AP Phvsics C 3d 6 Wks Take Home

1) A thin rod of mass M and length L has amoment of inertia through its center (ofmass) equal to (1 / l2)ltlz. A sphere, alsoof mass M and radius R=L/4 is glued toone end of the rod. The moment of inertiaof the sphere through its center of mass is(2/5)MR2. The combination now rotatesabout an axis through the center of massof the combined rod and sphere. What isthe moment of inertia of the rod andsphere about the center of mass of the two? Please express your answer in termsof a rational fraction (a numerical numerator over a numerical denominator)multiplied by MrL2. That is, if the answer is (x/y) l/4.Lz, find x and y.

Due L2 / IO /2OlO / , y'.

J- Irtop ( jrf

/"'--t* "n *.#,+rt"r.t {-#.*!. -u/u-#*E, A Y-&rqd t-xL:

(ryrlr- P{@}+-nn{**W} * '€Mt

I'/'I_

-'S?t

e) &*fd#* f

')=: ?*ir)'"

{#}

*f"ffit Fd** *ffieeq*

e,or rflr** ftmE'+ u {*&/'f t&v

tg{* :-, I *f L* *.

ev.fI

+ 5lr+r-* =Zc) !

Je" -* e{4 -*}Pdth*

t (a*f)It!*z L--/- {7 tb\-rf &e.ffG$ {fot xd

j

//L

.n? LJ r rt{L'16'- /b

e-tel! rFwt* =6

Lt4L

rl2J = HL* -r-120

I

AP Physics C 3d 6 Wks Take Home

0.30

G 0.20

; 0.t0>lEo6> -0.10

-4.20

Due 12 / lO /2OIO

ForceSensor

MotionSensor

50

40

z-, ?or- Jv,*{c)920v

IJ.10

.:'- * 0

0.30

1. (2000M1) A motion sensor and a force sensor record the motion of a cart along atrack, as shown

above. The cart is given a push so that it moves toward the force sensor and then collides with it. The two

sensors record the values shown in the following graphs.

a. Determine the cart's average acceleration between t : 0.33 s and t : 0.37 s.

b. Determine the magni*a" lf rn" change in the cart's momentum during the collision.

c. Determine the mass of the cart.

0.32 0.34 0.36 0.38 0.40Time r (s)

0.30 0.3? 0.34 0.36 0.38 0.40Time I (s)

\:w

d. Determine tJ:e energl lost in the collision between the force sensor and the cart

av vs -v; (-'ts::;;lq - ) "*Aa) Luo = ;E =T-161:G)> .o*

l+,i -'o *A;J

h) s;n*- ,,.se- dr-:* {',^,''q' *V'a*- f*"-/tt = al>

" I,1-olre * -f rdt -" A'eo-' L F uz t '

l:*{l-fl *oG or.*l *

f dpf =i*t4 - *lxti- r^('4 d"i

\ b 4::= :_q YtA:-r\.rd uf+

d) aK* K* * tr";'a/d'*r'

''" '

I

-2-

WlNI

I

I

--'l--I

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--TI

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+

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I

i/"i

I

I

I*t*I

I

_t_I

-FI

II

-l-I

L

Iilirr--I-Iiiri--r-lii.-- |- -t;lir--f-riti

{* s *8/fit1 1

LQ v.o1

u" l)

W*,**2{q

- [L^4

leT

*lo*f2

-/-rZ,3

AP Phvsics C 3d 6 Wks Take Home Due I2/LO/ZOLO

2. (1998M1) Two gliders move freely on an air track with negligible friction, as shown above. Glider Ahas a mass of 0.90 kg and glider B has a mass of 0.60 kg. Initially, glider A moves toward glider B, whichis at rest. A spring of negligible mass is attached to the right side of glider A. Strobe photography is used

to record successive positions of glider A at 0.10 s intervals over atotal time of 2.00 s, during which time itcollides with glider B.

The following diagram represents the data for the motion of glider A. Positions of gliderA at the end of each 0.10s interval are indicated by the s5rmbol A against a metric ruler.The total elapsed time t after each 0.50 s is also indicated.a. Determine the average speed of glider A for the following time intervals.

of glider A as a function of time t for the 2.00 s interval.

u(m/s)

C-) 2,)n.?'wt\=4

= l*/s

- t ^ + p 4 t t A^ot"to?t0.00s : ** LX*D,Lff 0.50s oqr-7f,15^.lu ,l,oot l t .50$ ?.9: t

i. o.Lo s to o.ffs it. o.go sto t.to s iii. t.7o sto l.9o Jl- rl?wr'I o,'gF.n^.b. On the axes below, sketch a graph, consistent with the ddta abovel of the speed

1.00

1.50

0.50

trltllllllllllllllllrrllllllllllllllllll

- -r- - -1- - -t - - + - - F - -l- - -l- - -l - - f - - t - - F - - F - -t- -'t - - |. - - t - - F - -l- - -l- - t_ _r_ - J _ - J - - r - - L - -r- - -!- - J - - r - - ! - - L - - t- - -l- - j - - t - - t- - - t- - -l- - J- - t--l---r---|--T--I- r -t--t I t | | | | | | | | | |

- -l- - {- - { - - + - - F - -r-- {- - -.1 - - +--+ - -F - -F - {-- + - - } - - + - - F - -.1- - {- - {rrtrlllllfllllllllll--t---'r---1--T--r--t---t---l--T--T--r--r--l--1--r--T--l---l---'l--l- -t-- J- -J -- l--L--l- - -l--J -- +--+--L--L- J-- -r--! -- I --L -J- -J- - |

: # : * -F- ff -#zig:x : i : : r : : i : : I : : r : i: - i - - i : : r : : r - i: : i: : i- -t- -.1 - - -i - - l - - l- - -t- - -i- 2f' -; - l - - t - - t - -l- - -l - - i- - - i - - l- - -l- - -l- - jI r I I | | | r)ttt I I I I | | | tl- -r---i---1--T- -l- --r---r- -'i-- T-1T.:-f --f- --t---T- - f -- T - -l- - -t---t- --l- -r- - i- - i - - r - - r- - -i- - -r- - i - - i - -Jr- -r- - - l- - -i- - i - - i - - t - - i - i- - -1- -

1- -r- -'1- - -I - - T - - r - -F - -r- -'1 - - T - - T - -it - -r - -l- - T - - r - - T - - r - -l- - -'l - - r- -t--J- -J--t-- L - -t---t- -J--I--r-4,--L-J--J--L-- !- -L- J--J- -JI I I I | | I I I I t1*,"1 t | | t | | |--t---1--1--t--t---t---t---1 --t- r-i----t--t--f--t---l---1--1- -i- -'1 - - t - - + - - | - -i- - -i- - -t - - + - - + - - l- - - F -+ -tc rle- +-rK- tlr -/,r - t--t--{--{ --+- - F--t-- -l---t--+--+--F- -F-+- - +--}--+--F-{--J- -{

t (s)2.00

,'{,b

,b

t &lx

t^*4( */,

| */,

t o/+

L M/*

i

$

f

fI*

{

-B i, .flf .L = *.3f,1, .rf/L* "?fI,D *12f'z c''$

t-1t.g

t, { .o.o1 fu { e +{s,u7lz

1,7{,4 "

o'fl,-fl- *- *$

Ifr ,t-/

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s r*f-f- { e-"t

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Air Track

0m 0.10 O.2O 0.30 0.40 0.50 0'60 0.70 0.80 0'90 1.00 1.10 1.2O 1.30 m

,a -YiZ. .O$f ,r %,Utft

frn*'t L} c.

{%q-,"b Hi

\;o l*{xlt

\r: 4-. gT:{F

va; ; D ^*l)

rt, t{ ' "**frrr"t{o

lf"y=-

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V*= I z*/*

_/, ,.I?) r{ er?,tLl -,*;

{" t--;- Ko

- h{ "b} */p

AP Phvsics C 3d 6 Wks Take Home Due r2/IO/2OIO

i. Use the data to calculate the speed of glider B immediately after it separates fromthe spring.ii. On the axes below, sketch a graph of the speed of glider B as a function of time t.

u(m/s)

1.50

ltllllllllllllllllllrrrrrrtlllllllllllll

- f - - F - -l- - -l- -'1 - - t - - f - - F - - F - -'t- - -t - - f - - t - - F - -l- - -1- - I__r- _ J__ J _ _1_ _L _ _r__ _r__J__J_ __ r__L_ _L _-l-_J_ _L__ l- _-!-_ _!_ _-!- - J--l---r--l--T--t---t---l I I I | | I I I | | | | |

1.00 --r- --1--1--t--r- -l-- -l--'1-- ?--T- -r-7f --1--1--r--?- -l- - 1--r,I1--

x* ln,{*+ Ftaffi''o

4."tblt, *p+.'

-- -*6Tt1) t,4f)

u

.L -to tdSLqs)

- -l- - -1 - - -1 - - + - - F - -l- - -l- - J - - + - - + - .iF -+*a-{- - { - - t- - - f - - F - -l-; -1- - {__r___l__ l__ r__r __r___r___r__ r - -Lj-t---r--\-r-- | L--t-6ffitd;ffi S 6::i::-j--]-:]--[--i--:i::]::I-,K-[_-[-jffi-\r--1--i--t---1---1--T--r--r- --l--1- -T7-7--r--r- -1--'r- -r--T --r - -1- --1-- I- -l- - J - - J - - I - - L - -l- - -l- - J - - Jr*: - l - - L - - L - J- - J - - L - - t - - L - J- - J- - J

I I I I I | | lJql | | | l I I I I | | |- -t- - 1- - 1 - - t - - r - -F - -l- - 1 ;.d- t - - t - - F - - r - a- - -t - - r - - t - - F - -1- - -1 - - I__,__r_ __r__ r __F __l_ __l_

=4_ _*__+__F_ _F__l__+ _ _t __

+ _ _F_ _l_ _ J_ _ {

r (s)

A graph of the total kinetic energ/ K for the two-glider system over the 2.0O s intervalhas the following shape. IG is the total kinetic enerS/ of the system at time t = 0.

t (s)

2.00

d. i. Is the collision elastic? Justiff your answer.

ii. Briefly explain why there is a minimum in the kinetic energy curve at t =1.00s.

{.) fuffr-r,',,;ppq* *s #fuC* *'*; '@:*4d & d:;# A

1id*pC-# r** .*#- *gr*s.*,.*-.ffi-" ;tu,,* T '&;'/'*rN

/"A#'

iu,) 7L kuj f &# d,fu d*..6*, E- *rd;,'6 d*f

#t***g t# ffiw*;wr*e%%**15 ffffi6lm-*&'e*l//fe/v@* n-gw-*-& f%rrye+#*-*&* $ Wfs# #- ffiqY*€*#s***d '{ "wffry '{*de*-/ &-tL f/*,fi_*tr^ {# ai;*r*r'ft,-}":a u,-#

/lht-sVriyr6

AP Physics C 3d 6 Wks Take Home Due 12/lO/2OrO

Before Collision

TOP VIEWS3. (2005M3) A system consists of a ball of mass Mz and a uniform rod of mass Mr andlength d. The rod is attached to a horizontal frictionless table by a pivot at point P andinitially rotates at an angular speed c0, as shown above left. The rotational inertia of the

rod about point n is ] Mrd2. The rod strikes the ball, which is initially at rest. As a3

result of this collision, the rod is stopped and the ball moves in the direction shownabove right. Express all answers in terms of Mt, M2, cD, d, and fundamental constants.a. Derive an expression for the angular momentum of the rod about point P before the

collision.b. Derive an expression for the speed u of the ball after the collision.c. Assuming that this collision is elastic, calculate the numerical value of the ratio Mt /

ib 6""{-*-: &"o ? V.u * Lk

s* (E#-#/Yk(*

/[a,{*H

t*{*)

.t. 11,r%.

S +:*du,*"fz;l-[,p-t

Before Collision

r{-

:;.,s

* Aa* J-L

pivot, as shown above. Again assuming the collision is elastic, for what value of xwill the rod stop moving after hitting the ball? Ldk

3KCr^^J 4 nr,u)cat'o"-*J-*t^" g ; r{.#X ;.} '}fa-L *--Ll-Y? *^d*o

St-"&r*.* +(*,e,r*J **&*' &r*nft'q-- k* 4,(L?)'

-b-

{H'j"*lMtt!

After Collision

l'--------- -l- -"'-""""" '""-'-l4. (2OO4M3) A uniform rod of mass M and length L is attached to a pivot of negligiblefriction as shown above. The pivot is located at a distance L/3 from the left end of therod. Express all answers in terms of the given quantities and fundamental constants.a. Calculate the rotational inertia of the rod about the pivot.b. The rod is then released from rest from the horizontal position shown above.

AP Phvsics C 3d 6 Wks Take Home

t-

swings.

b) h{-'q"e'

tA; *

Due rZ / IO l2OLO

gu:- l"t F $u, H $c-

: L iltL+tlL

$ A ** + j[x*d

c. The rod is brought to rest in the vertical position shown above and hangs freely.It is then displaced slightly from this position. Calculate the period of oscillation as it

*Y:=ilqc,F,r,^.s r*#**#*p# & ' V;* U; c- Wt {4f

*bd "(d* C;.-{' &'mtu(- we-*d*# s* W )W, & rag L/,* ) lAf *P

4

o{#i'*** (

^+ k)*r*

f , a-\( {** Je''h't$ + L- i -f *)a*- L(, b F,,-

7rl(A..t e

/;fr\* nl su'jfqrrvlfffi

VL

,fraL i .,t a r*f-eL*-;- x NL cJb df,5i

3sT)1

{iIfFsE

d

{tg

$

N

a

c1

frtJ\

AP Phvsics C 3d 6 Wks Take Home Due 12/ lO/2OlO

2000M2. An explorer plans a mission to place a satellite into a circular orbit aroundthe planetJupiter, which has mass Mt= 1.90 xlOzz kg and radius R.r= 7.14 x 107 m.a. If the radius of the planned orbit is R, use Newton's laws to show each of the

following.i. The orbital speed of the planned satellite is given by nt,r'

{;L

6*9 *&*

F &,"rE Yr* Cs,*l rw" rP#r:;'F{ a

?!'" - ^ tr#d r"id*'tr., P&JS{-Fixu q" Ae#J b HH

nffiafl.€-.#-*ii. The period of the orbit is given by

\v-L = GMa

w

r n. -l**

f=6*"oAFt r;:ti #{"W

;@

ie'!-

r"1

Ler*o*J

l*y:'lf ,,*o%,

4trz Rt'""eY&. Kr?*tr/?'

b. The explorer wants the satellit.', orUftffi synchronized with Jupiter's rotation. This requires an

eouatorial orbit whose oeriod eouals Juoiter's rotation period of t hr 51 min: 3.55 x l0a s. Deter

;{*qrequatorial orbit whose period equals Jupiter's rotation period of t hr 51 min: 3.55 x 10" s. Determine

A{# b.:r lt # lYt*

c. Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity thatdiffers from the desired value in each of the following ways, sketch the resulting orbit on the figure. (J

is the center of Jupiter, the dashed circle is the desired orbit, and P is the injection point.) Also,describe the resulting orbit qualitatively but specifically.

i. When the satellite is at the desired altitude over the equator, its velocityvector has the correct direction, but the speed is slightly faster than thecorrect speed for a circular orbit of that radius.

ii. When the satellite is at the desired altitude over the equator, itsvelocity vector has the conect direction, but the speed is slightlyslower than the correct speed for a circular orbit of that radius.

, f i,E{

n,fL&h,d

the required orbital radius in

s-- i; o ry- - iT'j"**2t-&u:I:#I(:1i,l,'lt('= ---:f ; \--**-" ---7;.r-

efl;Y\tuf *t Ne-

AP Physics C 3d 6 Wks Take Home

2000M3. A light string that is attached to a large block ofmass 4n passes over a pulley with negligible rotational inertiaand is wrapped around a vertical pole of radius r, as shown inExperiment A above. The system is released from rest, and as

the block descends the string unwinds and the vertical polewith its attached apparatus rotates. The apparatus consists of ahorizontal rod of length2L, with a small block of mass m

attached at each end. The rotational inertia of the pole and the

. \* < rodarenegligible.*l Jtlgo'rd a. Determine the rotational inertia of the rod-and-block

{v#nn, b trJ,:#;Jff:'H*"J},'.Ti;llT,:l'l;*. rarge brock

1, L\ c. When the large block has descended a distance D, how

i ),t't [L I do.s the instantaneous total kinetic energy of the three

around the pole to bring the large block back to itsoriginal location. The small blocks are detachedfrom the rod and then suspended from each end ofthe rod, using strings of length /. The system is againreleased from rest so that as the large block descends

and the apparatus rotates, the small blocks swingoutward, as shown in Experiment B above. This timethe downward acceleration of the block decreases

with time after the system is released.d. When the large block has descended a distance

D, how does the instantaneous total kineticenergy of the three blocks compare to that inpart c.? Check the appropriate space below andjustify your answer.

Greater before Equal to before

d\- -Fa "L * *L.r*, a

Due 12 / lO /2OlOt[\," t ;*"r*

a

T5

wJt'" I

d

.-. \i ., ti,r,: Ita 1 ,

+ D - fblocks compare with the value 4zgD ? Check the

.L f appropriate space below andjusti! your answer. ./ Experiment A

\f-' '^, Greaterth',4mgD- Equalto4 ^ro / Lessthan4mgD -

\ -''""-.-, /t" ; "J ^J 5.-,o1, * /,l4z:-": /c-..t' ̂-A" -. -tA l-jf a?r *'? ..Q

Tr-c {**. " u:u / .TT r: '-.,, q *-/

b g*- epp# y- t'p***n.s;d/ *;"e;r1;*+dl3ft-*-r.- $ *t, t"a#x,-.* 5 -€ln-*"4-**I

i** t iovk'fu*g rF de*sp{ee6 r GtaJ { * rlr"B bThe system is now reset. The*string is rewound 2L

-1

Experiment B

Less than before

j,"lt ed$;f

fC*gu-"

,)-T* q.-

=.J-?n "*J

*T "." L{ rva-

.+ ,*u$

"'*r3

1a

i

4 n*fr * ry z- "$ *t4*qt r*t, z] =+,w3el* kn. +

ye,&ff* /^**g*,*sk,* ds{ P:'ffir {**t{* 4**b d#.*tr-g .sw-* CC*

-,r'l4ru+#"* 2*Y

a\cll-

# |e ,* I\t 5* I7*V I

in,tii,f- {;rf #*ff ry rydry& s,r+a$,,€er $Y{J-8--lu; -rU H,&, -### t{*-

'rl -l

r,-radlus r r

bb*{q**

AP Physics C 3d 6 Wks Take Home Due 12 / IO /2OLO-\, \J F-h J; *Vt****slr\

I A f - q$ t@

:'\ d\ 'l*^-y' [a.lr

Yb,::sg# ",H #,i +:*.=l:"*u

3$Tlr -r-r_t-?^<-_D l\Ml ',lr; T, -t-_.o-;%',^ e-6vit; L,ir,."-._ (-1,; ffiffi - -. i G : Jij - *J =JtJ

2000M3. A pulley of radius Rr and rotational inertia 1r is mounted on an axle withnegligible friction. A light cord passing over the pulley has two blocks of mass mattached to either end, as shown above. Assume that the cord does not slip on thepulley. Determine the answers to parts (a) and (b) in terms of m" Rr, 1r, afldfundamental constants.a. Determine the tension T in the cord.b. One block is now removed from the right and hung on the left. When the system is

released from rest, the three blocks on the left accelerate downward with anacceleration g/3 . Determine the following.i. The tension Ts in the section of cord supporting the three blocks on the leftii. The tension Tr in the section of cord supporting the single block on the rightiii. The rotational inertia /r of the pulley

O;) Li,.* *;*i" *1

u,€ , - L)*8-t;z- L^:r t''l$-

**2,6,

: i r,-"'3 + { {rai i9 r, d3-i L*9-

*i*,* te1-J

e tL) L.= f^QzzRb 16llr ; (rct,){;3ia ' f '*l^-*'#Ig1

ffil,,.-, 1ffi lI.F 8s,*4fY#rrtu{rM

' ww#ffil r sprw

c. The blocks are now removed and the cord is tied lnto a loop, which ls passedaround tJre orlginal pulley and a second pulley of radius 2Rr and rotauonal inertia16Ir. The axis ofthe orignal pulley is attached to a motor that rotates it at anqularspeed <rr1, whlch in turn causes the larger pulley to rotate. The loop does not Jipon the pulleys. Determine the following in terms of Ir, Rr, and or.i, The angular speed ro2 of the larger pulteyii. The angular mornentum Ia of the larger pulleyiii'Thetotalkine,ujet"tw,"rti'esystem r .r ..,L= f, f ,-la:

" trrrx?l)a

K-*e - /\" lto- ? Z,u,-+- 7 J.-'z). - ;-)t-t -Ll

3-'} J,# i ?- -e- /''l iz4+)"t^ {

it

.l"rl/l

I

AP Phvsics C 3d 6 Wks Take Home Due 12 / lO /2OrO

1998M2. A space shuttle astronaut in a circular orbit around the Earth has anassembly consisting of two small dense spheres, each of mass m, whose centers areconnected by a rigid rod of length I and negligible mass. The astronaut also has adevice that will launch a small lump of clay of mass m at speed ve . Express youranswers in terms of m, vo l. and fundamental constants.

a-{ c) Li (aL* t*f c,s, &{, } 1;- {.}

mHmtlroI

om

G. et# 1-**'"f.,,,ea$ :**"* *ogi*k;#*

a. Initially, the assembly is "floating" freely at rest relative to the cabin, and theastronaut launches the clay lump so that it perpendicularly strikes and sticks to themidpoint of the rod, as shown above. i. Determine the total kinetic energ/ of the

b. The assembly ls brought to rest, the clay lump removed, and the experiment isrepeated as shown above, with t}re clay lump striking perpendicular to the rod butthls tlme sUcldng to one of tfle spheres of the assembly.i. Determine the distance from the left end of the rod to the center of mass of the

system (assembly and clay lump) immediately after the collision. (Assume thatthe radii of the spheres and clay lump are much smaller than the separauon oftJle spheres.)

ii. On the flgure above, indicate the direction of the motion of the center of massimmediately aff-er the ' t I

collislon. Va< 7,alteJ la {a

iii. Determine the speed of the center of mass immediately after the collision. (g* J.p-e-)iv- Determine the angular speed of the system (assembly and clay tump)immediately aft.er the colliston.

v. Determine the change in kinetic energ/ as a result of the collision,

(,^ ":{riw;*y *-d''*'*e'r

n f WrH)

€tr {lt{;'i "F-u* - -e qf f

d.,*,,!.{"r,** &rr{::bs'r, $* &t tS '*'' 3"4 dF J % xd 'DIs

xlt*sf,

',trfthi"Wl;ffb(*r

l{p - i(:*) v;* g * {€}L Yr"%t

- -**-.-,*.11\

.;- \-* -' r;rY]- 1

ild (,flr+t5 I *E* L s&-"\tb:*,o$

C*t *'Rn

LL* rvr{ti;(bt) = I *r

4L,

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