STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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Chapter 1 1-1 Calculate the mass m of a body that weighs 600 lb at the surface of the earth. SOLUTION

slug63.182.32

600g

Wm ....................................................Ans.

1-2 Calculate the weight W of a body at the surface of the earth if it has a mass m of 675 kg. SOLUTION

kN62.6N1062.681.9675 3mgW ..................................Ans.

1-3 If a man weighs 180 lb at sea level, determine the weight W of the man (a) At the top of Mt. McKinley (20,320 ft above sea level). (b) At the top of Mt. Everest (29,028 ft above sea level). SOLUTION

2e

e

rmGmW

Therefore: mGmrWrW ehh22

00

where ft10090.2 70r

(a) ft10092032.2100320.210090.2 7470 hrrh

lb7.17910092032.210090.2180

27

27

2

200

hh r

rWW ......................................Ans.

(b) ft100929028.2109028.210090.2 7470 hrrh

lb5.179100929028.2

10090.218027

27

2

200

hh r

rWW ....................................Ans.

1-4 Calculate the weight W of a navigation satellite at a distance of 20,200 km above the earth’s surface if the satellite weighs 9750 N at the earth’s surface.

SOLUTION

2e

e

rmGmW

Therefore: mGmrWrW ehh22

00

where 60 6.370 10 mr

(a) 0 6370 20,200 26,570 kmhr r h

22

0 022

9750 6370560 N

26,570h

h

W rWr

...........................................Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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1-5 Compute the gravitational force acting between two spheres that are touching each other if each sphere weighs 1125 lb and has a diameter of 20 in.

SOLUTION

2

2

2221

s

s

ss

ss

dGm

rrmGm

rmGmF

28

62

11253.439 10 32.2 15.11 10 lb20

12

..................................Ans.

1-6 Two spherical bodies have masses of 60 kg and 80 kg, respectively. Determine the gravitational force of attraction between the spheres if the distance from center to center is 600 mm.

SOLUTION

11

61 222

6.673 10 60 800.890 10 N

0.600Gm mF

r........................Ans.

1-7 Determine the weight W of a satellite when it is in orbit 8500 mi above the surface of the earth if the satellite weighs 7600 lb at the earth’s surface.

SOLUTION

221

rmGmW

Therefore: behh mGmrWrW 2200

where ft10090.2 70r

ft10578.6)5280(850010090.2 77hrr oh

272

22 7

7600 2.090 10768 lb

6.578 10o o

hh

W rWr

.....................................Ans.

1-8 Determine the weight W of a satellite when it is in orbit 20.2(106) m above the surface of the earth if the satellite weighs 8450 N at the earth’s surface.

SOLUTION

221

rmGmW

Therefore: behh mGmrWrW 2200

where 60 6.370 10 mr

6 6 66.370 10 20.2 10 26.570 10 mh or r h

262

22 6

8450 6.370 10486 N

26.570 10o o

hh

W rWr

.....................................Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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1-9 If a woman weighs 135 lb when standing on the surface of the earth, how much would she weigh when standing on the surface of the moon?

SOLUTION

221

rmGmW

Therefore on the surface of the earth where 234.095 10 slugsem and 3960 mir

23

2

4.095 10135

3960 5280

G m

7 21.4413 10 lb ft /slugGm

Then on the surface of the moon where 215.037 10 slugsmm and 1080 mir

7 21

2

1.4413 10 5.037 1022.33 lb

1080 5280W ...................................Ans.

1-10 Determine the weight W of a body that has a mass of 1000 kg (a.) At the surface of the earth. (b.) At the top of Mt. McKinley (6193 m above sea level). (c.) In a satellite at an altitude of 250 km. SOLUTION

221

rmGmW

(a) 11 24

23

6.673 10 5.976 10 10009830 N

6370 10W .............................Ans.

(b) 11 24

23

6.673 10 5.976 10 10009810 N

6370 10 6193W .............................Ans.

(c) 11 24

23 3

6.673 10 5.976 10 10009100 N

6370 10 250 10W .............................Ans.

1-11 If a man weighs 210 lb at sea level, determine the weight W of the man (a.) At the top of Mt. Everest (29,028 ft above sea level). (b.) In a satellite at an altitude of 200 mi. SOLUTION

221

rmGmW

Therefore 22103960 5280

eGm m

16 29.181 10 lb fteGm m

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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(a) 16

29.181 10 209.4 lb

3960 5280 29,028W .......................................Ans.

(b) 16

29.181 10 190.3 lb

3960 200 5280W ........................................Ans.

1-12 A space traveler weighs 800 N on earth. A planet having a mass of 5(1025) kg and a diameter of 30(106) m orbits a distant star. Determine the weight W of the traveler on the surface of this planet.

SOLUTION

221

rmGmW

Therefore on the surface of the earth where 245.976 10 kgem and 6370 kmr

24

23

5.976 10800

6370 10

G m

9 25.432 10 N m /kgGm

Then on the surface of the planet where 255 10 kgm and 615 10 mr

9 25

26

5.432 10 5 101207 N

15 10W .........................................Ans.

1-13 The planet Jupiter has a mass of 1.302(1026) slug and a visible diameter (top of the cloud layer) of 88,700 mi. Determine the gravitational acceleration g

(a.) At a point 100,000 miles above the top of the clouds. (b.) At the top of the cloud layers. SOLUTION

221

rmGmW

(a) 8 26

2

3.439 10 1.302 10

44,350 100,000 5280

mW mg

27.71 ft/sg .................................................................Ans.

(b) 8 26

2

3.439 10 1.302 10

44,350 5280

mW mg

281.7 ft/sg ................................................................Ans.

1-14 The planet Saturn has a mass of 5.67(1026) kg and a visible diameter (top of the cloud layer) of 120,000 km. The weight W of a planetary probe on earth is 4.50 kN. Determine

(a.) The weight of the probe when it is 600,000 km above the top of the clouds. (b.) The weight of the probe as it begins its penetration of the cloud layers. SOLUTION

221

rmGmW

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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Therefore on the surface of the earth where 245.976 10 kgem and 6370 kmr

24

23

5.976 104500

6370 10

G m

8 23.055 10 N m /kgGm

(a) 8 26

23

3.055 10 5.67 1039.8 N

60,000 600,000 10W .....................................Ans.

(b) 8 26

23

3.055 10 5.67 104810 N

60,000 10W ......................................Ans.

1-15 The first U.S. satellite, Explorer 1, had a mass of approximately 1 slug. Determine the force exerted on the satellite by the earth at the low and high points of its orbit, which were 175 mi and 2200 mi, respectively, above the surface of the earth.

SOLUTION

1 22

Gm mFr

8 23

2

3.439 10 4.095 10 129.5 lb

3960 175 5280F ...................................Ans.

8 23

2

3.439 10 4.095 10 113.31 lb

3960 2200 5280F ..................................Ans.

1-16 A neutron star has a mass of 2(1030) kg and a diameter of 10 km. Determine the gravitational force of attraction on a 10-kg space probe

(a.) When it is 1000 km from the center of the star. (b.) At the instant of impact with the surface of the star. SOLUTION

1 22

Gm mFr

11 30

923

6.673 10 2 10 101.335 10 N

1000 10F ...............................Ans.

11 30

1323

6.673 10 2 10 105.34 10 N

5 10F ...............................Ans.

1-17 Determine the weight W, in U.S. customary units, of a 75-kg steel bar under standard conditions (sea level at a latitude of 45 degrees).

SOLUTION

75 9.81 735.75 N 0.2248 lb/N 165.4 lbW mg ......................Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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1-18 Determine the mass m, in SI units, for a 500-lb steel beam under standard conditions (sea level at a latitude of 45 degrees).

SOLUTION W mg

500 15.528 slug 14.59 kg/slug 227 kg32.2

Wmg

........................Ans.

1-19 An automobile has a 440 cubic inch engine displacement. Determine the engine displacement in liters. SOLUTION

33 3 3 3 1 3 3440 in. 16.39 10 mm /in. 10 cm/mm 10 L/cmV

7.21 LV ...................................................................Ans.

1-20 How many barrels of oil are contained in 100 kL of oil? One barrel (petroleum) equals 42.0 gal. SOLUTION

3100 10 L 0.2642 gal/L 1 barrel 42 gal 629 barrelV ..................Ans.

1-21* Express the density, in SI units, of a specimen of material that has a specific weight of 0.025 lb/in.3 SOLUTION g

3 3

3 3 3

0.025 slug 14.59 kg in. 1000 mm32.2 in. slug 16.39 10 mm mg

3691 kg/m ...............................................................Ans.

1-22 The viscosity of crude oil under conditions of standard temperature and pressure is 7.13(10-3) N-s/m2. Determine the viscosity of crude oil in U.S. Customary units.

SOLUTION

2

3 32 2 2

N s 0.2248 lb 0.0929 m lb s7.13 10 0.1489 10 m N ft ft

...........Ans.

1-23 One acre equals 43,560 ft2. One gallon equals 231 in.3 Determine the number of liters of water required to cover 2000 acres to a depth of 1 foot.

SOLUTION

32

3

43,560 ft 12 in gal 3.785 L2000 acre ftacre ft 231 in. gal

V

92.47 10 LV ..............................................................Ans.

1-24 The stress in a steel bar is 150 MPa. Express the stress in appropriate U.S. Customary units (ksi) by using the values listed in Table 1-6 for length and force as defined values.

SOLUTION

2

6 2 2 2 ftstress 150 10 N/m 0.2248 lb/N 0.0929 m /ft12 in.

3 2stress 21.75 10 lb/in. 21.75 ksi .........................................Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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1-25 By definition, 1 hp = 33,000 ft-lb/min and 1 W = 1 N-m/s. Verify the conversion factors listed in Table 1-6 for converting power from U.S. Customary units to SI units by using the values listed for length and force as defined values.

SOLUTION

ft lb 4.448 N 0.3048 m min N m1 hp 33,000 745.7 min lb ft 60 s s

.............Ans.

1-26 The specific heat of air under standard atmospheric pressure, in SI units, is 1003 N-m/kg- K. Determine the specific heat of air under standard atmospheric pressure in U.S. customary units (ft-lb/slug- R).

SOLUTION

0.2248 lb 3.281 ft 14.59 kg 5 KN m1003 kg K N m slug 9 R

lb ft6000 slug R ..........................................................Ans.

1-27 Newton’s law of gravitation can be expressed in equation form as

221

rmmGF

If F is a force, m1 and m2 are masses, and r is a distance, determine the dimensions of G. SOLUTION

222 3

21 2

ML T LFr LGm m M M MT

..........................................Ans.

1-28 The elongation of a bar of uniform cross section subjected to an axial force is given by the equation EAPL . What are the dimensions of E if and L are lengths, P is a force, and A is an area?

SOLUTION

2

2 22

ML T LPL M FEA LT LL L

......................................Ans.

1-29 The period of oscillation of a simple pendulum is given by the equation gLkT , where T is in seconds, L is in feet, g is the acceleration due to gravity, and k is a constant. What are the dimensions of k for dimensional homogeneity?

SOLUTION

1 22

1 (dimensionless)L Tk T g L TL

...............................Ans.

1-30 An important parameter in fluid flow problems involving thin films is the Weber number (We) which can be expressed in equation form as

Lv2

We

where is the density of the fluid, v is a velocity, L is a length, and is the surface tension of the fluid. If the Weber number is dimensionless, what are the dimensions of the surface tension ?

SOLUTION

232

21e

M L L T Lv L M FW T L

................................Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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1-31 In the dimensionally homogeneous equation

I

McAP

is a stress, A is an area, M is a moment of a force, and c is a length. Determine the dimensions of P and I. SOLUTION

2 2

2 2

ML T LM PLT IL

Therefore

22 2

M MLP L FLT T

...............................................Ans.

2 2

42

ML T LI L

M LT......................................................Ans.

1-32 In the dimensionally homogeneous equation 2 21 1

2 2Pd mv I

d is a length, m is a mass, v is a linear velocity, and is an angular velocity. Determine the dimensions of P and I.

SOLUTION

2

2

1LP L M IT T

Therefore

2 2

2

ML T MLP FL T

................................................Ans.

2

2 22

MLI T MLT

......................................................Ans.

1-33 In the dimensionally homogeneous equation

bI

VQJ

Tr

is a stress, T is a torque (moment of a force), V is a force, r and b are lengths and I is a second moment of an area. Determine the dimensions of J and Q.

SOLUTION

2 2 2

2 4

ML T L ML T QMLT J L L

Therefore

2 2

42

ML T LJ L

M LT.....................................................Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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2 5

32

M LT LQ L

ML T.....................................................Ans.

1-34 In the dimensionally homogeneous equation

J

TrAP

is a stress, A is an area, T is a torque (moment of a force), and r is a length. Determine the dimensions of P and J.

SOLUTION

2 2

2 2

ML T LM PLT JL

Therefore

22 2

M MLP L FLT T

...............................................Ans.

2 2

42

ML T LJ L

M LT.....................................................Ans.

1-35 The equation atAex bt sin is dimensionally homogeneous. If A is a length and t is time, determine the dimensions of x, a, b, and .

SOLUTION

sinT bx L e a T

Therefore

1 1x L L ...........................................................Ans.

b T ......................................................................Ans.

1a T ....................................................................Ans.

1 (dimensionless) ........................................................Ans.

1-36 In the dimensionally homogeneous equation xbabxaxxw 223 , if x is a length, what are the dimensions of a, b, and w?

SOLUTION

If x L , then each term has the dimension 3L . Therefore

3w L .....................................................................Ans.

3

2

La L

L...............................................................Ans.

3

2L

b LL

...............................................................Ans.

Using the last term as a check,

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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2 22

3L La b L

x L

1-37 Determine the dimensions of a, b, c, and y in the dimensionally homogeneous equation

cbtaAey bt 21cos

in which A is a length and t is time. SOLUTION

2cos 1b Ty L e a b T c

Therefore

1 1y L L ..........................................................Ans.

1b T 1b T ....................................................................Ans.

1 (dimensionless)a ........................................................Ans.

1 (dimensionless)c ........................................................Ans.

1-38 Determine the dimensions of c, , k and P in the differential equation

tPkxdtdxc

dtxdm cos2

2

in which m is a mass, x is a length, and t is time. SOLUTION

2cos

M L Lc k L P T

TT

Therefore

2ML T Mc

L T T.........................................................Ans.

2

2

ML T M FkL T L

..................................................Ans.

2

MLP FT

.............................................................Ans.

1 T ....................................................................Ans.

1-39 Round off the following numbers to two significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference.

(a) 0.015362 (b) 55.33682 (c) 63,746.27 SOLUTION

(a) 0.015 0.015362 100 2.36 %

0.015362.............................................Ans.

(b) 55 55.33682 100 0.609 %

55.33682...............................................Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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(c) 64,000 63,746.27 100 0.398 %

63,746.27.........................................Ans.

1-40 Round off the following numbers to two significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference.

(a) 0.837482 (b) 374.9371 (c) 937,284.9 SOLUTION

(a) 0.84 0.837482 100 0.301%

0.837482.............................................Ans.

(b) 370 374.9371 100 1.317 %

374.9371.............................................Ans.

(c) 940,000 937,284.9 100 0.290%

937,284.9.........................................Ans.

1-41 Round off the following numbers to three significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference.

(a) 0.034739 (b) 26.39473 (c) 55,129.92 SOLUTION

(a) 0.0347 0.034739 100 0.1123 %

0.034739.........................................Ans.

(b) 26.4 26.39473 100 0.01997 %

26.39473..........................................Ans.

(c) 55,100 55,129.92 100 0.0543 %

55,129.92.........................................Ans.

1-42 Round off the following numbers to three significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference.

(a) 0.472916 (b) 826.4836 (c) 339,872.8 SOLUTION

(a) 0.473 0.472916 100 0.01776 %

0.472916.........................................Ans.

(b) 826 826.4836 100 0.0585 %

826.4836............................................Ans.

(c) 340,000 339,872.8 100 0.0374 %

339,872.8.......................................Ans.

1-43 Round off the following numbers to four significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference.

(a) 0.056623 (b) 74.82917 (c) 27,382.84 SOLUTION

(a) 30.05662 0.056623 100 5.30 10 %0.056623

.....................................Ans.

(b) 374.83 74.82917 100 1.109 10 %74.82917

......................................Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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(c) 327,380 27,382.84 100 10.37 10 %27,382.84

....................................Ans.

1-44 Round off the following numbers to four significant figures. Find the percent difference between each rounded-off number and the original number by using the original number as the reference.

(a) 0.664473 (b) 349.3378 (c) 274,918.2 SOLUTION

(a) 30.6645 0.664473 100 4.06 10 %0.664473

......................................Ans.

(b) 3349.3 349.3378 100 10.82 10 %349.3378

......................................Ans.

(c) 3274,900 274,918.2 100 6.62 10 %274,918.2

....................................Ans.

1-45 The weight of the first Russian satellite, Sputnik I, was 184 lb on the surface of the earth. Determine the force exerted on the satellite by the earth at the low and high points of its orbit which were 149 mi and 597 mi, respectively, above the surface of the earth.

SOLUTION

1 22

Gm mFr

Therefore 21843960 5280

eGm m

16 28.044 10 lb fteGm m

16

28.044 10 170.9 lb

3960 149 5280F .........................................Ans.

16

28.044 10 138.9 lb

3960 597 5280F .........................................Ans.

1-46 The planet Jupiter has a mass of 1.90(1027) kg and a radius of 7.14(107) m. Determine the force of attraction between the earth and Jupiter when the minimum distance between the two planets is 6(1011) m.

SOLUTION

1 22

Gm mFr

11 24 27

18211

6.673 10 5.976 10 1.90 102.10 10 N

6 10F ....................Ans.

1-47 Convert 640 acres (1 square mile) to hectares if 1 acre equals 4840 yd2 and 1 hectare equals 104 m2. SOLUTION

22 2

2 4

4840 yd 3 ft 0.0929 m hect640 acres 259 hectareacre yd ft 10 m

............Ans.

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

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1-48 Determine the dimension of c in the dimensionally homogeneous equation

mctec

mgv 1

in which v is a velocity, m is a mass, t is time, and g is the gravitational acceleration. SOLUTION

2

1 c T MM L TL eT c

Therefore

2ML T Mc

L T T.........................................................Ans.

As a check, 1 (dimensionless)M T Tct

m M

1-49 Develop an expression for the change in gravitational acceleration g between the surface of the earth and a height h when h << Re.

SOLUTION

2eGm mW mg

r

Therefore

2e e emg Gm m r

2

e emg Gm m r h

2 2

2 2 22 2

22 2

1 1

2

2

e eee

ee e e

e e

ee

e e

g g g Gmrr h

Gm r r r h hr h r

Gm r h hr h r

4 3

22e ee

ee

Gm Gm hg r hrr

..........................................................Ans.