zero of gravitational potential energy be 2.5 m below the initial position of the 2-kg block and R represent the radius of the pulley. Let the system include both blocks, the shelf and pulley, and the earth. The initial potential energy of the 2-kg block will be transformed into the translational kinetic energy of both blocks plus rotational kinetic energy of the pulley. (a) Use energy conservation to relate the speed of the 2 kg block when it has fallen a distance ∆h to its initial potential energy and the kinetic energy of the system:
0=∆+∆ UK or, because Ki = Uf = 0,
( ) 02pulley2
1221 =−++ mghIvMm ω
Substitute for Ipulley and ω to obtain: ( ) ( ) 02
22
21
212
21 =−++ mgh
RvMRvMm
Solve for v:
pMmMmghv
21
2++
=
Substitute numerical values and evaluate v:
( )( )( )( )
m/s3.95
kg0.6kg2kg4m2.5m/s9.81kg22
21
2
=
++=v
(b) Find the angular velocity of the pulley from its tangential speed:
rad/s49.3m0.08
m/s3.95===
Rvω
69 •• Picture the Problem The diagrams show the forces acting on each of the masses and the pulley. We can apply Newton’s 2nd law to the two blocks and the pulley to obtain three equations in the unknowns T1, T2, and a.
Chapter 9
662
Apply Newton’s 2nd law to the two blocks and the pulley:
∑ == amTFx 41 , (1)
( )∑ =−= ατ pp IrTT 12 , (2)
and
∑ =−= amTgmFx 222 (3)
Eliminate α in equation (2) to obtain:
aMTT p21
12 =− (4)
Eliminate T1 and T2 between equations (1), (3) and (4) and solve for a:
pMmmgma
21
42
2
++=
Substitute numerical values and evaluate a:
( )( )( )
2
21
2
m/s3.11kg0.6kg4kg2
m/s9.81kg2=
++=a
Using equation (1), evaluate T1: ( )( ) N12.5m/s3.11kg4 21 ==T
Solve equation (3) for T2: ( )agmT −= 22
Substitute numerical values and evaluate T2:
( )( )N13.4
m/s3.11m/s9.81kg2 222
=
−=T
70 •• Picture the Problem We’ll solve this problem for the general case in which the mass of the block on the ledge is M, the mass of the hanging block is m, the mass of the pulley is Mp, and R is the radius of the pulley. Let the zero of gravitational potential energy be 2.5 m below the initial position of the 2-kg block. The initial potential energy of the 2-kg block will be transformed into the translational kinetic energy of both blocks plus rotational kinetic energy of the pulley plus work done against friction. (a) Use energy conservation to relate the speed of the 2 kg block when it has fallen a distance ∆h to its initial potential energy, the kinetic energy of the system and the work done against friction:
0f =+∆+∆ WUK
or, because Ki = Uf = 0, ( )
0k
2pulley2
1221
=+−
++
Mghmgh
IvMm
µ
ω
Substitute for Ipulley and ω to obtain: ( ) ( )0k
2
2
21
212
21
=+−
++
MghmghRvMvMm p
µ
Rotation
663
Solve for v: ( )pMmM
Mmghv21
k2++−
=µ
Substitute numerical values and evaluate v:
( )( ) ( )( )[ ]( ) m/s79.2
kg0.6kg2kg4kg425.0kg2m2.5m/s9.812
21
2
=++
−=v
(b) Find the angular velocity of the pulley from its tangential speed:
rad/s9.43m0.08
m/s79.2===
Rvω
71 •• Picture the Problem Let the zero of gravitational potential energy be at the water’s surface and let the system include the winch, the car, and the earth. We’ll apply energy conservation to relate the car’s speed as it hits the water to its initial potential energy. Note that some of the car’s initial potential energy will be transformed into rotational kinetic energy of the winch and pulley. Use energy conservation to relate the car’s speed as it hits the water to its initial potential energy:
0=∆+∆ UK or, because Ki = Uf = 0,
02pp2
12ww2
1221 =∆−++ hmgIImv ωω
Express ωw and ωp in terms of the speed v of the rope, which is the same throughout the system:
2p
2
p2w
2
w andrv
rv
== ωω
Substitute to obtain: 02
p
2
p21
2w
2
w212
21 =∆−++ hmg
rvI
rvImv
Solve for v:
2p
p2
w
w
2
rI
rIm
hmgv++
∆=
Substitute numerical values and evaluate v:
( )( )( )
( ) ( )m/s21.8
m0.3mkg4
m0.8mkg320kg1200
m5m/s9.81kg12002
2
2
2
2
2
=
⋅+
⋅+
=v
Chapter 9
664
*72 •• Picture the Problem Let the system include the blocks, the pulley and the earth. Choose the zero of gravitational potential energy to be at the ledge and apply energy conservation to relate the impact speed of the 30-kg block to the initial potential energy of the system. We can use a constant-acceleration equations and Newton’s 2nd law to find the tensions in the strings and the descent time.
(a) Use conservation of energy to relate the impact speed of the 30-kg block to the initial potential energy of the system:
0=∆+∆ UK or, because Ki = Uf = 0,
03020
2pp2
12202
12302
1
=∆−∆+
++
hgmhgm
Ivmvm ω
Substitute for ωp and Ip to obtain: ( )
03020
2
22
p21
212
20212
3021
=∆−∆+
⎟⎟⎠
⎞⎜⎜⎝
⎛++
hgmhgmrvrMvmvm
Solve for v: ( )
p21
3020
20302Mmm
mmhgv++−∆
=
Substitute numerical values and evaluate v:
( )( )( )( )
m/s73.2
kg5kg30kg20kg20kg30m2m/s9.812
21
2
=
++−
=v
(b) Find the angular speed at impact from the tangential speed at impact and the radius of the pulley:
rad/s27.3m0.1m/s2.73
===rvω
(c) Apply Newton’s 2nd law to the blocks:
∑ =−= amgmTFx 20201 (1)
∑ =−= amTgmFx 30230 (2)
Using a constant-acceleration equation, relate the speed at impact to the fall distance and the
havv ∆+= 220
2
or, because v0 = 0,
Rotation
665
acceleration and solve for and evaluate a:
( )( )
222
m/s1.87m22m/s2.73
2==
∆=
hva
Substitute in equation (1) to find T1: ( )
( )( )N234
m/s1.87m/s9.81kg20 22201
=
+=
+= agmT
Substitute in equation (2) to find T2: ( )
( )( )N238
m/s1.87m/s9.81kg30 22302
=
−=
−= agmT
(d) Noting that the initial speed of the 30-kg block is zero, express the time-of-fall in terms of the fall distance and the block’s average speed:
vh
vh
vht ∆
=∆
=∆
=∆2
21
av
Substitute numerical values and evaluate ∆t:
( ) s1.47m/s2.73m22
==∆t
73 •• Picture the Problem The force diagram shows the forces acting on the sphere and the hanging object. The tension in the string is responsible for the angular acceleration of the sphere and the difference between the weight of the object and the tension is the net force acting on the hanging object. We can use Newton’s 2nd law to obtain two equations in a and T that we can solve simultaneously.
(a)Apply Newton’s 2nd law to the sphere and the hanging object:
∑ == ατ sphere0 ITR (1)
and
∑ =−= maTmgFx (2)
Substitute for Isphere and α in equation (1) to obtain:
( )RaMRTR 2
52= (3)
Chapter 9
666
Eliminate T between equations (2) and (3) and solve for a to obtain:
mM
ga
521+
=
(b) Substitute for a in equation (2) and solve for T to obtain: Mm
mMgT25
2+
=
74 •• Picture the Problem The diagram shows the forces acting on both objects and the pulley. By applying Newton’s 2nd law of motion, we can obtain a system of three equations in the unknowns T1, T2, and a that we can solve simultaneously.
(a) Apply Newton’s 2nd law to the pulley and the two objects:
∑ =−= amgmTFx 111 , (1)
( )∑ =−= ατ 0120 IrTT , (2)
and
∑ =−= amTgmFx 222 (3)
Substitute for I0 = Ipulley and α in equation (2) to obtain:
( ) ( )ramrrTT 2
21
12 =− (4)
Eliminate T1 and T2 between equations (1), (3) and (4) and solve for a to obtain:
( )mmm
gmma21
21
12
++−
=
Substitute numerical values and evaluate a:
( )( )( )
2
21
2
cm/s9.478
g50g510g500cm/s981g500g510
=
++−
=a
(b) Substitute for a in equation (1) and solve for T1 to obtain:
( )( )( )
N4.9524
m/s0.09478m/s9.81kg0.500 2211
=
+=
+= agmT
Rotation
667
Substitute for a in equation (3) and solve for T2 to obtain:
( )( )( )
N4.9548
m/s0.09478m/s9.81kg0.510 2222
=
−=
−= agmT
Find ∆T:
N0.0024
N4.9524N.9548412
=
−=−=∆ TTT
(c) If we ignore the mass of the pulley, our acceleration equation is:
( )21
12
mmgmma
+−
=
Substitute numerical values and evaluate a:
( )( )
2
2
cm/s9.713
g510g500cm/s981g500g510
=
+−
=a
Substitute for a in equation (1) and solve for T1 to obtain:
( )agmT += 11
Substitute numerical values and evaluate T1:
( )( ) N4.9536m/s0.09713m/s9.81kg0.500 221 =+=T
From equation (4), if m = 0:
21 TT =
*75 •• Picture the Problem The diagram shows the forces acting on both objects and the pulley. By applying Newton’s 2nd law of motion, we can obtain a system of three equations in the unknowns T1, T2, and α that we can solve simultaneously.
(a) Express the condition that the system does not accelerate:
02211net =−= gRmgRmτ
Chapter 9
668
Solve for m2:
2
112 R
Rmm =
Substitute numerical values and evaluate m2:
( ) kg72.0m0.4m1.2
kg242 ==m
(b) Apply Newton’s 2nd law to the objects and the pulley:
∑ =−= amTgmFx 111 , (1)
∑ =−= ατ 022110 IRTRT , (2)
and
∑ =−= amgmTFx 222 (3)
Eliminate a in favor of α in equations (1) and (3) and solve for T1 and T2:
( )α111 RgmT −= (4)
and ( )α222 RgmT += (5)
Substitute for T1 and T2 in equation (2) and solve for α to obtain:
( )0
222
211
2211
IRmRmgRmRm
++−
=α
Substitute numerical values and evaluate α:
( )( ) ( )( )[ ]( )( )( ) ( )( )
2222
2
rad/s37.1mkg40m0.4kg72m1.2kg36
m/s9.81m0.4kg72m1.2kg36=
⋅++−
=α
Substitute in equation (4) to find T1:
( )[ ( )( )] N294rad/s1.37m1.2m/s9.81kg36 221 =−=T
Substitute in equation (5) to find T2:
( )[ ( )( )] N467rad/s1.37m4.0m/s9.81kg27 222 =+=T
Rotation
669
76 •• Picture the Problem Choose the coordinate system shown in the diagram. By applying Newton’s 2nd law of motion, we can obtain a system of two equations in the unknowns T and a. In (b) we can use the torque equation from (a) and our value for T to findα. In (c) we use the condition that the acceleration of a point on the rim of the cylinder is the same as the acceleration of the hand, together with the angular acceleration of the cylinder, to find the acceleration of the hand.
(a) Apply Newton’s 2nd law to the cylinder about an axis through its center of mass:
∑ ==RaITR 00τ (1)
and
∑ =−= 0TMgFx (2)
Solve for T to obtain:
MgT =
(b) Rewrite equation (1) in terms of α:
α0ITR =
Solve for α:
0ITR
=α
Substitute for T and I0 to obtain:
Rg
MRMgR 2
221
==α
(c) Relate the acceleration a of the hand to the angular acceleration of the cylinder:
αRa =
Substitute for α to obtain: gRgRa 22
=⎟⎠⎞
⎜⎝⎛=
Chapter 9
670
77 •• Picture the Problem Let the zero of gravitational potential energy be at the bottom of the incline. By applying Newton’s 2nd law to the cylinder and the block we can obtain simultaneous equations in a, T, and α from which we can express a and T. By applying the conservation of energy, we can derive an expression for the speed of the block when it reaches the bottom of the incline.
(a) Apply Newton’s 2nd law to the cylinder and the block:
∑ == ατ 00 ITR (1)
and
∑ =−= amTgmFx 22 sinθ (2)
Substitute for α in equation (1), solve for T, and substitute in equation (2) and solve for a to obtain:
2
1
21
sin
mm
ga+
=θ
(b) Substitute for a in equation (2) and solve for T:
2
1
121
21
sin
mm
gmT
+=
θ
(c) Noting that the block is released from rest, express the total energy of the system when the block is at height h:
ghmKUE 2=+=
(d) Use the fact that this system is conservative to express the total energy at the bottom of the incline:
ghmE 2bottom =
(e) Express the total energy of the system when the block is at the bottom of the incline in terms of its kinetic energies:
202
1222
1
rottranbottom
ωIvm
KKE
+=
+=
Rotation
671
Substitute for ω and I0 to obtain: ( ) ghmrvrmvm 22
22
121
212
221 =+
Solve for v to obtain:
2
1
21
2
mm
ghv+
=
(f) For θ = 0: 0== Ta
For θ = 90°:
2
1
21
mm
ga+
= ,
am
mmgm
T 121
2
1
121
21
=+
= ,
and
2
1
21
2
mm
ghv+
=
For m1 = 0: θsinga = , 0=T , and
ghv 2=
*78 •• Picture the Problem Let r be the radius of the concentric drum (10 cm) and let I0 be the moment of inertia of the drum plus platform. We can use Newton’s 2nd law in both translational and rotational forms to express I0 in terms of a and a constant-acceleration equation to express a and then find I0. We can use the same equation to find the total moment of inertia when the object is placed on the platform and then subtract to find its moment of inertia.
(a) Apply Newton’s 2nd law to the platform and the weight:
∑ == ατ 00 ITr (1)
∑ =−= MaTMgFx (2)
Chapter 9
672
Substitute a/r for α in equation (1) and solve for T:
arIT 2
0=
Substitute for T in equation (2) and solve for a to obtain:
( )a
agMrI −=
2
0 (3)
Using a constant-acceleration equation, relate the distance of fall to the acceleration of the weight and the time of fall and solve for the acceleration:
( )221
0 tatvx ∆+∆=∆
or, because v0 = 0 and ∆x = D,
( )22
tDa
∆=
Substitute for a in equation (3) to obtain:
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−
∆=⎟
⎠⎞
⎜⎝⎛ −= 1
21
222
0 DtgMr
agMrI
Substitute numerical values and evaluate I0:
( )( )( )( )
( )2
22
20
mkg1.177
1m1.82
s4.2m/s9.81
m0.1kg2.5
⋅=
⎥⎦
⎤⎢⎣
⎡−×
=I
(b) Relate the moments of inertia of the platform, drum, shaft, and pulley (I0) to the moment of inertia of the object and the total moment of inertia:
( )⎟⎟⎠
⎞⎜⎜⎝
⎛−
∆=
⎟⎠⎞
⎜⎝⎛ −=+=
12
1
22
20tot
DtgMr
agMrIII
Substitute numerical values and evaluate Itot:
( )( )( )( )
( )2
22
2tot
mkg125.3
1m1.82
s8.6m/s9.81
m0.1kg2.5
⋅=
⎥⎦
⎤⎢⎣
⎡−×
=I
Solve for and evaluate I:
2
2
20tot
mkg1.948
mkg1.177
mkg3.125
⋅=
⋅−
⋅=−= III
Rotation
673
Objects Rolling Without Slipping *79 •• Picture the Problem The forces acting on the yo-yo are shown in the figure. We can use a constant-acceleration equation to relate the velocity of descent at the end of the fall to the yo-yo’s acceleration and Newton’s 2nd law in both translational and rotational form to find the yo-yo’s acceleration.
Using a constant-acceleration equation, relate the yo-yo’s final speed to its acceleration and fall distance:
havv ∆+= 220
2
or, because v0 = 0, hav ∆= 2 (1)
Use Newton’s 2nd law to relate the forces that act on the yo-yo to its acceleration:
∑ =−= maTmgFx (2)
and ατ 00 ITr ==∑ (3)
Use αra = to eliminate α in equation (3) r
aITr 0= (4)
Eliminate T between equations (2) and (4) to obtain:
maarI
mg =− 20 (5)
Substitute 2
21 mR for I0 in equation
(5): maa
rmR
mg =− 2
221
Solve for a:
2
2
21
rR
ga+
=
Substitute numerical values and evaluate a: ( )
( )
2
2
2
2
m/s0.0864
m0.12m1.51
m/s9.81=
+=a
Substitute in equation (1) and evaluate v:
( )( )m/s3.14
m57m/s0.08642 2
=
=v
Chapter 9
674
80 •• Picture the Problem The diagram shows the forces acting on the cylinder. By applying Newton’s 2nd law of motion, we can obtain a system of two equations in the unknowns T, a, and α that we can solve simultaneously.
(a) Apply Newton’s 2nd law to the cylinder:
∑ == ατ 00 ITR (1)
and
∑ =−= MaTMgFx (2)
Substitute for α and I0 in equation (1) to obtain:
( ) ⎟⎠⎞
⎜⎝⎛=
RaMRTR 2
21
Solve for T:
MaT 21= (3)
Substitute for T in equation (2) and solve for a to obtain:
ga 32=
(b) Substitute for a in equation (3) to obtain:
( ) MggMT 31
32
21 ==
81 •• Picture the Problem The forces acting on the yo-yo are shown in the figure. Apply Newton’s 2nd law in both translational and rotational form to obtain simultaneous equations in T, a, and α from which we can eliminate α and solve for T and a.
Apply Newton’s 2nd law to the yo-yo: ∑ =−= maTmgFx (1)
and ατ 00 ITr ==∑ (2)
Use αra = to eliminate α in equation (2) r
aITr 0= (3)
Rotation
675
Eliminate T between equations (1) and (3) to obtain:
maarI
mg =− 20 (4)
Substitute 221 mR for I0 in equation
(4): maa
rmR
mg =− 2
221
Solve for a:
2
2
21
rR
ga+
=
Substitute numerical values and evaluate a: ( )
( )
2
2
2
2
m/s0.192
m0.012m1.01
m/s9.81=
+=a
Use equation (1) to solve for and evaluate T:
( )( )( )
N0.962
m/s0.192m/s9.81kg0.1 22
=
−=
−= agmT
*82 • Picture the Problem We can determine the kinetic energy of the cylinder that is due to its rotation about its center of mass by examining the ratio KK rot .
Express the rotational kinetic energy of the homogeneous solid cylinder:
( ) 241
2
22
21
212
cyl21
rot mvrvmrIK === ω
Express the total kinetic energy of the homogeneous solid cylinder:
2432
212
41
transrot mvmvmvKKK =+=+=
Express the ratio K
K rot : 31
243
241
rot ==mvmv
KK
and correct. is )(b
83 • Picture the Problem Any work done on the cylinder by a net force will change its kinetic energy. Therefore, the work needed to give the cylinder this motion is equal to its kinetic energy. Express the relationship between the work needed to stop the cylinder and its kinetic energy:
2212
21 ωImvKW +=∆=
Chapter 9
676
Because the cylinder is rolling without slipping, its translational and angular speeds are related according to:
ωrv =
Substitute for I (see Table 9-1) and ω and simplify to obtain:
( )2
43
2
22
21
212
21
2212
21
mvrvmrmv
ImvW
=
+=
+= ω
Substitute for m and v to obtain: ( )( ) kJ1.13m/s5kg60 2
43 ==W
84 • Picture the Problem The total kinetic energy of any object that is rolling without slipping is given by rottrans KKK += . We can find the percentages associated with each
motion by expressing the moment of inertia of the objects as kmr2 and deriving a general expression for the ratios of rotational kinetic energy to total kinetic energy and translational kinetic energy to total kinetic energy and substituting the appropriate values of k. Express the total kinetic energy associated with a rotating and translating object: ( )
( )kmvkmvmvrvkmrmv
ImvKKK
+=+=
+=
+=+=
12212
212
21
2
22
212
21
2212
21
rottrans ω
Express the ratio K
K rot : ( )
kk
kkmv
kmvK
K11
1112
21
221
rot
+=
+=
+=
Express the ratio K
K trans : ( ) kkmv
mvK
K+
=+
=1
112
21
221
trans
(a) Substitute k = 2/5 for a uniform sphere to obtain:
%6.28286.0
4.011
1rot ==+
=K
K
and
%4.71714.04.01
1trans ==+
=K
K
Rotation
677
(b) Substitute k = 1/2 for a uniform cylinder to obtain:
%3.33
5.011
1rot =+
=K
K
and
%7.665.01
1trans =+
=K
K
(c) Substitute k = 1 for a hoop to obtain: %0.50
111
1rot =+
=K
K
and
%0.5011
1trans =+
=K
K
85 • Picture the Problem Let the zero of gravitational potential energy be at the bottom of the incline. As the hoop rolls up the incline its translational and rotational kinetic energies are transformed into gravitational potential energy. We can use energy conservation to relate the distance the hoop rolls up the incline to its total kinetic energy at the bottom of the incline. Using energy conservation, relate the distance the hoop will roll up the incline to its kinetic energy at the bottom of the incline:
0=∆+∆ UK or, because Kf = Ui = 0,
0fi =+− UK (1)
Express Ki as the sum of the translational and rotational kinetic energies of the hoop:
2212
21
rottransi ωImvKKK +=+=
When a rolling object moves with speed v, its outer surface turns with a speed v also. Hence ω = v/r. Substitute for I and ω to obtain:
( ) 22
22
212
21
i mvrvmrmvK =+=
Letting ∆h be the change in elevation of the hoop as it rolls up the incline and ∆L the distance it rolls along the incline, express Uf:
θsinf LmghmgU ∆=∆=
Substitute in equation (1) to obtain:
0sin2 =∆+− θLmgmv
Chapter 9
678
Solve for ∆L: θsin
2
gvL =∆
Substitute numerical values and evaluate ∆L:
( )( ) m45.9
sin30m/s9.81m/s15
2
2
=°
=∆L
*86 •• Picture the Problem From Newton’s 2nd law, the acceleration of the center of mass equals the net force divided by the mass. The forces acting on the sphere are its weight
grm downward, the normal force nFr
that balances the normal component of the weight,
and the force of friction fr
acting up the incline. As the sphere accelerates down the incline, the angular velocity of rotation must increase to maintain the nonslip condition. We can apply Newton’s 2nd law for rotation about a horizontal axis through the center of mass of the sphere to find α, which is related to the acceleration by the nonslip condition. The only torque about the center of mass is due to f
rbecause both grm and nF
ract through
the center of mass. Choose the positive direction to be down the incline.
Apply aF rr
m=∑ to the sphere: cmsin mafmg =−θ (1)
Apply ατ cmI=∑ to the sphere: αcmIfr =
Use the nonslip condition to eliminate α and solve for f:
raIfr cm
cm=
and
cm2cm a
rIf =
Substitute this result for f in equation (1) to obtain:
cmcm2cmsin maa
rImg =−θ
From Table 9-1 we have, for a solid sphere:
252
cm mrI =
Rotation
679
Substitute in equation (1) and simplify to obtain:
cmcm52sin maamg =−θ
Solve for and evaluate θ :
( )°=⎥
⎦
⎤⎢⎣
⎡=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
−
−
3.165
2.07sin
57sin
1
cm1
gg
gaθ
87 •• Picture the Problem From Newton’s 2nd law, the acceleration of the center of mass equals the net force divided by the mass. The forces acting on the thin spherical shell are its weight grm downward, the normal force nF
rthat balances the normal component of the
weight, and the force of friction fr
acting up the incline. As the spherical shell accelerates down the incline, the angular velocity of rotation must increase to maintain the nonslip condition. We can apply Newton’s 2nd law for rotation about a horizontal axis through the center of mass of the sphere to find α, which is related to the acceleration by the nonslip condition. The only torque about the center of mass is due to f
rbecause both grm and
nFr
act through the center of mass. Choose the positive direction to be down the incline.
Apply aF rr
m=∑ to the thin spherical shell:
cmsin mafmg =−θ (1)
Apply ατ cmI=∑ to the thin spherical shell:
αcmIfr =
Use the nonslip condition to eliminate α and solve for f:
raIfr cm
cm= and cm2cm a
rIf =
Substitute this result for f in equation (1) to obtain:
cmcm2cmsin maa
rImg =−θ
From Table 9-1 we have, for a thin 232
cm mrI =
Chapter 9
680
spherical shell:
Substitute in equation (1) and simplify to obtain:
cmcm32sin maamg =−θ
Solve for and evaluate θ :
( )°==
=
−
−
5.193
2.05sin
35sin
1
cm1
gg
gaθ
Remarks: This larger angle makes sense, as the moment of inertia for a given mass is larger for a hollow sphere than for a solid one. 88 •• Picture the Problem The three forces acting on the basketball are the weight of the ball, the normal force, and the force of friction. Because the weight can be assumed to be acting at the center of mass, and the normal force acts through the center of mass, the only force which exerts a torque about the center of mass is the frictional force. We can use Newton’s 2nd law to find a system of simultaneous equations that we can solve for the quantities called for in the problem statement.
(a) Apply Newton’s 2nd law in both translational and rotational form to the ball:
∑ =−= mafmgFx ssinθ , (1)
∑ =−= 0cosn θmgFFy (2)
and
∑ == ατ 0s0 Irf (3)
Because the basketball is rolling without slipping we know that:
ra
=α
Substitute in equation (3) to obtain: r
aIrf 0s = (4)
From Table 9-1 we have:
232
0 mrI =
Substitute for I0 and α in equation (4) and solve for fs:
( ) maframrrf 3
2s
232
s =⇒= (5)
Rotation
681
Substitute for fs in equation (1) and solve for a:
θsin53 ga =
(b) Find fs using equation (5): ( ) θθ sinsin 5
253
32
s mggmf ==
(c) Solve equation (2) for Fn:
θcosn mgF =
Use the definition of fs,max to obtain:
maxsnsmaxs, cosθµµ mgFf ==
Use the result of part (b) to obtain: maxsmax52 cossin θµθ mgmg =
Solve for θmax: ( )s2
51max tan µθ −=
89 •• Picture the Problem The three forces acting on the cylinder are the weight of the cylinder, the normal force, and the force of friction. Because the weight can be assumed to be acting at the center of mass, and the normal force acts through the center of mass, the only force which exerts a torque about the center of mass is the frictional force. We can use Newton’s 2nd law to find a system of simultaneous equations that we can solve for the quantities called for in the problem statement.
(a) Apply Newton’s 2nd law in both translational and rotational form to the cylinder:
∑ =−= mafmgFx ssinθ , (1)
∑ =−= 0cosn θmgFFy (2)
and
∑ == ατ 0s0 Irf (3)
Because the cylinder is rolling without slipping we know that:
ra
=α
Substitute in equation (3) to obtain: r
aIrf 0s = (4)
From Table 9-1 we have:
221
0 mrI =
Chapter 9
682
Substitute for I0 and α in equation (4) and solve for fs:
( ) maframrrf 2
1s
221
s =⇒= (5)
Substitute for fs in equation (1) and solve for a:
θsin32 ga =
(b) Find fs using equation (5): ( ) θθ sinsin 3
132
21
s mggmf ==
(c) Solve equation (2) for Fn:
θcosn mgF =
Use the definition of fs,max to obtain:
maxsnsmaxs, cosθµµ mgFf ==
Use the result of part (b) to obtain: maxsmax31 cossin θµθ mgmg =
Solve for θmax: ( )s
1max 3tan µθ −=
*90 •• Picture the Problem Let the zero of gravitational potential energy be at the elevation where the spheres leave the ramp. The distances the spheres will travel are directly proportional to their speeds when they leave the ramp. Express the ratio of the distances traveled by the two spheres in terms of their speeds when they leave the ramp:
vv'
tvtv'
LL'
=∆∆
= (1)
Use conservation of mechanical energy to find the speed of the spheres when they leave the ramp:
0=∆+∆ UK or, because Ki = Uf = 0,
0if =−UK (2)
Express Kf for the spheres:
( )
( ) 221
2212
21
2
22
212
21
2cm2
1221
rottransf
1 mvk
kmvmvRvkmRmv
Imv
KKK
+=
+=
+=
+=
+=
ω
where k is 2/3 for the spherical shell and 2/5 for the uniform sphere.
Substitute in equation (2) to obtain: ( ) mgHmvk =+ 2211
Rotation
683
Solve for v:
kgH
v+
=12
Substitute in equation (1) to obtain:
09.111
11
5232
=++
=++
=k'k
LL'
or LL' 09.1=
91 •• Picture the Problem Let the subscripts u and h refer to the uniform and thin-walled spheres, respectively. Because the cylinders climb to the same height, their kinetic energies at the bottom of the incline must be equal. Express the total kinetic energy of the thin-walled cylinder at the bottom of the inclined plane: ( ) 2
h2
22
h212
h21
2h2
12h2
1rottransh
vmrvrmvm
IvmKKK
=+=
+=+= ω
Express the total kinetic energy of the solid cylinder at the bottom of the inclined plane: ( ) 2
u43
2
22
u21
212
u21
2u2
12u2
1rottransu '
v'mrv'rmv'm
IvmKKK
=+=
+=+= ω
Because the cylinders climb to the same height:
ghmvm
ghmv'm
h2
h
u2
u43
and=
=
Divide the first of these equations by the second: ghm
ghmvmv'm
h
u2
h
2u4
3=
Simplify to obtain:
143
2
2
=vv'
Solve for v′:
vv'34
=
Chapter 9
684
92 •• Picture the Problem Let the subscripts s and c refer to the solid sphere and thin-walled cylinder, respectively. Because the cylinder and sphere descend from the same height, their kinetic energies at the bottom of the incline must be equal. The force diagram shows the forces acting on the solid sphere. We’ll use Newton’s 2nd law to relate the accelerations to the angle of the incline and use a constant acceleration to relate the accelerations to the distances traveled down the incline.
Apply Newton’s 2nd law to the sphere:
sssin∑ =−= mafmgFx θ , (1)
∑ =−= 0cosn θmgFFy , (2)
and
∑ == ατ 0s0 Irf (3)
Substitute for I0 and α in equation (3) and solve for fs:
( ) s52
s2
52
s maframrrf =⇒=
Substitute for fs in equation (1) and solve for a:
θsin75
s ga =
Proceed as above for the thin-walled cylinder to obtain:
θsin21
c ga =
Using a constant-acceleration equation, relate the distance traveled down the incline to its acceleration and the elapsed time:
( )221
0 tatvs ∆+∆=∆
or, because v0 = 0, ( )2
21 tas ∆=∆ (4)
Because ∆s is the same for both objects: 2
cc2ss tata =
where ( ) 76.58.44.2 s
2s
2s
2c ++=+= tttt
provided tc and ts are in seconds.
Substitute for as and ac to obtain the quadratic equation:
2s7
10s
2s 76.58.4 ttt =++
Rotation
685
Solve for the positive root to obtain:
s3.12s =t
Substitute in equation (4), simplify, and solve for θ : ⎥
⎦
⎤⎢⎣
⎡ ∆= −
2s
1
514sin
gtsθ
Substitute numerical values and evaluate θ :
( )( )( )°=
⎥⎦
⎤⎢⎣
⎡= −
324.0
s12.3m/s9.815m314sin 22
1θ
93 ••• Picture the Problem The kinetic energy of the wheel is the sum of its translational and rotational kinetic energies. Because the wheel is a composite object, we can model its moment of inertia by treating the rim as a cylindrical shell and the spokes as rods. Express the kinetic energy of the wheel:
2
2
cm212
tot21
2cm2
12tot2
1
rottrans
RvIvM
IvM
KKK
+=
+=
+=
ω
where Mtot = Mrim + 4Mspoke
Express the moment of inertia of the wheel: ( )
( ) 2spoke3
4rim
2spoke3
12rim
spokesrimcm
4
RMM
RMRM
III
+=
+=
+=
Substitute for Icm in the equation for K:
( )[ ]( )[ ] 2
spoke32
rimtot21
2
22
spoke34
rim212
tot21
vMMMRvRMMvMK
++=
++=
Substitute numerical values and evaluate K:
( ) ( )[ ]( )J223
m/s6kg1.2kg3kg7.8 232
21
=
++=K
Chapter 9
686
94 ••• Picture the Problem Let M represent the combined mass of the two disks and their connecting rod and I their moment of inertia. The object’s initial potential energy is transformed into translational and rotational kinetic energy as it rolls down the incline. The force diagram shows the forces acting on this composite object as it rolls down the incline. Application of Newton’s 2nd law will allow us to derive an expression for the acceleration of the object.
(a) Apply Newton’s 2nd law to the disks and rod:
∑ =−= MafMgFx ssinθ , (1)
∑ =−= 0cosn θMgFFy , (2)
and
∑ == ατ Irf s0 (3)
Eliminate fs and α between equations (1) and (3) and solve for a to obtain: 2
sin
rIM
Mga+
=θ
(4)
Express the moment of inertia of the two disks plus connecting rod: ( )
2rod2
12disk
2rod2
12disk2
1
roddisk
2
2
rmRm
rmRm
III
+=
+=
+=
Substitute numerical values and evaluate I:
( )( ) ( )( )2
2212
mkg1.80
m0.02kg1m0.3kg20
⋅=
+=I
Substitute in equation (4) and evaluate a:
( )( )
( )2
2
2
2
m/s0.0443
m0.02mkg1.80kg41
sin30m/s9.81kg41
=
⋅+
°=a
(b) Find α from a: 2
2
rad/s2.21m0.02m/s0.0443
===raα
Rotation
687
(c) Express the kinetic energy of translation of the disks-plus-rod when it has rolled a distance ∆s down the incline:
221
trans MvK =
Using a constant-acceleration equation, relate the speed of the disks-plus-rod to their acceleration and the distance moved:
savv ∆+= 220
2
or, because v0 = 0, sav ∆= 22
Substitute to obtain: ( )( )( )
J3.63
m2m/s0.0443kg41 2trans
=
=
∆= sMaK
(d) Express the rotational kinetic energy of the disks after rolling 2 m in terms of their initial potential energy and their translational kinetic energy:
transtransirot KMghKUK −=−=
Substitute numerical values and evaluate Krot:
( )( )( )
J399
J3.63sin30m2m/s9.81kg41 2
rot
=
−°=K
95 ••• Picture the Problem We can express the coordinates of point P as the sum of the coordinates of the center of the wheel and the coordinates, relative to the center of the wheel, of the tip of the vector 0r
r. Differentiation of these expressions with respect to time
will give us the x and y components of the velocity of point P. (a) Express the coordinates of point P relative to the center of the wheel:
θ
θ
sinand
cos
0
0
ry
rx
=
=
Because the coordinates of the center of the circle are X and R:
( ) ( )θθ sin,cos, 00 rRrXyx PP ++=
Chapter 9
688
(b) Differentiate xP to obtain: ( )
dtdr
dtdX
rXdtdvPx
θθ
θ
⋅−=
+=
sin
cos
0
0
Note that
RV
dtdV
dtdX
−=−== ωθand so: θsin0
RVr
VvPx +=
Differentiate yP to obtain: ( )
dtdrrR
dtdvPy
θθθ ⋅=+= cossin 00
BecauseRV
dtd
−=−= ωθ: θcos0
RVrvPy −=
(c) Calculate rv rr
⋅ :
( )
( )
0
sincos
cossin
00
00
=
+⎟⎠⎞
⎜⎝⎛−
⎟⎠⎞
⎜⎝⎛ +=
+=⋅
θθ
θθ
rRRVr
rRVrV
rvrv yPyxPxrv rr
(d) Express v in terms of its components:
2
200
20
20
22
sin21
cossin
Rr
Rr
V
RVr
RVr
V
vvv yx
++=
⎟⎠⎞
⎜⎝⎛−+⎟
⎠⎞
⎜⎝⎛ +=
+=
θ
θθ
Express r in terms of its components:
( ) ( )
2
200
20
20
22
sin21
sincos
Rr
Rr
R
rRr
rrr yx
++=
++=
+=
θ
θθ
Divide v by r to obtain:
RV
rv
==ω
Rotation
689
*96 ••• Picture the Problem Let the letter B identify the block and the letter C the cylinder. We can find the accelerations of the block and cylinder by applying Newton’s 2nd law and solving the resulting equations simultaneously. Apply xx maF =∑ to the block: B' mafF =− (1)
Apply xx maF =∑ to the cylinder: CMaf = , (2)
Apply ατ CMCM I=∑ to the cylinder:
αCMIfR = (3)
Substitute for ICM in equation (3) and solve for f = f ′ to obtain:
αMRf 21= (4)
Relate the acceleration of the block to the acceleration of the cylinder:
CBBC aaa +=
or, because aCB = −Rα is the acceleration of the cylinder relative to the block,
αRaa −= BC
and CB aaR −=α (5)
Equate equations (2) and (4) and substitute from (5) to obtain:
CB 3aa =
Substitute equation (4) in equation (1) and substitute for aC to obtain:
BB31 maMaF =−
Solve for aB: mM
Fa3
3B +
=
97 ••• Picture the Problem Let the letter B identify the block and the letter C the cylinder. In this problem, as in Problem 97, we can find the accelerations of the block and cylinder by applying Newton’s 2nd law and solving the resulting equations simultaneously.
Chapter 9
690
Apply xx maF =∑ to the block: BmafF =− (1)
Apply xx maF =∑ to the cylinder: CMaf = , (2)
Apply ατ CMCM I=∑ to the cylinder:
αCMIfR = (3)
Substitute for ICM in equation (3) and solve for f:
αMRf 21= (4)
Relate the acceleration of the block to the acceleration of the cylinder:
CBBC aaa +=
or, because aCB = −Rα, αRaa −= BC
and CB aaR −=α (5)
(a) Solve for α and substitute for aB to obtain:
( )mMRF
Ra
Raa
Raa
32
23 CCCCB
+=
=−
=−
=α
direction. ckwisecounterclo thein is , thereforeand, torquethat the
evident isit diagram force theFromα
(b) Equate equations (2) and (4) and substitute (5) to obtain:
CB 3aa =
From equations (1) and (4) we obtain:
BB31 maMaF =−
Solve for aB: mM
Fa3
3B +
=
Substitute to obtain the linear acceleration of the cylinder relative to the table:
mMFaa B 33
1C +
==
Rotation
691
(c) Express the acceleration of the cylinder relative to the block:
mMF
aaaaaa
32
23 CCCBCCB
+−=
−=−=−=
98 ••• Picture the Problem Let the system include the earth, the cylinder, and the block. Then F
ris an external force that
changes the energy of the system by doing work on it. We can find the kinetic energy of the block from its speed when it has traveled a distance d. We can find the kinetic energy of the cylinder from the sum of its translational and rotational kinetic energies. In part (c) we can add the kinetic energies of the block and the cylinder to show that their sum is the work done by Fr
in displacing the system a distance d.
(a) Express the kinetic energy of the block: 2
B21
blockonB mvWK ==
Using a constant-acceleration equation, relate the velocity of the block to its acceleration and the distance traveled:
davv B20
2B 2+=
or, because the block starts from rest, dav B
2B 2=
Substitute to obtain:
( ) dmadamK BB21
B 2 == (1)
Apply xx maF =∑ to the block: BmafF =− (2)
Apply xx maF =∑ to the cylinder: CMaf = , (3)
Apply ατ CMCM I=∑ to the
cylinder:
αCMIfR = (4)
Substitute for ICM in equation (4) and solve for f:
αMRf 21= (5)
Relate the acceleration of the block to the acceleration of the cylinder:
CBBC aaa +=
or, because aCB = −Rα,
Chapter 9
692
αRaa −= BC
and CB aaR −=α (6)
Equate equations (3) and (5) and substitute in (6) to obtain:
CB 3aa =
Substitute equation (5) in equation (2) and use CB 3aa = to obtain:
BC maMaF =−
or BB3
1 maMaF =−
Solve for aB:
MmFa
31B +
=
Substitute in equation (1) to obtain: Mm
mFdK31B +
=
(b) Express the total kinetic energy of the cylinder:
2
2CB
CM212
C21
2CM2
12C2
1rottranscyl
RvIMv
IMvKKK
+=
+=+= ω(7)
where BCCB vvv −= .
In part (a) it was established that: CB 3aa =
Integrate both sides of the equation with respect to time to obtain:
constant3 CB += vv
where the constant of integration is determined by the initial conditions that vC = 0 when vB = 0.
Substitute the initial conditions to obtain: 0constant = and
CB 3vv =
Substitute in our expression for vCB to obtain:
CCCBCCB 23 vvvvvv −=−=−=
Substitute for ICM and vCB in equation (7) to obtain: ( )( )
2C2
3
2
2C2
21
212
C21
cyl2
MvRvMRMvK
=
−+=
(8)
Rotation
693
Because B31
C vv = : 2B9
12C vv =
It part (a) it was established that: dav B
2B 2=
and
MmFa
31B +
=
Substitute to obtain: ( )
( )MmFd
dMm
Fdav
31
319
2B9
12C
92
2
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
==
Substitute in equation (8) to obtain:
( )
( )MmMFd
MmFdMK
31
312
3cyl
3
92
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
(c) Express the total kinetic energy of the system and simplify to obtain:
( )( )( ) FdFd
MmMm
MmMFd
MmmFd
KKK
=++
=
++
+=
+=
31
31
31
cylBtot
33
3
99 •• Picture the Problem The forces responsible for the rotation of the gears are shown in the diagram to the right. The forces acting through the centers of mass of the two gears have been omitted because they produce no torque. We can apply Newton’s 2nd law in rotational form to obtain the equations of motion of the gears and the not slipping condition to relate their angular accelerations.
(a) Apply ατ I=∑ to the gears to obtain their equations of motion:
111mN 2 αIFR =−⋅ (1) and
222 αIFR = (2) where F is the force keeping the gears from slipping with respect to each other.
Because the gears do not slip 2211 αα RR =
Chapter 9
694
relative to each other, the tangential accelerations of the points where they are in contact must be the same:
or
121
12
12 ααα ==
RR
(3)
Divide equation (1) by R1 to obtain:
11
1
1
mN 2 αRIF
R=−
⋅
Divide equation (2) by R2 to obtain:
22
2 αRIF =
Add these equations to obtain:
22
21
1
1
1
mN 2 ααRI
RI
R+=
⋅
Use equation (3) to eliminate α2:
12
21
1
1
1 2mN 2 αα
RI
RI
R+=
⋅
Solve for α1 to obtain:
22
11
1
2
mN2
IRRI +
⋅=α
Substitute numerical values and evaluate α1:
( ) ( )2
221
rad/s400.0
mkg16m12m0.5mkg1
mN2
=
⋅+⋅
⋅=α
Use equation (3) to evaluate α2: ( ) 22
21
2 rad/s0.200rad/s0.400 ==α
(b) To counterbalance the 2-N·m torque, a counter torque of 2 N·m must be applied to the first gear. Use equation (2) with α1 = 0 to find F:
0mN 2 1 =−⋅ FR and
N4.00m0.5mN2mN2
1
=⋅
=⋅
=R
F
Rotation
695
*100 •• Picture the Problem Let r be the radius of the marble, m its mass, R the radius of the large sphere, and v the speed of the marble when it breaks contact with the sphere. The numeral 1 denotes the initial configuration of the sphere-marble system and the numeral 2 is configuration as the marble separates from the sphere. We can use conservation of energy to relate the initial potential energy of the marble to the sum of its translational and rotational kinetic energies as it leaves the sphere. Our choice of the zero of potential energy is shown on the diagram.
(a) Apply conservation of energy:
0=∆+∆ KU or
01212 =−+− KKUU
Because U2 = K1 = 0: ( )[ ]0
cos2
212
21 =++
+−+−
ω
θ
ImvrRrRmg
or ( )( )[ ]
0cos1
2212
21 =++
−+−
ω
θ
ImvrRmg
Use the rolling-without-slipping condition to eliminate ω:
( )( )[ ]
0
cos1
2
2
212
21 =++
−+−
rvImv
rRmg θ
From Table 9-1 we have: 2
52 mrI =
Substitute to obtain: ( )( )[ ]
( ) 0
cos1
2
22
52
212
21 =++
−+−
rvmrmv
rRmg θ
or ( )( )[ ]
0cos1
2512
21 =++
−+−
mvmvrRmg θ
Solve for v2 to obtain: ( )( )θcos1
7102 −+= rRgv
Apply rr maF =∑ to the marble as it separates from the sphere:
rRvmmg+
=2
cosθ
or
Chapter 9
696
( )rRgv
+=
2
cosθ
Substitute for v2:
( ) ( )( )
( )⎥⎦⎤
⎢⎣⎡ −=
⎥⎦⎤
⎢⎣⎡ −+
+=
θ
θθ
cos17
10
cos17
101cos rRgrRg
Solve for and evaluate θ :
°=⎟⎠⎞
⎜⎝⎛= − 0.541710cos 1θ
(b)
sphere. theleavesit before slippingwithout rolling ball thekeep toneeded force than theless bemust friction
of force that themeaning sphere, theleaves ball theepoint wher at the 0 todecreases force normal theHowever, marble. on the force
normal by the multiplied than less always isfriction of force The sµ
Rolling With Slipping 101 • Picture the Problem Part (a) of this problem is identical to Example 9-16. In part (b) we can use the definitions of translational and rotational kinetic energy to find the ratio of the final and initial kinetic energies. (a) From Example 9-16:
gv
sk
20
1 4912
µ= ,
gv
tk
01 7
2µ
= , and
01k25
1 75 vgtv == µ
(b) When the ball rolls without slipping, v1 = rω. Express the final kinetic energy of the ball:
( )2014
52110
7
2
212
52
212
121
2212
121
rottransf
MvMvrvMrMv
IMv
KKK
==
+=
+=
+=
ω
Rotation
697
Express the ratio of the final and initial kinetic energies: 7
5202
1
2014
5
i
f ==MvMv
KK
(c) Substitute in the expressions in (a) to obtain:
( )( )( ) m6.26
m/s9.810.06m/s8
4912
2
2
1 ==s
( )( ) s3.88m/s9.810.06
m/s872
21 ==t
( ) m/s5.71m/s8
75
1 ==v
*102 •• Picture the Problem The cue stick’s blow delivers a rotational impulse as well as a translational impulse to the cue ball. The rotational impulse changes the angular momentum of the ball and the translational impulse changes its linear momentum. Express the rotational impulse Prot as the product of the average torque and the time during which the rotational impulse acts:
tP ∆= avrot τ
Express the average torque it produces about an axis through the center of the ball:
( ) ( )rhPrhP −=−= 00av sinθτ
where θ (= 90°) is the angle between F and the lever arm h − r.
Substitute in the expression for Prot to obtain:
( ) ( )( )( ) 0trams
00rot
ωILrhPrhtPtrhPP
=∆=−=−∆=∆−=
The translational impulse is also given by:
00trans mvptPP =∆=∆=
Substitute to obtain: ( ) 02
52
0 ωmrrhmv =−
Solve for ω0: ( )
20
0 25
rrhv −
=ω
Chapter 9
698
103 •• Picture the Problem The angular velocity of the rotating sphere will decrease until the condition for rolling without slipping is satisfied and then it will begin to roll. The force diagram shows the forces acting on the sphere. We can apply Newton’s 2nd law to the sphere and use the condition for rolling without slipping to find the speed of the center of mass when the sphere begins to roll without slipping. Relate the velocity of the sphere when it begins to roll to its acceleration and the elapsed time:
tav ∆= (1)
Apply Newton’s 2nd law to the sphere:
∑ == mafFx k , (2)
∑ =−= 0n mgFFy , (3)
and
∑ == ατ 0k0 Irf (4)
Using the definition of fk and Fn from equation (3), substitute in equation (2) and solve for a:
ga kµ=
Substitute in equation (1) to obtain: tgtav ∆=∆= kµ (5)
Solve for α in equation (4):
rg
mrmar
Irf k
252
0
k
25 µα ===
Express the angular speed of the sphere when it has been moving for a time ∆t:
trgt ∆−=∆−=
25 k
00µωαωω (6)
Express the condition that the sphere rolls without slipping:
ωrv =
Substitute from equations (5) and (6) and solve for the elapsed time until the sphere begins to roll:
gr
tk
0
72
µω
=∆
Rotation
699
Use equation (4) to find v when the sphere begins to roll: 7
272 0
k
k0k
ωµ
µωµ rg
grtgv ==∆=
104 •• Picture the Problem The sharp force delivers a rotational impulse as well as a translational impulse to the ball. The rotational impulse changes the angular momentum of the ball and the translational impulse changes its linear momentum. In parts (c) and (d) we can apply Newton’s 2nd law to the ball to obtain equations describing both the translational and rotational motion of the ball. We can then solve these equations to find the constant accelerations that allow us to apply constant-acceleration equations to find the velocity of the ball when it begins to roll and its sliding time.
(a) Relate the translational impulse delivered to the ball to its change in its momentum:
0avtrans mvptFP =∆=∆=
Solve for v0: m
tFv ∆= av
0
Substitute numerical values and evaluate v0:
( )( ) m/s200kg0.02
s102kN20 4
0 =×
=−
v
(b) Express the rotational impulse Prot as the product of the average torque and the time during which the rotational impulse acts:
tP ∆= avrot τ
Letting h be the height at which the impulsive force is delivered, express the average torque it produces about an axis through the center of the ball:
θτ sinav lF=
where θ is the angle between F and the lever arm l .
Substitute h − r for l and 90° for θ ( )rhF −=avτ
Chapter 9
700
to obtain: Substitute in the expression for Prot to obtain:
( ) trhFP ∆−=rot
Because Ptrans = F∆t:
( )0
252
0transrot
ω
ω
mr
ILrhPP
=
=∆=−=
Express the translational impulse delivered to the cue ball:
00trans mvptPP =∆=∆=
Substitute for Ptrans to obtain:
002
52 mvmr =ω
Solve for ω0: ( )2
00 2
5r
rhv −=ω
Substitute numerical values and evaluate ω0:
( )( )( )
rad/s8000
m.052m0.05m0.09m/s2005
20
=
−=ω
(c) and (d) Relate the velocity of the ball when it begins to roll to its acceleration and the elapsed time:
tav ∆= (1)
Apply Newton’s 2nd law to the ball: ∑ == mafFx k , (2)
∑ =−= 0n mgFFy , (3)
and
∑ == ατ 0k0 Irf (4)
Using the definition of fk and Fn from equation (3), substitute in equation (2) and solve for a:
ga kµ=
Substitute in equation (1) to obtain: tgtav ∆=∆= kµ (5)
Solve for α in equation (4):
rg
mrmar
Irf k
252
0
k
25 µα ===
Rotation
701
Express the angular speed of the ball when it has been moving for a time ∆t:
trgt ∆−=∆−=
25 k
00µωαωω (6)
Express the speed of the ball when it has been moving for a time ∆t:
tgvv ∆+= k0 µ (7)
Express the condition that the ball rolls without slipping:
ωrv =
Substitute from equations (6) and (7) and solve for the elapsed time until the ball begins to roll:
gvrt
k
00
72
µω −
=∆
Substitute numerical values and evaluate ∆t:
( )( )( )( )
s11.6
m/s9.810.5m/s200rad/s8000m0.05
72
2
=
⎥⎦
⎤⎢⎣
⎡ −=∆t
Use equation (4) to express v when the ball begins to roll:
tgvv ∆+= k0 µ
Substitute numerical values and evaluate v:
( )( )( )m/s572
s11.6m/s9.810.5m/s200 2
=
+=v
105 •• Picture the Problem Because the impulse is applied through the center of mass, ω0 = 0. We can use the results of Example 9-16 to find the rolling time without slipping, the distance traveled to rolling without slipping, and the velocity of the ball once it begins to roll without slipping. (a) From Example 9-16 we have:
gvtk
01 7
2µ
=
Substitute numerical values and evaluate t1:
( )( ) s0.194m/s9.810.6
m/s472
21 ==t
(b) From Example 9-16 we have: g
vsk
20
1 4912
µ=
Chapter 9
702
Substitute numerical values and evaluate s1:
( )( )( ) m0.666
m/s9.810.6m/s4
4912
2
2
1 ==s
(c) From Example 9-16 we have: 01 7
5 vv =
Substitute numerical values and evaluate v1:
( ) m/s2.86m/s475
1 ==v
106 •• Picture the Problem Because the impulsive force is applied below the center line, the spin is backward, i.e., the ball will slow down. We’ll use the impulse-momentum theorem and Newton’s 2nd law to find the linear and rotational velocities and accelerations of the ball and constant-acceleration equations to relate these quantities to each other and to the elapsed time to rolling without slipping. (a) Express the rotational impulse delivered to the ball:
( ) 02
52
0cm00rot 32
ω
ω
mR
IRmvrmvP
=
===
Solve for ω0:
Rv0
0 35
=ω
(b) Apply Newton’s 2nd law to the ball to obtain:
∑ == ατ cmk0 IRf , (1)
∑ =−= 0n mgFFy , (2)
and
∑ =−= mafFx k (3)
Using the definition of fk and Fn
from equation (2), solve for α: Rg
mRmgR
ImgR
25 k
252
k
cm
k µµµα ===
Using a constant-acceleration equation, relate the angular speed of the ball to its acceleration:
tR
gt ∆+=∆+=2
5 k00
µωαωω
Rotation
703
Using the definition of fk and Fn
from equation (2), solve equation (3) for a:
ga kµ−=
Using a constant-acceleration equation, relate the speed of the ball to its acceleration:
tgvtavv ∆−=∆+= k00 µ (4)
Impose the condition for rolling without slipping to obtain:
tgvtR
gR ∆−=⎟⎠⎞
⎜⎝⎛ ∆+ k0
k0 2
5µ
µω
Solve for ∆t:
gv
tk
0
2116
µ=∆
Substitute in equation (4) to obtain:
0
0k
0k0
238.0
215
2116
v
vgvgvv
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛−=
µµ
(c) Express the initial kinetic energy of the ball:
( )20
20
202
52
212
021
202
1202
1rottransi
056.1
1819
35
mv
mvRvmRmv
ImvKKK
=
=⎟⎠⎞
⎜⎝⎛+=
+=+= ω
(d) Express the work done by friction in terms of the initial and final kinetic energies of the ball:
fifr KKW −=
Express the final kinetic energy of the ball:
( )( ) 2
02
0107
2107
2
22
52
212
21
2cm2
1221
f
0397.0238.0 mvvm
mvRvmRmv
ImvK
==
=+=
+= ω
Substitute to find Wfr:
20
20
20fr
016.1
0397.0056.1
mv
mvmvW
=
−=
Chapter 9
704
107 •• Picture the Problem The figure shows the forces acting on the bowling during the sliding phase of its motion. Because the ball has a forward spin, the friction force is in the direction of motion and will cause the ball’s translational speed to increase. We’ll apply Newton’s 2nd law to find the linear and rotational velocities and accelerations of the ball and constant-acceleration equations to relate these quantities to each other and to the elapsed time to rolling without slipping.
(a) and (b) Relate the velocity of the ball when it begins to roll to its acceleration and the elapsed time:
tavv ∆+= 0 (1)
Apply Newton’s 2nd law to the ball: ∑ == mafFx k , (2)
∑ =−= 0n mgFFy , (3)
and
∑ == ατ 0k0 IRf (4)
Using the definition of fk and Fn from equation (3), substitute in equation (2) and solve for a:
ga kµ=
Substitute in equation (1) to obtain: tgvtavv ∆+=∆+= k00 µ (5)
Solve for α in equation (4):
Rg
mRmaR
IRf k
252
0
k
25 µα ===
Relate the angular speed of the ball to its acceleration:
tR
g∆−= k
0 25 µ
ωω
Apply the condition for rolling without slipping:
⎟⎠⎞
⎜⎝⎛ ∆−=
⎟⎠⎞
⎜⎝⎛ ∆−==
tR
gRvR
tR
gRRv
k0
k0
253
25
µ
µωω
Rotation
705
∴ tgvv ∆−= k0 253 µ (6)
Equate equations (5) and (6) and solve ∆t: g
vt
k
0
74
µ=∆
Substitute for ∆t in equation (6) to obtain: 00 57.1
711 vvv ==
(c) Relate ∆x to the average speed of the ball and the time it moves before beginning to roll without slipping:
( )
gv
gv
gvvv
tvvtvx
k
20
k
20
k
0002
1
021
av
735.04936
74
711
µµ
µ
==
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛ +=
∆+=∆=∆
*108 •• Picture the Problem The figure shows the forces acting on the cylinder during the sliding phase of its motion. The friction force will cause the cylinder’s translational speed to decrease and eventually satisfy the condition for rolling without slipping. We’ll use Newton’s 2nd law to find the linear and rotational velocities and accelerations of the ball and constant-acceleration equations to relate these quantities to each other and to the distance traveled and the elapsed time until the satisfaction of the condition for rolling without slipping.
(a) Apply Newton’s 2nd law to the cylinder:
∑ =−= MafFx k , (1)
∑ =−= 0n MgFFy , (2)
and
∑ == ατ 0k0 IRf (3)
Use fk = µkFn to eliminate Fn between equations (1) and (2) and solve for a:
ga kµ−=
Chapter 9
706
Using a constant-acceleration equation, relate the speed of the cylinder to its acceleration and the elapsed time:
tgvtavv ∆−=∆+= k00 µ
Similarly, eliminate fk between equations (2) and (3) and solve for α:
Rgk2µ
α =
Using a constant-acceleration equation, relate the angular speed of the cylinder to its acceleration and the elapsed time:
tR
gt ∆=∆+= k0
2µαωω
Apply the condition for rolling without slipping:
tg
tR
gRRtgvv
∆=
⎟⎠⎞
⎜⎝⎛ ∆==∆−=
k
kk0
2
2
µ
µωµ
Solve for ∆t:
gv
tk
0
3µ=∆
Substitute for ∆t in the expression for v: 0
k
0k0 3
23
vg
vgvv =−=µ
µ
(b) Relate the distance the cylinder travels to its average speed and the elapsed time:
( )
gv
gvvvtvx
k
20
k
003
202
1av
185
3
µ
µ
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=∆=∆
(c) Express the ratio of the energy dissipated in friction to the cylinder’s initial mechanical energy:
i
fi
i
fr
KKK
KW −
=
Express the kinetic energy of the cylinder as it begins to roll without slipping: ( )
20
2
02
2
22
21
212
21
2cm2
1221
f
31
32
43
43 MvvMMv
RvMRMv
IMvK
=⎟⎠⎞
⎜⎝⎛==
+=
+= ω
Rotation
707
Substitute for Ki and Kf and simplify to obtain: 3
1202
1
203
1202
1
i
fr =−
=Mv
MvMvKW
109 •• Picture the Problem The forces acting on the ball as it slides across the floor are its weight ,mg
r the normal force nF
rexerted by
the floor, and the friction force .fv
Because the weight and normal force act through the center of mass of the ball and are equal in magnitude, the friction force is the net (decelerating) force. We can apply Newton’s 2nd law in both translational and rotational form to obtain a set of equations that we can solve for the acceleration of the ball. Once we have determined the ball’s acceleration, we can use constant-acceleration equations to obtain its velocity when it begins to roll without slipping.
(a) Apply aF rr
m=∑ to the ball: ∑ =−= mafFx (1) and
∑ =−= 0n mgFFy (2)
From the definition of the coefficient of kinetic friction we have:
nk Ff µ= (3)
Solve equation (2) for Fn: mgF =n
Substitute in equation (3) to obtain: mgf kµ=
Substitute in equation (1) to obtain: mamg =− kµ or
ga kµ−=
Apply ατ I=∑ to the ball: αIfr =
Solve for α to obtain: Imgr
Ifr kµα ==
Assuming that the coefficient of kinetic friction is constant*, we can use constant-acceleration equations to describe how long it will take the ball to begin
tgtavv k ∆−=∆=− µf (4) and
tIgmrk ∆=
µωf (5)
Chapter 9
708
rolling without slipping:
Once rolling without slipping has been established, we also have: r
vff =ω (6)
Equate equations (5) and (6):
tIgmr
rv k ∆=
µf
Solve for ∆t:
2f
gmrIvt
kµ=∆
Substitute in equation (4) to obtain:
f2
2f
f
vmr
Igmr
Ivgvvk
k
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=−
µµ
Solve for vf:
v
mrIv f
21
1
+=
(b) Express the total kinetic energy of the ball:
2f
2f 2
121 ωImvK +=
Because the ball is now rolling without slipping, fωrv = and:
( )
⎟⎠⎞
⎜⎝⎛
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
++=⎟
⎠⎞
⎜⎝⎛
++⎟
⎠⎞
⎜⎝⎛
+=
22
2
222
2
22
22
2
2
/11
21
/11/1
21
/11
21
/11
21
mrImv
mrImrImv
rv
mrIIv
mrImK
* Remarks: This assumption is not necessary. One can use the impulse-momentum theorem and the related theorem for torque and change in angular momentum to prove that the result holds for an arbitrary frictional force acting on the ball, so long as the ball moves along a straight line and the force is directed opposite to the direction of motion of the ball. General Problems *110 • Picture the Problem The angular velocity of an object is the ratio of the number of revolutions it makes in a given period of time to the elapsed time.
Rotation
709
The moon’s angular velocity is:
rad/s102.66
s3600h1
h24day1
revrad2
days27.3rev1days27.3
rev1
6−×=
×××=
=
π
ω
111 • Picture the Problem The moment of inertia of the hoop, about an axis perpendicular to the plane of the hoop and through its edge, is related to its moment of inertia with respect to an axis through its center of mass by the parallel axis theorem. Apply the parallel axis theorem: 2222
cm 2mRMRMRMhII =+=+=
112 •• Picture the Problem The force you exert on the rope results in a net torque that accelerates the merry-go-round. The moment of inertia of the merry-go-round, its angular acceleration, and the torque you apply are related through Newton’s 2nd law. (a) Using a constant-acceleration equation, relate the angular displacement of the merry-go-round to its angular acceleration and acceleration time:
( )221
0 tt ∆+∆=∆ αωθ
or, because ω0 = 0, ( )2
21 t∆=∆ αθ
Solve for and evaluate α: ( )
( )( )
222 rad/s0873.0
s12rad222
==∆∆
=πθα
t
(b) Use the definition of torque to obtain: ( )( ) mN572m2.2N260 ⋅=== Frτ
(c) Use Newton’s 2nd law to find the moment of inertia of the merry-go-round: 23
2net
mkg106.55
rad/s0.0873mN572
⋅×=
⋅==
ατI
Chapter 9
710
113 • Picture the Problem Because there are no horizontal forces acting on the stick, the center of mass of the stick will not move in the horizontal direction. Choose a coordinate system in which the origin is at the horizontal position of the center of mass. The diagram shows the stick in its initial raised position and when it has fallen to the ice. Express the displacement of the right end of the stick ∆x as the difference between the position coordinates x2 and x2:
12 xxx −=∆
Using trigonometry, find the initial coordinate of the right end of the stick:
( ) m0.866cos30m1cos1 =°== θlx
Because the center of mass has not moved horizontally:
m12 == lx
Substitute to find the displacement of the right end of the stick:
m0.134m0.866m1 =−=∆x
114 •• Picture the Problem The force applied to the string results in a torque about the center of mass of the disk that accelerates it. We can relate these quantities to the moment of inertia of the disk through Newton’s 2nd law and then use constant-acceleration equations to find the disk’s angular velocity the angle through which it has rotated in a given period of time. The disk’s rotational kinetic energy can be found from its definition. (a) Use the definition of torque to obtain:
( )( ) mN2.40m0.12N20 ⋅==≡ FRτ
(b) Use Newton’s 2nd law to express the angular acceleration of the disk in terms of the net torque acting on it and its moment of inertia:
221
netnet
MRIττα ==
Substitute numerical values and evaluate α:
( )( )( )
22 rad/s66.7
m0.12kg5mN2.402
=⋅
=α
(c) Using a constant-acceleration equation, relate the angular velocity of the disk to its angular
t∆+= αωω 0
or, because ω0 = 0, t∆= αω
Rotation
711
acceleration and the elapsed time:
Substitute numerical values and evaluate ω:
( )( ) rad/s333s5rad/s66.7 2 ==ω
(d) Use the definition of rotational kinetic energy to obtain:
( ) 2221
212
21
rot ωω MRIK ==
Substitute numerical values and evaluate Krot:
( )( ) ( )kJ2.00
rad/s333m0.12kg5 2241
rot
=
=K
(e) Using a constant-acceleration equation, relate the angle through which the disk turns to its angular acceleration and the elapsed time:
( )221
0 tt ∆+∆=∆ αωθ
or, because ω0 = 0, ( )2
21 t∆=∆ αθ
Substitute numerical values and evaluate ∆θ :
( )( ) rad834s5rad/s66.7 2221 ==∆θ
(f) Express K in terms of τ and θ : ( ) ( )
θτ
αταατω
∆=
∆=∆⎟⎠⎞
⎜⎝⎛== 2
212
212
21 ttIK
115 •• Picture the Problem The diagram shows the rod in its initial horizontal position and then, later, as it swings through its vertical position. The center of mass is denoted by the numerals 0 and 1. Let the length of the rod be represented by L and its mass by m. We can use Newton’s 2nd law in rotational form to find, first, the angular acceleration of the rod and then, from α, the acceleration of any point on the rod. We can use conservation of energy to find the angular velocity of the center of mass of the rod when it is vertical and then use this value to find its linear velocity.
(a) Relate the acceleration of the center of the rod to the angular
αα2La == l
Chapter 9
712
acceleration of the rod: Use Newton’s 2nd law to relate the torque about the suspension point of the rod (exerted by the weight of the rod) to the rod’s angular acceleration:
Lg
ML
LMg
I 232
231
===τα
Substitute numerical values and evaluate α:
( )( )
22
rad/s18.4m0.82
m/s9.813==α
Substitute numerical values and evaluate a:
( )( ) 2221 m/s7.36rad/s18.4m0.8 ==a
(b) Relate the acceleration of the end of the rod to α:
( )( )2
2end
m/s14.7
rad/s18.4m0.8
=
== αLa
(c) Relate the linear velocity of the center of mass of the rod to its angular velocity as it passes through the vertical:
Lhv ωω 21=∆=
Use conservation of energy to relate the changes in the kinetic and potential energies of the rod as it swings from its initial horizontal orientation through its vertical orientation:
00101 =−+−=∆+∆ UUKKUK
or, because K0 = U1 = 0, 001 =−UK
Substitute to obtain:
hmgI P ∆=221 ω
Substitute for ∆h and solve for ω:
Lg3
=ω
Substitute to obtain: gL
LgLv 33
21
21 ==
Substitute numerical values and evaluate v: ( )( ) m/s2.43m0.8m/s9.813 2
21 ==v
Rotation
713
116 •• Picture the Problem Let the zero of gravitational potential energy be at the bottom of the track. The initial potential energy of the marble is transformed into translational and rotational kinetic energy as it rolls down the track to its lowest point and then, because the portion of the track to the right is frictionless, into translational kinetic energy and, eventually, into gravitational potential energy. Using conservation of energy, relate h2 to the kinetic energy of the marble at the bottom of the track:
0=∆+∆ UK or, because Kf = Ui = 0,
0fi =+− UK
Substitute for Ki and Uf to obtain: 022
21 =−− MghMv
Solve for h2:
gvh2
2
2 = (1)
Using conservation of energy, relate h1 to the kinetic energy of the marble at the bottom of the track:
0=∆+∆ UK or, because Ki = Uf = 0,
0if =−UK
Substitute for Kf and Ui to obtain: 01
2212
21 =−+ MghIMv ω
Substitute for I and solve for v2 to obtain:
17102 ghv =
Substitute in equation (1) to obtain:
17517
10
2 2h
ggh
h ==
*117 •• Picture the Problem To stop the wheel, the tangential force will have to do an amount of work equal to the initial rotational kinetic energy of the wheel. We can find the stopping torque and the force from the average power delivered by the force during the slowing of the wheel. The number of revolutions made by the wheel as it stops can be found from a constant-acceleration equation. (a) Relate the work that must be done to stop the wheel to its kinetic energy:
( ) 224122
21
212
21 ωωω mrmrIW ===
Chapter 9
714
Substitute numerical values and evaluate W:
( )( )
kJ780
s60min1
revrad2
minrev1100
m1.4kg1202
241
=
⎥⎦
⎤⎢⎣
⎡×××
=
π
W
(b) Express the stopping torque is terms of the average power required:
avav τω=P
Solve for τ :
av
av
ωτ P
=
Substitute numerical values and evaluate τ : ( )( )
( )( )( )
mN3.902
smin/601rad/rev2rev/min1100s/min60min2.5
kJ780
⋅=
= πτ
Relate the stopping torque to the magnitude of the required force and solve for F:
N151m0.6
mN90.3=
⋅==
RF τ
(c) Using a constant-acceleration equation, relate the angular displacement of the wheel to its average angular velocity and the stopping time:
t∆=∆ avωθ
Substitute numerical values and evaluate ∆θ:
( )
rev1380
min2.52
rev/min1100
=
⎟⎠⎞
⎜⎝⎛=∆θ
118 •• Picture the Problem The work done by the four children on the merry-go-round will change its kinetic energy. We can use the work-energy theorem to relate the work done by the children to the distance they ran and Newton’s 2nd law to find the angular acceleration of the merry-go-round.
Rotation
715
(a) Use the work-kinetic energy theorem to relate the work done by the children to the kinetic energy of the merry-go-round:
f
forcenet
K
KW
=
∆=
or 2
214 ωIsF =∆
Substitute for I and solve for ∆s to obtain:
Fmr
Fmr
FIs
1688
2222212 ωωω
===∆
Substitute numerical values and evaluate ∆s: ( )( )
( )m6.11
N2616rev
rad2s2.8
rev1m2kg2402
2
=
⎥⎦
⎤⎢⎣
⎡×
=∆
π
s
(b) Apply Newton’s 2nd law to express the angular acceleration of the merry-go-round:
mrF
mrFr
I84
221
net ===τα
Substitute numerical values and evaluate α:
( )( )( )
2rad/s0.433m2kg240
N268==α
(c) Use the definition of work to relate the force exerted by each child to the distance over which that force is exerted:
( )( ) J302m11.6N26 ==∆= sFW
(d) Relate the kinetic energy of the merry-go-round to the work that was done on it:
sFKKW ∆=−=∆= 40fforcenet
Substitute numerical values and evaluate Wnet force:
( )( ) kJ1.21m11.6N264forcenet ==W
119 •• Picture the Problem Because the center of mass of the hoop is at its center, we can use Newton’s second law to relate the acceleration of the hoop to the net force acting on it. The distance moved by the center of the hoop can be determined using a constant-acceleration equation, as can the angular velocity of the hoop. (a) Using a constant-acceleration equation, relate the distance the
( )2cm2
1 tas ∆=∆
Chapter 9
716
center of the travels in 3 s to the acceleration of its center of mass: Relate the acceleration of the center of mass of the hoop to the net force acting on it:
mF
a netcm =
Substitute to obtain: ( )mtFs
2
2∆=∆
Substitute numerical values and evaluate ∆s:
( )( )( ) m15.0
kg1.52s3N5 2
==∆s
(b) Relate the angular velocity of the hoop to its angular acceleration and the elapsed time:
t∆= αω
Use Newton’s 2nd law to relate the angular acceleration of the hoop to the net torque acting on it:
mRF
mRFR
I=== 2
netτα
Substitute to obtain: mR
tF∆=ω
Substitute numerical values and evaluate ω:
( )( )( )( ) rad/s15.4
m0.65kg1.5s3N5
==ω
120 •• Picture the Problem Let R represent the radius of the grinding wheel, M its mass, r the radius of the handle, and m the mass of the load attached to the handle. In the absence of information to the contrary, we’ll treat the 25-kg load as though it were concentrated at a point. Let the zero of gravitational potential energy be where the 25-kg load is at its lowest point. We’ll apply Newton’s 2nd law and the conservation of mechanical energy to determine the initial angular acceleration and the maximum angular velocity of the wheel. (a) Use Newton’s 2nd law to relate the acceleration of the wheel to the net torque acting on it:
2221
net
mrMRmgr
I +==
τα
Rotation
717
Substitute numerical values and evaluate α:
( )( )( )( )( ) ( )( )
2
2221
2
rad/s9.58
m0.65kg25m0.45kg60m0.65m/s9.81kg25
=
+=α
(b) Use the conservation of mechanical energy to relate the initial potential energy of the load to its kinetic energy and the rotational kinetic energy of the wheel when the load is directly below the center of mass of the wheel:
0=∆+∆ UK or, because Ki = Uf = 0,
0irotf,transf, =−+ UKK .
Substitute and solve for ω: ( ) 02221
212
21 =−+ mgrMRmv ω ,
0224122
21 =−+ mgrMRmr ωω ,
and
2224
MRmrmgr+
=ω
Substitute numerical values and evaluate ω:
( )( )( )( )( ) ( )( )
rad/s38.4
m0.45kg60m0.65kg252m0.65m/s9.81kg254
22
2
=
+=ω
*121 •• Picture the Problem Let the smaller block have the dimensions shown in the diagram. Then the length, height, and width of the larger block are ,lS h,S and w,S respectively. Let the numeral 1 denote the smaller block and the numeral 2 the larger block and express the ratios of the surface areas, masses, and moments of inertia of the two blocks.
(a) Express the ratio of the surface areas of the two blocks:
( )( ) ( )( ) ( )( )
( )
2
21
2
S222222S
222SS2SS2SS2
=
++++
=
++++
=
whhwwhhw
whhwhwhw
AA
ll
ll
ll
ll
Chapter 9
718
(b) Express the ratio of the masses of the two blocks:
( )( )( )
( ) 33
1
2
1
2
1
2
SS
SSS
==
===
hwhw
hwhw
VV
VV
MM
l
l
l
l
ρρ
(c) Express the ratio of the moments of inertia, about the axis shown in the diagram, of the two blocks:
( ) ( )[ ][ ]
[ ][ ] ( )2
1
222
222
1
2
22112
1
22212
1
1
2
SS
SS
⎟⎟⎠
⎞⎜⎜⎝
⎛=
++
=
++
=
MM
hh
MM
hMhM
II
l
l
l
l
In part (b) we showed that: 3
1
2 S=MM
Substitute to obtain: ( )( ) 523
1
2 SSS ==II
122 •• Picture the Problem We can derive the perpendicular-axis theorem for planar objects by following the step-by-step procedure outlined in the problem. (a) and (b) ( )
yx
z
II
dmydmx
dmyxdmrI
+=
+=
+==
∫ ∫∫∫
22
222
(c) Let the z axis be the axis of rotation of the disk. By symmetry:
yx II =
Express Iz in terms of Ix:
xz II 2=
Letting M represent the mass of the disk, solve for Ix:
( ) 2412
21
21
21 MRMRII zx ===
123 •• Picture the Problem Let the zero of gravitational potential energy be at the center of the disk when it is directly below the pivot. The initial gravitational potential energy of the disk is transformed into rotational kinetic energy when its center of mass is directly below the pivot. We can use Newton’s 2nd law to relate the force exerted by the pivot to the weight of the disk and the centripetal force acting on it at its lowest point.
Rotation
719
(a) Use the conservation of mechanical energy to relate the initial potential energy of the disk to its kinetic energy when its center of mass is directly below the pivot:
0=∆+∆ UK or, because Ki = Uf = 0,
0irotf, =−UK
Substitute for rotf,K and iU :
0221 =− MgrIω (1)
Use the parallel-axis theorem to relate the moment of inertia of the disk about the pivot to its moment of inertia with respect to an axis through its center of mass:
2cm MhII +=
or 2
2322
21 MrMrMrI =+=
Solve equation (1) for ω and substitute for I to obtain: r
g34
=ω
(b) Letting F represent the force exerted by the pivot, use Newton’s 2nd law to express the net force acting on the swinging disk as it passes through its lowest point:
2net ωMrMgFF =−=
Solve for F and simplify to obtain:
MgMgMgrgMrMgMrMgF
37
34
2
34
=+=
+=+= ω
124 •• Picture the Problem The diagram shows a vertical cross-piece. Because we’ll need to take moments about the point of rotation (point P), we’ll need to use the parallel-axis theorem to find the moments of inertia of the two parts of this composite structure. Let the numeral 1 denote the vertical member and the numeral 2 the horizontal member. We can apply Newton’s 2nd law in rotational form to the structure to express its angular acceleration in terms of the net torque causing it to fall and its moment of inertia with respect to point P.
Chapter 9
720
(a) Taking clockwise rotation to be positive (this is the direction the structure is going to rotate), apply
ατ PI=∑ :
αPIwgmgm =⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛
22 12
2l
Solve for α to obtain:
PIgwmgm
2122 −
=lα
or ( )
( )PP IIwmmg
21
122
2 +−
=lα (1)
Convert wand,, 21 ll to SI units:
m3.66ft3.281
m1ft121 =×=l ,
m83.1ft3.281
m1ft62 =×=l , and
m610.0ft3.281
m1ft2 =×=w
Using Table 9-1 and the parallel-axis theorem, express the moment of inertia of the vertical member about an axis through point P:
( )2412
131
1
2
12113
11 2
wm
wmmI P
+=
⎟⎠⎞
⎜⎝⎛+=
l
l
Substitute numerical values and evaluate I1P:
( ) ( ) ( )[ ]23
2412
31
1
mkg1060.1
m0.610m3.66kg350
⋅×=
+=PI
Using the parallel-axis theorem, express the moment of inertia of the horizontal member about an axis through point P:
22cm,22 dmII P += (2)
where ( ) ( )2
2212
21
12 wwd −++= ll
Solve for d: ( ) ( )2
2212
21
1 wwd −++= ll
Substitute numerical values and evaluate d:
( )[ ] ( )[ ] m86.3m0.610m1.83m0.610m3.66 2212
21 =−++=d
From Table 9-1 we have: 2
22121
cm,2 lmI =
Substitute in equation (2) to obtain: ( )22
2121
2
22
22212
12
dm
dmmI P
+=
+=
l
l
Rotation
721
Evaluate I2P: ( ) ( ) ( )[ ]23
22121
2
mkg1066.2
m3.86m1.83kg175
⋅×=
+=PI
Substitute in equation (1) and evaluate α:
( ) ( )( ) ( )( )[ ]( )
223
2
rad/s123.0mkg102.661.602
m0.61kg350m1.83kg175m/s9.81=
⋅×+−
=α
(b) Express the magnitude of the acceleration of the sparrow:
Ra α= where R is the distance of the sparrow from the point of rotation and
( ) ( )22
21
2 wwR −++= ll
Solve for R: ( ) ( )22
21 wwR −++= ll
Substitute numerical values and evaluate R:
( ) ( ) m4.44m0.610m1.83m0.610m3.66 22 =−++=R
Substitute numerical values and evaluate a:
( )( )2
2
m/s0.546
m4.44rad/s0.123
=
=a
(c) Refer to the diagram to express ax in terms of a: R
waaax+
== 1cos lθ
Substitute numerical values and evaluate ax: ( )
2
2
m/s0.525
m4.44m0.61m3.66m/s0.546
=
+=xa
Chapter 9
722
125 •• Picture the Problem Let the zero of gravitational potential energy be at the bottom of the incline. The initial potential energy of the spool is transformed into rotational and translational kinetic energy when the spool reaches the bottom of the incline. We can apply the conservation of mechanical energy to find an expression for its speed at that location. The force diagram shows the forces acting on the spool when there is enough friction to keep it from slipping. We’ll use Newton’s 2nd law in both translational and rotational form to derive an expression for the static friction force.
(a) In the absence of friction, the forces acting on the spool will be its weight, the normal force exerted by the incline, and the tension in the string. A component of its weight will cause the spool to accelerate down the incline and the tension in the string will exert a torque that will cause counterclockwise rotation of the spool.
unwinds. string asdirection ckwisecounterclo a
in spinning on,acceleraticonstant at plane down the move willspool The
Use the conservation of mechanical energy to relate the speed of the center of mass of the spool at the bottom of the slope to its initial potential energy:
0=∆+∆ UK or, because Ki = Uf = 0,
0irotf,transf, =−+ UKK .
Substitute for transf,K , rotf,K and iU :
0sin2212
21 =−+ θω MgDIMv (1)
Substitute for ω and solve for v to obtain:
0sin2
2
212
21 =−+ θMgD
rvIMv
and
2
sin2
rIM
MgDv+
=θ
Rotation
723
(b) Apply Newton’s 2nd law to the spool: ∑ =−−= 0sin sfTMgFx θ
∑ =−= 0s0 RfTrτ
Eliminate T between these equations to obtain:
rR
Mgf+
=1
sins
θ, up the incline.
126 •• Picture the Problem While the angular acceleration of the rod is the same at each point along its length, the linear acceleration and, hence, the force exerted on each coin by the rod, varies along its length. We can relate this force the linear acceleration of the rod through Newton’s 2nd law and the angular acceleration of the rod. Letting x be the distance from the pivot, use Newton’s 2nd law to express the force F acting on a coin:
( ) ( )xmaxFmgF =−=net
or ( ) ( )( )xagmxF −= (1)
Use Newton’s 2nd law to relate the angular acceleration of the system to the net torque acting on it: L
gML
LMg
I 232
231
net ===τα
Relate a(x) and α: ( ) ( ) gxgxxxa ===
m5.123α
Substitute in equation (1) to obtain:
( ) ( ) ( )xmggxgmxF −=−= 1
Evaluate F(0.25 m): ( ) ( ) mgmgF 75.0m25.01m25.0 =−=
Evaluate F(0.5 m): ( ) ( ) mgmgF 5.0m5.01m5.0 =−=
Evaluate F(0.75 m): ( ) ( ) mgmgF 25.0m75.01m75.0 =−=
Evaluate F(1 m): ( ) ( ) ( ) 0m5.1m25.1m1 === FFF
*127 •• Picture the Problem The diagram shows the force the hand supporting the meterstick exerts at the pivot point and the force the earth exerts on the meterstick acting at the center of mass. We can relate the angular acceleration to the acceleration of the end of the meterstick using αLa = and use Newton’s 2nd law in rotational form to relate α to the moment of inertia of the meterstick.
Chapter 9
724
(a) Relate the acceleration of the far end of the meterstick to the angular acceleration of the meterstick:
αLa = (1)
Apply ατ PP I=∑ to the meterstick:
αPILMg =⎟⎠⎞
⎜⎝⎛
2
Solve for α:
PIMgL2
=α
From Table 9-1, for a rod pivoted at one end, we have:
2
31 MLIP =
Substitute to obtain: L
gMLMgL
23
23
2 ==α
Substitute in equation (1) to obtain: 2
3ga =
Substitute numerical values and evaluate a:
( ) 22
m/s14.72m/s9.813
==a
(b) Express the acceleration of a point on the meterstick a distance x from the pivot point:
xLgxa
23
== α
Express the condition that the meterstick leaves the penny behind:
ga >
Substitute to obtain: gx
Lg
>23
Solve for and evaluate x: ( ) cm7.66
3m12
32
==>Lx
Rotation
725
128 •• Picture the Problem Let m represent the 0.2-kg mass, M the 0.8-kg mass of the cylinder, L the 1.8-m length, and x + ∆x the distance from the center of the objects whose mass is m. We can use Newton’s 2nd law to relate the radial forces on the masses to the spring’s stiffness constant and use the work-energy theorem to find the work done as the system accelerates to its final angular speed. (a) Express the net inward force acting on each of the 0.2-kg masses:
( )∑ ∆+=∆= 2radial ωxxmxkF
Solve for k: ( ) 2
xxxmk
∆∆+
=ω
Substitute numerical values and evaluate k:
( )( )( )
N/m230
m0.4rad/s24m0.8kg0.2 2
=
=k
(b) Using the work-energy theorem, relate the work done to the change in energy of the system:
( )2212
21
springrot
xkI
UKW
∆+=
∆+=
ω (1)
Express I as the sum of the moments of inertia of the cylinder and the masses:
m
mM
IMLMr
III
221212
21
2
++=
+=
From Table 9-1 we have, for a solid cylinder about a diameter through its center:
21212
41 mLmrI +=
where L is the length of the cylinder.
For a disk (thin cylinder), L is small and:
241 mrI =
Apply the parallel-axis theorem to obtain:
2241 mxmrIm +=
Substitute to obtain: ( )( )22
412
1212
21
22412
1212
21
2
2
xrmMLMr
mxmrMLMrI
+++=
+++=
Substitute numerical values and evaluate I:
Chapter 9
726
( )( ) ( )( ) ( ) ( ) ( )[ ]2
22412
1212
21
mN492.0
m0.8m0.2kg0.22m1.8kg0.8m0.2kg0.8
⋅=
+++=I
Substitute in equation (1) to obtain:
( )( ) ( )( ) J160m0.4N/m230rad/s24mN0.492 22122
21 =+⋅=W
129 •• Picture the Problem Let m represent the 0.2-kg mass, M the 0.8-kg mass of the cylinder, L the 1.8-m length, and x + ∆x the distance from the center of the objects whose mass is m. We can use Newton’s 2nd law to relate the radial forces on the masses to the spring’s stiffness constant and use the work-energy theorem to find the work done as the system accelerates to its final angular speed. Using the work-energy theorem, relate the work done to the change in energy of the system:
( )2212
21
springrot
xkI
UKW
∆+=
∆+=
ω (1)
Express I as the sum of the moments of inertia of the cylinder and the masses:
m
mM
IMLMr
III
221212
21
2
++=
+=
From Table 9-1 we have, for a solid cylinder about a diameter through its center:
21212
41 mLmrI +=
where L is the length of the cylinder.
For a disk (thin cylinder), L is small and:
241 mrI =
Apply the parallel-axis theorem to obtain:
2241 mxmrIm +=
Substitute to obtain: ( )( )22
412
1212
21
22412
1212
21
2
2
xrmMLMr
mxmrMLMrI
+++=
+++=
Substitute numerical values and evaluate I:
( )( ) ( )( ) ( ) ( ) ( )[ ]2
22412
1212
21
mN492.0
m0.8m0.2kg0.22m1.8kg0.8m0.2kg0.8
⋅=
+++=I
Rotation
727
Express the net inward force acting on each of the 0.2-kg masses:
( )∑ ∆+=∆= 2radial ωxxmxkF
Solve for ω:
( )xxmxk∆+
∆=ω
Substitute numerical values and evaluate ω:
( )( )( )( ) rad/s12.2
m0.8kg0.2m0.4N/m60
==ω
Substitute numerical values in equation (1) to obtain:
( )( )( )( )J4.14
m0.4N/m06
rad/s2.21mN0.4922
21
2221
=
+
⋅=W
130 •• Picture the Problem The force diagram shows the forces acting on the cylinder. Because F causes the cylinder to rotate clockwise, f, which opposes this motion, is to the right. We can use Newton’s 2nd law in both translational and rotational forms to relate the linear and angular accelerations to the forces acting on the cylinder. (a) Use Newton’s 2nd law to relate the angular acceleration of the center of mass of the cylinder to F:
MRF
MRFR
I2
221
net ===τα
Use Newton’s 2nd law to relate the acceleration of the center of mass of the cylinder to F:
MF
MFa == net
cm
Express the rolling-without-slipping condition to the accelerations:
αα 2cm ===MRF
Ra'
(b) Take the point of contact with the floor as the ″pivot″ point, express the net torque about that point, and solve for α:
ατ IFR == 2net
and
IFR2
=α
Chapter 9
728
Express the moment of inertia of the cylinder with respect to the pivot point:
22322
21 MRMRMRI =+=
Substitute to obtain: MRF
MRFR
342
223
==α
Express the linear acceleration of the cylinder: M
FRa34
cm == α
Apply Newton’s 2nd law to the forces acting on the cylinder:
∑ =+= cmMafFFx
Solve for f:
direction. positive in the 3
4
31
cm
xF
FFFMaf
=
−=−=
131 •• Picture the Problem As the load falls, mechanical energy is conserved. As in Example 9-7, choose the initial potential energy to be zero. Apply conservation of mechanical energy to obtain an expression for the speed of the bucket as a function of its position and use the given expression for t to determine the time required for the bucket to travel a distance y. Apply conservation of mechanical energy:
000iiff =+=+=+ KUKU (1)
Express the total potential energy when the bucket has fallen a distance y:
⎟⎠⎞
⎜⎝⎛−−=
++=
2c
wfcfbff
y'gmmgy
UUUU
where 'mc is the mass of the hanging part of the cable.
Assume the cable is uniform and express 'mc in terms of mc, y, and L: L
my'm cc = or y
Lm'm c
c =
Substitute to obtain: L
gymmgyU
2
2c
f −−=
Rotation
729
Noting that bucket, cable, and rim of the winch have the same speed v, express the total kinetic energy when the bucket is falling with speed v: ( )
2412
c212
21
2
22
21
212
c212
21
2f2
12c2
1221
wfcfbff
MvvmmvRvMRvmmv
Ivmmv
KKKK
++=
++=
++=
++=
ω
Substitute in equation (1) to obtain:
02
2412
c21
221
2c
=++
+−−
Mvvm
mvLgymmgy
Solve for v:
c
c
mmMLgymmgy
v22
242
++
+=
A spreadsheet solution is shown below. The formulas used to calculate the quantities in the columns are as follows:
Cell Formula/Content Algebraic Form D9 0 y0
D10 D9+$B$8 y + ∆y E9 0 v0
E10 ((4*$B$3*$B$7*D10+2*$B$7*D10^2/(2*$B$5))/ ($B$1+2*$B$3+2*$B$4))^0.5
c
c
mmMLgymmgy
22
242
++
+
F10 F9+$B$8/((E10+E9)/2) yvvt nn
n ∆⎟⎠⎞
⎜⎝⎛ +
+ −− 2
11
J9 0.5*$B$7*H9^2 2
21 gt
A B C D E F G H I J
1 M= 10 kg 2 R= 0.5 m 3 m= 5 kg 4 mc= 3.5 kg 5 L= 10 m 6 7 g= 9.81 m/s^2 8 dy= 0.1 m y v(y) t(y) t(y) y 1/2gt^29 0.0 0.00 0.00 0.00 0.0 0.00
10 0.1 0.85 0.23 0.23 0.1 0.27 11 0.2 1.21 0.33 0.33 0.2 0.54 12 0.3 1.48 0.41 0.41 0.3 0.81 13 0.4 1.71 0.47 0.47 0.4 1.08 15 0.5 1.91 0.52 0.52 0.5 1.35
Chapter 9
730
105 9.6 9.03 2.24 2.24 9.6 24.61 106 9.7 9.08 2.25 2.25 9.7 24.85 107 9.8 9.13 2.26 2.26 9.8 25.09 108 9.9 9.19 2.27 2.27 9.9 25.34 109 10.0 9.24 2.28 2.28 10.0 25.58
The solid line on the graph shown below shows the position y of the bucket when it is in free fall and the dashed line shows y under the conditions modeled in this problem.
0
2
4
6
8
10
12
14
16
18
20
0.0 0.4 0.8 1.2 1.6 2.0
t (s)
y (m
)
y'free fall
132 •• Picture the Problem The pictorial representation shows the forces acting on the cylinder when it is stationary. First, we note that if the tension is small, then there can be no slipping, and the system must roll. Now consider the point of contact of the cylinder with the surface as the “pivot” point. If τ about that point is zero, the system will not roll. This will occur if the line of action of the tension passes through the pivot point. From the diagram we see that:
⎟⎠⎞
⎜⎝⎛= −
Rr1cosθ
Rotation
731
*133 •• Picture the Problem Free-body diagrams for the pulley and the two blocks are shown to the right. Choose a coordinate system in which the direction of motion of the block whose mass is M (downward) is the positive y direction. We can use the given relationship θµ ∆= s
max' TeT to relate the tensions in the rope on either side of the pulley and apply Newton’s 2nd law in both rotational form (to the pulley) and translational form (to the blocks) to obtain a system of equations that we can solve simultaneously for a, T1, T2, and M. (a) Use θµ ∆= s
max' TeT to evaluate the maximum tension required to prevent the rope from slipping on the pulley:
( ) ( ) N7.25N10' 3.0max == πeT
(c) Given that the angle of wrap is π radians, express T2 in terms of T1:
13.0
12 57.2 TeTT == π (1)
Because the rope doesn’t slip, we can relate the angular acceleration, α, of the pulley to the acceleration, a, of the hanging masses by:
ra
=α
Apply yy maF =∑ to the two blocks to obtain:
mamgT =−1 (2) and
MaTMg =− 2 (3)
Apply ∑ = ατ I to the pulley to obtain:
( )raIrTT =− 12 (4)
Substitute for T2 from equation (1) in equation (4) to obtain:
( )raIrTT =− 1157.2
Solve for T1 and substitute numerical values to obtain: ( )
( )a
aar
IT
kg91.9m0.151.57mkg0.35
57.1 2
2
21
=
⋅==
(5)
Substitute in equation (2) to obtain:
( ) mamga =−kg91.9
Chapter 9
732
Solve for and evaluate a:
22
m/s10.11
kg1kg9.91m/s9.81
1kg91.9kg91.9
=−
=
−=
−=
m
gm
mga
(b) Solve equation (3) for M:
agTM−
= 2
Substitute in equation (5) to find T1: ( )( ) N10.9m/s1.10kg91.9 2
1 ==T
Substitute in equation (1) to find T2: ( )( ) N28.0N10.9.5722 ==T
Evaluate M: kg21.3
m/s1.10m/s9.81N28.0
22 =−
=M
134 ••• Picture the Problem When the tension is horizontal, the cylinder will roll forward and the friction force will be in the direction of .T
r We can use Newton’s 2nd law to obtain
equations that we can solve simultaneously for a and f. (a) Apply Newton’s 2nd law to the cylinder:
∑ =+= mafTFx (1)
and
∑ =−= ατ IfRTr (2)
Substitute for I and α in equation (2) to obtain:
mRaRamRfRTr 2
1221 ==− (3)
Solve equation (3) for f: ma
RTrf 2
1−= (4)
Substitute equation (4) in equation (1) and solve for a:
⎟⎠⎞
⎜⎝⎛ +=
Rr
mTa 1
32
(5)
Substitute equation (5) in equation (4) to obtain: ⎟
⎠⎞
⎜⎝⎛ −= 12
3 RrTf
(b) Equation (4) gives the acceleration of the center of mass: ⎟
⎠⎞
⎜⎝⎛ +=
Rr
mTa 1
32
Rotation
733
(c) Express the condition that mTa > : 11
321
32
>⎟⎠⎞
⎜⎝⎛ +⇒>⎟
⎠⎞
⎜⎝⎛ +
Rr
mT
Rr
mT
or Rr 2
1>
(d) If Rr 2
1> : . ofdirection in the i.e., ,0 Tr
>f
135 ••• Picture the Problem The system is shown in the drawing in two positions, with angles θ0 and θ with the vertical. The drawing also shows all the forces that act on the stick. These forces result in a rotation of the stick—and its center of mass—about the pivot, and a tangential acceleration of the center of mass. We’ll apply the conservation of mechanical energy and Newton’s 2nd law to relate the radial and tangential forces acting on the stick. Use the conservation of mechanical energy to relate the kinetic energy of the stick when it makes an angle θ with the vertical and its initial potential energy:
0ifif =−+− UUKK
or, because Kf = 0,
0cos2
cos2 0
221 =−+− θθω LMgLMgI
Substitute for I and solve for ω2: ( )02 coscos3 θθω −=
Lg
Express the centripetal force acting on the center of mass:
( )
( )0
0
2c
coscos2
3
coscos32
2
θθ
θθ
ω
−=
−=
=
MgLgLM
LMF
Express the radial component of g
rM :
( ) θcosradial MgMg =
Express the total radial force at the hinge:
F|| = Fc + (Mg)radial
Chapter 9
734
( )
( )021
0
cos3cos5
coscoscos2
3
θθ
θθθ
−=
+−=
Mg
MgMg
Relate the tangential acceleration of the center of mass to its angular acceleration:
a⊥= 21 Lα
Use Newton’s 2nd law to relate the angular acceleration of the stick to the net torque acting on it: L
gML
LMg
I 2sin3sin
22
31
net θθτα ===
Express a⊥ in terms of α: a⊥= 2
1 Lα = 43 gsinθ = gsinθ + F⊥/M
Solve for F⊥ to obtain: F⊥ θsin4
1 Mg−= where the minus sign
indicates that the force is directed oppositely to the tangential component of
.gr
M
735
Chapter 10 Conservation of Angular Momentum Conceptual Problems *1 • (a) True. The cross product of the vectors A
rand B
ris defined to be .ˆsin nBA φAB=×
rr
If Ar
and Br
are parallel, sinφ = 0. (b) True. By definition, ωr is along the axis. (c) True. The direction of a torque exerted by a force is determined by the definition of the cross product.
2 • Determine the Concept The cross product of the vectors A
rand B
ris defined to be
.ˆsin nBA φAB=×rr
Hence, the cross product is a maximum when sinφ = 1. This
condition is satisfied provided Ar
and Br
are perpendicular. correct. is )(c
3 • Determine the Concept L
rand p
r are related according to .prL
rrr×= From this
definition of the cross product, Lr
and pr
are perpendicular; i.e., the angle between them
is 90°.
4 • Determine the Concept L
rand p
r are related according to .prL
rrr×= Because the
motion is along a line that passes through point P, r = 0 and so is L. correct. is )(b
*5 •• Determine the Concept L
rand p
r are related according to .prL
rrr×=
(a) Because L
r is directly proportional
to :pr
. doubles Doubling Lprr
(b) Because Lr
is directly proportional to :rr
. doubles Doubling Lrrr
Chapter 10
736
6 •• Determine the Concept The figure shows a particle moving with constant speed in a straight line (i.e., with constant velocity and constant linear momentum). The magnitude of L is given by rpsinφ = mv(rsinφ).
Referring to the diagram, note that the distance rsinφ from P to the line along which the
particle is moving is constant. Hence, mv(rsinφ) is constant and so constant. is Lr
7 • False. The net torque acting on a rotating system equals the change in the system’s angular momentum; i.e., dtdL=netτ , where L = Iω. Hence, if netτ is zero, all we can say
for sure is that the angular momentum (the product of I and ω) is constant. If I changes, so mustω. *8 •• Determine the Concept Yes, you can. Imagine rotating the top half of your body with arms flat at sides through a (roughly) 90° angle. Because the net angular momentum of the system is 0, the bottom half of your body rotates in the opposite direction. Now extend your arms out and rotate the top half of your body back. Because the moment of inertia of the top half of your body is larger than it was previously, the angle which the bottom half of your body rotates through will be smaller, leading to a net rotation. You can repeat this process as necessary to rotate through any arbitrary angle. 9 • Determine the Concept If L is constant, we know that the net torque acting on the system is zero. There may be multiple constant or time-dependent torques acting on the system as long as the net torque is zero. correct. is )(e
10 •• Determine the Concept No. In order to do work, a force must act over some distance. In each ″inelastic collision″ the force of static friction does not act through any distance. 11 •• Determine the Concept It is easier to crawl radially outward. In fact, a radially inward force is required just to prevent you from sliding outward. *12 •• Determine the Concept The pull that the student exerts on the block is at right angles to its motion and exerts no torque (recall that Frτ rrr
×= and θτ sinrF= ). Therefore, we
Conservation of Angular Momentum
737
can conclude that the angular momentum of the block is conserved. The student does, however, do work in displacing the block in the direction of the radial force and so the block’s energy increases. correct. is )(b
*13 •• Determine the Concept The hardboiled egg is solid inside, so everything rotates with a uniform velocity. By contrast, it is difficult to get the viscous fluid inside a raw egg to start rotating; however, once it is rotating, stopping the shell will not stop the motion of the interior fluid, and the egg may start rotating again after momentarily stopping for this reason. 14 • False. The relationship dtdLτ
rr= describes the motion of a gyroscope independently of
whether it is spinning.
15 • Picture the Problem We can divide the expression for the kinetic energy of the object by the expression for its angular momentum to obtain an expression for K as a function of I and L. Express the rotational kinetic energy of the object:
221 ωIK =
Relate the angular momentum of the object to its moment of inertia and angular velocity:
ωIL =
Divide the first of these equations by the second and solve for K to obtain:
ILK2
2
= and so correct. is )(b
16 • Determine the Concept The purpose of the second smaller rotor is to prevent the body of the helicopter from rotating. If the rear rotor fails, the body of the helicopter will tend to rotate on the main axis due to angular momentum being conserved. 17 •• Determine the Concept One can use a right-hand rule to determine the direction of the torque required to turn the angular momentum vector from east to south. Letting the fingers of your right hand point east, rotate your wrist until your fingers point south. Note that your thumb points downward. correct. is )(b
Chapter 10
738
18 •• Determine the Concept In turning east, the man redirects the angular momentum vector from north to east by exerting a clockwise torque (viewed from above) on the gyroscope. As a consequence of this torque, the front end of the suitcase will dip downward.
correct. is )(d
19 •• (a) The lifting of the nose of the plane rotates the angular momentum vector upward. It veers to the right in response to the torque associated with the lifting of the nose. (b) The angular momentum vector is rotated to the right when the plane turns to the right. In turning to the right, the torque points down. The nose will move downward. 20 •• Determine the Concept If L
r points up and the car travels over a hill or through a
valley, the force on the wheels on one side (or the other) will increase and car will tend to tip. If L
r points forward and car turns left or right, the front (or rear) of the car will tend
to lift. These problems can be averted by having two identical flywheels that rotate on the same shaft in opposite directions. 21 •• Determine the Concept The rotational kinetic energy of the woman-plus-stool system is given by .222
21
rot ILIK == ω Because L is constant (angular momentum is conserved)
and her moment of inertia is greater with her arms extended, correct. is )(b
*22 •• Determine the Concept Consider the overhead view of a tether pole and ball shown in the adjoining figure. The ball rotates counterclockwise. The torque about the center of the pole is clockwise and of magnitude RT, where R is the pole’s radius and T is the tension. So L must decrease and correct. is )(e
23 •• Determine the Concept The center of mass of the rod-and-putty system moves in a straight line, and the system rotates about its center of mass.
Conservation of Angular Momentum
739
24 • (a) True. The net external torque acting a system equals the rate of change of the angular
momentum of the system; i.e.,dtdLτr
r=∑
iexti, .
(b) False. If the net torque on a body is zero, its angular momentum is constant but not necessarily zero. Estimation and Approximation *25 •• Picture the Problem Because we have no information regarding the mass of the skater, we’ll assume that her body mass (not including her arms) is 50 kg and that each arm has a mass of 4 kg. Let’s also assume that her arms are 1 m long and that her body is cylindrical with a radius of 20 cm. Because the net external torque acting on her is zero, her angular momentum will remain constant during her pirouette. Express the conservation of her angular momentum during her pirouette:
fi LL =
or inarmsinarmsoutarmsoutarms ωω II = (1)
Express her total moment of inertia with her arms out:
armsbodyoutarms III +=
Treating her body as though it is cylindrical, calculate its moment of inertia of her body, minus her arms:
( )( )2
2212
21
body
mkg00.1
m0.2kg50
⋅=
== mrI
Modeling her arms as though they are rods, calculate their moment of inertia when she has them out:
( )( )[ ]2
231
arms
mkg67.2
m1kg42
⋅=
=I
Substitute to determine her total moment of inertia with her arms out: 2
22outarms
mkg67.3
mkg67.2mkg00.1
⋅=
⋅+⋅=I
Express her total moment of inertia with her arms in: ( )( )[ ]
2
22
armsbodyiarms
mkg32.1m0.2kg42mkg00.1
⋅=
+⋅=
+= III n
Chapter 10
740
Solve equation (1) for inarmsω and
substitute to obtain:
( )
rev/s17.4
rev/s5.1mkg32.1mkg3.67
2
2
outarmsinarms
outarmsinarms
=
⋅⋅
=
= ωωII
26 •• Picture the Problem We can express the period of the earth’s rotation in terms of its angular velocity of rotation and relate its angular velocity to its angular momentum and moment of inertia with respect to an axis through its center. We can differentiate this expression with respect to I and then use differentials to approximate the changes in I and T. Express the period of the earth’s rotation in terms of its angular velocity of rotation:
ωπ2
=T
Relate the earth’s angular velocity of rotation to its angular momentum and moment of inertia:
IL
=ω
Substitute to obtain: L
IT π2=
Find dT/dI:
IT
LdIdT
==π2
Solve for dT/T and approximate ∆T:
IdI
TdT
= or TIIT ∆
≈∆
Substitute for ∆I and I to obtain:
TMmT
RMmrT
E2EE5
2
232
35
=≈∆
Substitute numerical values and evaluate ∆T:
( )( ) ( )
s552.0h
s3600d
h24d1039.6
d1039.6
d1kg1063kg102.35
6
6
24
19
=
×××=
×=
××
=∆
−
−
T
Conservation of Angular Momentum
741
27 • Picture the Problem We can use L = mvr to find the angular momentum of the particle. In (b) we can solve the equation ( )hll 1+=L for ( )1+ll and the approximate value of
l . (a) Use the definition of angular momentum to obtain: ( )( )( )
/smkg102.40
m104m/s103kg10228
333
⋅×=
×××=
=
−
−−−
mvrL
(b) Solve the equation
( )hll 1+=L for ( )1+ll : ( ) 2
2
1h
llL
=+
Substitute numerical values and evaluate ( )1+ll : ( )
52
2
34
28
1022.5
sJ011.05/smkg102.401
×=
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅×
⋅×=+ −
−
ll
Because l >>1, approximate its value with the square root of
( )1+ll :
261029.2 ×≈l
(c)
.1102 and102between atedifferentican experiment no because physics cmacroscopiin noticednot is momentumangular ofon quantizati The
26
26
+×=
×=
l
l
*28 •• Picture the Problem We can use conservation of angular momentum in part (a) to relate the before-and-after collapse rotation rates of the sun. In part (b), we can express the fractional change in the rotational kinetic energy of the sun as it collapses into a neutron star to decide whether its rotational kinetic energy is greater initially or after the collapse. (a) Use conservation of angular momentum to relate the angular momenta of the sun before and after its collapse:
aabb ωω II = (1)
Using the given formula, approximate the moment of inertia Ib of the sun before collapse: ( ) ( )
246
2530
2sunb
mkg1069.5km106.96kg1099.1059.0
059.0
⋅×=
××=
= MRI