APPhysics C Rotational Kinematics and Dynamics C Rotational Kinematics and Dynamics OBJECTIVES:...

AP Physics C Rotational Kinematics and Dynamics

OBJECTIVES: Upon the completion of this unit you should be able to:a. Use rotational dynamics variables and be familiar with the vector dircctions of each.b. Apply moments of inertia (rotational inertia) to torque problems.c. Conserve angular momentum.

TEXTBOOK READING:Chapter 12 Rotation of a Rigid BodyReview: 4.5 - 4.7 Angular Kinematics

II. APPROXIMATE SCHEDULEW 10/23 Rotational Kinematics & Rotational Dynamics: Rotational KE and Rotational InertiaTh 10/24 Rotational Inertia of Extended ObjectsF 10/25 Work on book problems (pg 378: 3ab, 7, 11, 13,25,29, 35b,c, 37a,b)

M 10/28 Rotation and TorqueT 10/29 Homework quiz on book problems/Independent practiceW 10/30 Combined Rotation and Translation,Th 10/31 Independcnt practiceF 11/01 No School

M I 1/04 Angular MomentumT I1/0S Homework Quiz: 04,83,89,01,00W 11/06 Indepcndent PracticeTb 11/07 Homework Quiz:, 08, 03, 06, 86, 02F 11/08 Independent Practice

M 11/11 Homework Quiz: 82, 92, 87, OST 11/12 ReviewW 11/13 Rotational Motion Test (toO pt)

III. ZERO HOUR LABSF 10/25 Rolling Motion LabF 1111 NO SCHOOLF 11/8 Spin Torquc Lab

IV. ASSIGNI\IENTS:Ch. 12 Rotational Book problems pg 378: 3ab. 7, 11. 13.25.29, 35b.c. 37a.bHomework Quiz: 04. 83, 89, 0 I. 00Homework Quiz:. 08, 03, 06, 86. 02Homework Quiz: 82.92,87,05

DUE DATES:Mon 11/4

Fri 11115

DUE DATES:T 10/29T 11/05Th 11/07M 11111

Rotational Motion AP problems

TI"tJ

2004M2. A solid disk of unknown mass and known mdius R is used as a pulley in a lab experiment, as shown above. Asmall block of mass m is attached to a string, the other end of which is attached to the pulley and wmpped around itseveral times. The block of mass m is released from rest and takes a time t to fall the distance D to the floor.

a. Calculate the linear accelemtion a of the falling block in tenns of the given quantities.D= \/~o.{'J.. O-:::;:;(j)-6i)..

o (C{(p l--1/ r;l{II V)...I,qoCf

D (In) I (5)

0.5 0.68

t 1,02

~

1.19

1.38

b. The time I is measured for various heights D and the data are recorded in the following table.'20 k.....•..--\

1. What quantities should be graphed in order to best determine the accelemtion of the block? Explain your reasoning.

/J.}) V S (i-

11. On the grid below, ~ot the quantities determined in (b) i.: label theaxes, ~nddmw the best-fit line to the data.DCM) ...•. ~.,.I.. ,. "'_.j &_'" •.•. I~••,...••.••..••••• "

.~~L:L~~-~L~_~t~i:t:~~_..J~ML~t:l::'-'r'.' .:..

3--;".1-- : H" •• ~ ••• , ".~. ". ~~. ~ •••••• + _.- ... ~ ... ' I. --; ..• -l ..~,",' •. _il l,~.; •. ~ .•••._f •. _,..J.. •• ,..l __ I.~ •• _~_ ••...1.___ ._I...~_...•_,.' ..

• ~ :: • I ,; • I I I"""""' '" ••,••••••• '1""""" .- '~~""r ,',-.,...••• "1'-"~"'J-" ,-,.

'.' l •• ,;,." •••••• "" •••••• "" .. ,J ••.•~•.••;...••••,..... •••.I., ••...l ••.•••••• "' •• -'N~"'._ .•• ~ I, };, I It.' 1 1

._.)..~l ....(._'~•..~ "'r'-''1l''''..,''''':-"'~ I-~T.~f'''''~'' -"r .-,4'" ~•..

I.Y

, , ,• '1'--'"'' I__ 1••. '4 ••••• _.

_..i--1"""1-~-"'~'-""~--~--

: ..j.-..; •. ~••. ,..- ~.... ~._.J_..., , ... ...•...~,....,~..... ,..

.1 .• _! .,4, , ,

. , . ,r-'- "-'~""I •.. t--'r"-_...•. _ ...•... _ ... _ ' .. _.M .. "._" •.. ~ .• 4_

.,..~.-....1.~..r-. -~-1-..+-.f.~-i~..;;~. _~_ ••• ~._~_ ••••• 4 •• _"' ••• N~ __

• j • ,.•..,..,,. '"'~ ".,. ...•...•. ,. .. "',. .~. __ ••. ....L ••. J •••.••.•.•••••. _.1_Ll_.J_ ••

, I I, I I I •""~p .•..•••••.•.,» ..•._ .... ",.....r •••••r' .•":.•.••, •. J. ! •.• I •."L .:... . . , ~

~... --_ ...•. --' ....- .;- .-to> 'Of +..

n.t; /o calculate the magnitude of the acceleration.

S(0""j0Q...- -=- 'd. I~IV\-I sL--Usc your gyJiii.

••

Calculate the rotational inertia of tIle pulley in tc.-:.msofm,~, and fundamental constants._ T""r;; = 1:' 0\. (.UC.0 5 - YYla.-") - T 4') 1= -;~q _~

The value of acceleration found in (b)iii, along with numerical values for the given quantities and your answer to (c), can beused to determine the rotational inertia of the pulley. The pulley is removed from its support and its rotational inertia isfound to be greater than.this value. Give one explanation fa this .discrcpanc~ (_ L ~ .tf YIt<.

-{'t\.R.... ~-tv."''1 t.,.A.)a4 -I.?" elL 0-0 ~ ~

J/;~;l~ ,/LO-dAV S

all Rot FR

1983M2. A uniform solid cylinder of mass m, and radius R is mounted on frictionless bearings about a fixed axisthrough O. The moment of inertia of the cylinder about the axis is I = Y2m,R'. A block of mass m,. suspended by acord wrapped around the cylinder as shown above, is released at time t = O.

a. On the diagram below draw and identify all of the forces acting on the cylinder and on the block.

~M l e-?;/ ,!:m 1-Cj \

... \2'" -+ 2.#\ 1- )

...- e.

t'Y\l-CJ b

The angular momentum orthe disk as a function of time l.

ivOf 0v ~<Ato J-t¥\2.-~ \

LJ -.: yYI. -t / lYi 'J,.Jr<-

11. The tension in the cord

t-=Y¥'I,-• iii.

b. In temlS of fit. 012. R. and g, determine each of the following.i. The acceleration of the block

0'-: T 0<Tj1 co J.:[rn /l. ~ T ':'

(J(

d- «h...~='~~o. + :t1Y1~C(

rn, ,"1-')1" . ~

rn I tz.1111.

--Page 6 01'30

"

C 4M

B 3M

A 2M

1989M2. Block A of mass 2M hangs from a cord that passes over a pulley and is conneeted to block B of mass 3Mthat is free to move on a frictionless horizontal surface, as shown above. The pulley is a disk with frictionlessbearings, having a radius R and moment of inertia 3MR'. Block C of mass 4M is on top of block B. The surfacebetween blocks Band C is NOT frictionless. Shortly after the system is released from rest, block A moves with adownward acceleration a, and the two blocks on the table move relative to each other.

In terms of M, g, and a, determine the3. tension T...in the vertical section of the cord

d. acceleration of block C

'F"\( :::'mOl-e:6\-)'~c\.2 ~W CA

If a = 2 meters per second squared, determine the

e. ~:e~of;:ti:f~;;~eenbloc;~d(q'6) _ 5'ry1(l)A~ LJjil{q.%) ::3Til LZ,)\q.lo-10 - )AI<. (3'1.2.) ~ Co

- -M,,-l?''1,'l.) " - 3, ("1Q:: o,~In/s?-

Page II 01'30


m

radius rF

m

Ellpcrimenl A

2001M3. A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertiaand is wrapped around a vertical pole of radius r, as shown in Experiment A above. The system is released from rest, and asthe block descends the string unwinds and the vertical pole with its attached apparatus rotates. The apparatus consists of ahorizontal rod oflength 2L, with asmall block of mass m attadied at each end. The rotational inertia of the pole and the rodare negligible.a. Determine the rotational inertia of the rod-and-block apparatus attached to the top of the pole.

I ~£..iYI (" 1. ;: 'J ii'\ L "1.

b.

c.

Determine the downward acceleration of the large block. rr I -c;;(. I roJ. -=- .1.-l' vL=o - T-t ifrvla -= t.fm C\

~ .:J Tr :: ;;t1Y\ L'l.- -J 1E0L'Zo, -+ 717"Cl -=1r1et- -0. (Ll.--'l.r1.-}"=''lt:3".'l-l//Yl~ r"2- :.J 1 1--- •.. :29(2-

L."l.C\ +- 2rJ..ik- ~( r0.:: zgr \or ,....~ ~ (L1.-'~r~) L

When the large block has descendecra distance D, how does the in~tantaneous total kinetic-energyof t e three blockscompare with the value 4mgp? Check the appropriate space below and justify your answer.

Greater than 4mgD .... Equal to 4mgD Less than 4mgD _ .

~.d) 0 ~

The system is now reset. The string is rewound around the pole to bring the large block back to its original location. Thesmall blocks are detached from the rod and then suspended from each end of the rod, using strings of length I. The system isagain released from rest so that as the large block descends and the apparatus rotates, the small blocks swing outward, asshown in Experiment B above. This time the downward acceleration of the block decreases with time after the system isreleased.d. When the large block has descended a distance D, how does the instantaneous total kinetic energy of the three blocks

compare to that in part c.? Check the appropriate space below and justify your answer.

Greater before Equal to before _._ Less than before ~

cao;." +\l ~-{Y\...L


~2000M3. A pulley of radius R, and rotational inertia I, is mounted on an axle with negligible friction. A light cord passingover the pulley has two blocks of mass m attached 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 ofm, R" 1" and fundamental constants.a. Detenninc 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 onthe left accelerate downward with an acceleration g/3 . Determine the following. 0. ~ Visi. The tensio~ ..TJ in the section of co~ supporting the thre~blocks on the left _

3~ - 1$ -:: ~rn( (jIg) T3 " 310"1(3 - rY"\ 9 - 2nr1~

iii. The rotational inertia II oCthe pulley

~ry =:- I C{ 'IJ~I -1\ rei

eft =.2l " 2L(,l" ~,

~h'I~e,-~~~)Se(2 ~ 2. -

!rJ fr\ f-, - 2 rY"\ f2. I J\ ~(Y\(2.:-- ;":r J

-Z•

The blocks are now removed and the cord is tied into a loop, which is passed around the original pulley and a secondpulley of radius 2R1 nnd rotational inertia 1611• The axis of the original pulley is attached to a motor that rotates it atangular speed 00], which in tum causes the larger pulley to rotate. The loop does not slip on the pulleys. Determine thefollowing in terms of II. Rio and COl_

i. The angular speed "'2 of the larger pulleyV I :::.V'l.. .f)!

()J11-,-:=: W .•.~r~\ii. The angular momentum L2 of tile larger pulley

:: {CD:I , (!If)

c.

1- tuJ£.z.- ( \


Hinge 2.0 kg./

0.60m---10.50 kg

2008M2. The horizontal unifonn rod shown above has length 0.60 m and mass 2.0 kg. The left end of the rod isattached to a vertical support by a fiictionless hinge that allows the rod to swing up or down. The right end of therod is supported by a cord that makes an angle onoo with the rod. A spring scale of negligible mass measuresthe tension in the cord. A 0.50 kg block is arso attached to the right end ofthe rod.

(a) On the diagram below, draw and label vectors to represt.'l1tall the forces acting on the rod. Show each forcevector originating at its point of application.

(b) Calculate the reading on the spring scale. 1Y1~ tYI~ l\ I .'\JI-::O 'F1"(.(,,~-;: (l,}C9.B)C3) + S)[q)?)l'~J

Sio1yf 5. <&1 t d .'1C{ eEl 0- d- 9, YAi :((c) The rotational inertia of a rod about its center is J'o ML', where M is the mass of the rod and L is its length.Calculate the rotational inertia of the rod-block system about the hinge.-r. :::TeM + tYlL'l +- mIt. -3 [2.)( .G)2+ (.5X.foyZ.

/ -;:tt n'll.. 'Z- + '((1 ~ 2. -+ f{\ IIL2. -, 2. <.j -\- • 1 IS -=CO-, l{-2-~5-,.,,-1...-Jlrf\ L'- 4- ((\'0 \.."1-3

(d) If the cord that supports the rod is cut nc<'lI' the md of the rod, calculate the initial angular acceleration of therod-block system about the hinge.

-£!1~:E-~C~)(q,'G)~:,)+-Cts )C'l.cc.)(,<e) == .4?.- 0(

S. ~ t c? .'1Y -=. 'i L<:><-d.-t (Ac{/<;'


,0J",\ A---'------

'-.•...-:

"

\.

~._--.---.

1000!)flO(.00200

=r=r= . 'T, -__ ,I --" '.' --_._- --

-- "-"" '-'1"._..

I -- --- -".__ ..~-' .._---

I --"- ..., ..'_._._- -----_ .-.. ., '-

.._.~- ,__ 1-...._--_ . ._- - - ..._-_ •. .-~-_.- -1------- .._. __ ._ ....... .._ ..•..•. _._ .. -•• _n' .•... -'-'."-'T --

J--I------r, , ,

()

III

", 20

-. 30S

40

:'ri)

h~~l A~1l,..1

2003M3. Some physics students build a catapult, as shown above. The supporting platform is fixed fimlly to the ground.The projectile, of mass 10 kg, is placed in cup A at one end of the rotating arm. A counterweight bucket B that is to beloaded with various masses greater than 10 kg is located at the other end of the arm. The arm is released from thehorizontal position, shown in Figure I, and begins rotating. There is a mechanism (not shown) that stops the ann in thevertical position, allowing the projectile to be launched with a horizontal velocity as shown in Figure 2.

a. The students load five different masses in the counterweight bucket, release the catapult, and measure the resultingdistance x traveled by the 10 kg projectile, recordin the followin data.

Mass (k) 100 300 500 700x(m) 18 37 45 48 51

I. The data are plotted on the axes below. Sketch a best-fit curve for these data points .

M••• (k~)

II. Using your best-fit curve, detemline the distance x traveled by the projectile if250 kg is placed in the counterweightbucket.

b. The students assume Ihat the mass of the rotating arm, the cup, and the counterweight bucket can be neglected. With thisassumption, they develop a theoretical model for x as a function of the counterweight mass using the relationship.r = ~\t,where 1.', is the horizontal velocity of the projectile as it leaves the cup and I is the time after launch.i. How many seconds after leaving the cup will the projectile strike the ground?

11. Derive the equation that describes the gravitational potential energy of the system relative to the ground when in theposition shown in Figure I, assuming the mass in the counterweight bucket is AI.

111. Derive the equation for the velocity of the projectile as it leaves the cup, as shown in Figure 2.

C. I. Complete the theoretical model by writing the relationship for x as a function of the counterweight mass using (heresults from (b) i and (b) iii.

II. Compare the experimental and theoretical values of x for a counterv,,'eight bucket mass of 300 kg. Offer a reason forany difference.


TH,

•

c.

2006M3. A thin hoop of mass M, radius R, and rotational inertia MR' is released from rest from the top of the ramp ofleugth L above. The ramp makes an angle 0 with respect to a horizontal tabletop to which the ramp is fixed. The table is aheightH abovc the 110or.Assume that the hoop rolls without slipping down the ramp and across the table. Express allalgebraic answers in terms of given quantities and fundamental constants.r-

I

a. Derive an expression for the acceleration ofll1C center of mass of the hoop as it rolls down the ramp. L.

C\.CM -=. d- I' fY\g~ lAG - £' :=c. t'YIOCYV' ry-=I""" J'rz =. m~C\ 01. f'I11,'t,,,,,,yY\I051",G -,tYI,ac .•-=- M,a I"t\ • j'f-"«If2. ~ oS

J /'I n. _ "$I,,,e95\'1\ &"' 6l LACM ""'O'\'\ - ?-

b. Derive an expression for the spced of the center of maOf.-ft 1e oop w en 1 reac es the boltom of the ramp.

rY'<3h z Y-z-.I'Y\V 1.- +- '/"2- T W 1- \Ie~1. ~ I/C/:" f ~a. ~ ~

~i~L~I-&~ -!-z.,~'\/ ""," t- J. II"~ ~~ 'J 1.- ");;l (~IVI ) t-. '2. " I'" !' 2-

a LS,./\ e ~V t1i"L' .a.. , _0' I '" 5"";1\9J V -=- I~gsI(\V J - If.,;:)Denve an expression for the horizontal distance from the edge of the table to where the hoop lands on the floor.

ts~";V Y. I ~ H ~ 11.g{~ -t '-1~::.TL.8{'i~(J-1-~-.- ~TJiI-LSI'r16

BricOy justify your response.

d. Suppose that the hoop is now replaced by a disk having the same mass M and radius R. How will the distance from the edgeof the table to where the disk lands on the floor cOlnpare with the distance detenilined iii. part (c) for the hoop?

Less than __ Thc some as ~ter than I ~ L." J. noop..5:> mffL "rfi\ cJ- h4 t/.-n"~ y'~ SDJeJD ~(yM'f\j1 t--V) ~f~M~

all Rot FR

1986M2. An inclined plane makes an angle of8 with the horizontal, as shown above. A solid sphere of radius Rand mass M is initially at rest in the position shown, such that the lowest point of the sphere is a vertical height Itabove the base of the plane. The sphere is released and rolls down the plane without slipping. The moment of inertiaof the sphere about an axis through its center is 2MR'/5. Express your answers in terms ofM, R.It, g, and 8.a. Determine the following for the sphere when it is at the bOllom of the plane:

i. Its translational kinetic energy

ii. Its rotational kinetic energy

b. Determine the following for the sphere when it is on the plane.I. Its linear acceleration

ii. The magnitude orthe frictional force acting on it

The solid sphere is replaced by a hollow sphere of identical radius R and mass M. The hollow sphere, which isreleased from the same location as the solid sphere, rolls down the incline without slipping.c. What is the total kinetic energy of the hollow sphere at the bottol1l of the plane?

d. State whether the rotational kinetic energy of the hollow sphere is greater than, less than, or equal to that ofthe solid sphere at the bottom orthe plane. Justify your answer

Page 8 of30


TI".1

2002M2. The cart shown above is made of a block of mass m and four solid rubber tires each of mass m/4 and radius r.Each tire may be considered to be a disk. (A disk has rotational inertia y, MLl, where M is the mass and L is the radius ofthe disk.) The cart is released from rest and rolls without slipping from the top of an inclined plane of height h. Express allalgebraic answers in terms of the given quantities and fundamental constants.a. Determine the total rotational inertia of all four tires.

b. Determine the speed of the cart when it reaches the bottom of the incline.

c. After rolling down the incline and across the horizontal surface, the cart collides with a bumper of negligible massattached to an ideal spring, which has a spring constant k. Determine the distance Xm the spring is compressed beforethe cart and bumper come to rest.

d. Now assume that the bumper has a non-negligible mass. After the collision with the bumper, the spring is compressedto a maximum distance of about 90% of the value ofxm in part (c). Give a reasonable explanation for this decrease.

all Rot FR

f-L 21-i2m ,p

IIIIII

1982M3. A system consists nftwo small disks, of masses m and 2m, attached to a rod of negligible mass oflength3/ as shown above. The rod is free to turn about a vertical axis through point P. The two disks rest on a roughhorizontal surface; the coefficient of friction between the disks and the surface is ~. At time t = 0, the rod has aninitial counterclockwise angular velocity 00, about P. The system is gradually brought to rest by friction. Developexpressions for the following quantities in terms of ~ m, /, g. and rooa. The initial angular momentum of the system about the axis through P

b. The frictional torque acting on the system about the axis through P

c. The time T at which the system will come to rest.

Page 5 of30

all Rot FR

.l:' Axis MBug ~>--- •

3M ~I'------ 2£ .,1992M2. Two identical spheres, each of mass M and negligible radius, arc fastened to opposite ends of a rod ofnegligible mass and length 21. This system is initially at rest with the rod horizontal, as shown above, and is free torotate about a frictionless, horizontal axis through the center of the rod and perpendicular to the plane of the page. Abug, of mass 3M, lands gently on the sphere on the left. Assume that the size of the bug is small compared to thelength of the rod. Express your answers to all parts oCtile qllestion in terms orM, I, and physical constants.a. Determine the torque about the axis immediately after the bug lands on the sphere.

t 'Y -= Y fYljt - rY'1- :: 3rrg e

!~', ,,,,,,

\\I

AxisI\'\\,,,,,,

',,- M--

Determine the angular acceleration of the rod-spheres-bug system immediately after the bug lands. Z ~

<f '/ To<.. 7. :;)fYI~ £ML

3JrSt ~ t(~ 1.-- 0( ~ :lfr1 e-

-ll- ~D(

b.

3M

The angular momentum of the system

The rod-sphcres-bug system swings about the axis. At the instant that the rod is vCI1ical, as shown above,determine each of the following.

e. TheMq~r ~ee~gt~e SugJ J W 1-

'-lrf\9f' + .y(~l-:;::~~ -;;.X +--kd.

e.

L-<I'w ~~ 5'rf\Q}- Iit.

The magnitude and direction of~e lorce that must be exerted on the bug by the sphere to keep the bug frombeing thrown off the sphere

Page 14 of30

all Rot FR

BefortCollision

3kg

"g•-10 mfs

lIIII

//

I//

//'

/

~~-------

AfterCollision

,Note: You may use J:" 10 m/s~.

1987M3. A l.O-kilogram object is moving horizontally with a velocity of 10 meters per second, as shown above, whenit makes a glancing collision with the lower end of a bar that was hanging vertically at rest before the collision. Forthe system consisting of the object and bar, linear momentum is not conserved in this collision, but kinetic energy isconserved. The bar, which has a length I of 1.2 meters and a mass m of 3.0 kilograms, is pivoted about the upperend. Immediately after the collision the object moves with speed v at an angle e relative to its original direction.The bar swings freely, and after the collision reaches a maximum angle of 90° with respect to the vertical. Themoment of inertia of the bar about the pivot is Ibar::: mPI3 Ignore all friction.a. Detennine the angular velocity of the bar immediately after the collision.

b. Detemline the speed v of the I-kilogram object immediately after the collision.

c. Determine the magnitude of the angular momentum of the object about the pivot just before the collision.

d. Detenninc the angle 8.

Page 9 01'30


All •• Collisi""

/

j~1

L ~ :; L.('

TO? VIEWS2005M3. A system consists of a ball of mass M, and a unifonn rod of mass MI and length d. The rod is attached to ahorizontal frictionless table by a pivot at point P and initially rotates at an angular speed Cll, as shown above left. The

rotational inertia of the rod about point P is .!. Mid' . The rod strikes the ball, which is initially at rest. As a result of this3

collision, the rod is stopped and the ball moves in the direction shown above rig~. Express all answers in tenns of MI, M" W,

d, and fundamental constants. - -a. Derive an expression for the angular momentum of the rod about point P before the collision.

L -= T(j) -

b. Derive an expression for the speed l' of the ball after the collision.

-j, YY\ .& ZtAj -:- 1112 vi/

c. Assuming that this collision is elastic, calculate the numerical value of the ratioMJ / M]

~ 3Vl.-YY\, :---- - duJ/YIv

Bdhrc: Colli,.;on

I

d. A new ball wIth the same mass j\1j as the rod IS now placed a distance x from the pivot, as shown above. Again assuming ]...

the C~10:/Js;c;;;r w:a~ 7;111 the rod stop moving after hitting the ball? i ~ ~V 1-:;. t (t t<t.')lJ3 IY' Ii I ~ \j ?= ~o,\,.')l.-J-- d2w - ~l'i.! X-;; r3 o~ 3,

;; '1--.,1 ---

~d~'~1~4::--'i-~ f-I

rf1Ldi iJ- +0 IfjIIt Advanced Physics Rotational Motion Tcst Name: _

2. Which oCthe live lettcred points represents the centeroCmass oCthe sphere-rod combination'!(A) A (11)B (C) C (D) D (E) E

I. A turntable that is initially at rest is sct in motion witha constant angular acceleration a. What is the angularvelocity oCthe turntable after it has made one compkterevolution?

A C D E10kg0 ~ .. 85kg

A 5-kilogram sphere is connected to a IO-kilogramsphere by a rigid rod oCnegligible mass, as shownabove.

g(E) I

5g(0) 71

2F

,.F

g(C) 41

g(11) 51

g(A) 71

F

6. A system of two wheels fixed to each other is free torotate about a frictionless axis through the commoneemer oCthe wheels and perpendicular to the page.Four forces are exerted tangentially to the rims of thewheels, as shown above. The magnitude of the nettorque on the system aboutlhe axis is(A) zero (11) foR (C) 2FR (0) 5FR (E) 14FR

1--.£ - •••..-- 2£----<• r •M l..Pivot M3 0 Point n

5. A light rigid rod with masses attached to its ends ispivoled about a horizontal axis as shown above. Whenrekased Cromrest in a horizontal orientation, the rodbegins to rotate with an angular acceleration ofmagnitude

(C) .J4JW

(E)4JW

Questions 2 - 3

(0) 2a

(A) .ha (11) .J2JW

3. The sphere-rod combination can be pivoted about anaxis that is perpendicular to the plane oCthe page andthat passes through onc oCthe live lettered points.Through which point should the axis pass Corthcmoment of inertia oCthe sphere-rod combination aboutthis axis to be greatest?(A) A (B) 11 (C) C (D) D (E) E

4. A uniConn stick has length L. The moment of inertiaabout the center of the stick is I". A particle of mass Mis attached to one end of the stick. The moment oCinertia oCthe combined system aboutlhe center oCthestick is

Kinetic EnergvIncreasesRemains ConstantIncreasesRemains ConstantRemains Constant

(A) 10 +}.,,\lL"4

(D) 10 +elf!."

I ,(Il) lo+-ML-2

(1-) 5 ,: 'n +-AJL-4

(C) 3 ,10+-AIL-

4

7. An ice skater is spinning about a vertical axis withanns Cullvextended. ICthe arn1Sare pulled in closer tothe body,~in which oCthe following ways are theangular momentum and kinetic energy oCthe skaterafleetcd'!Angular Momentum(A) Increascs(11) Incrcases(C) Rcmains Constant(D) Remains Constant(E) Dcereases

(B) 2 kg m'/s (C) 3 kg m'/s

(E) It cannot be detennined withoutknowing the radius of the cylinder.

2009 Advanced Physics Rotational Motion Test

Questions 8-10A cylinder rotates with constant angular accelerationabout a fixed axis. The cvlinder's moment of inertiaabout the axis is 4 kg m': At time t = 0 the cylinder isat rest. At time t = 2 seconds its angular velocity is1 radian per second.

8. What is the angular acceleration of the cylinderbetween t = 0 and t = 2 seconds?(A) 0.5 radian/s' (13) I radian/s2 (C) 2 radian/s'

(D) 4 radian/s' (E) 5 radian/s'

9, What is the angular momentum of the cylinder at timet = 2 seconds?(A) I kg m'/s

(D) 4 kg m'/5

10, What is the kinetic energy of the cylinder at time t = 2seconds?

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12. A particle of mass m moves with a constant speed valong the dashed line y = a. When the x-coordinate ofthe particle is x"' the magnitude of the angularmomentum of the particle with respect to the origin ofthe system is(A) zero (B) mm (e) mvx.

(D) r",)"o' + ,,'

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(E) cannot be detennined without knowing the radiusof the cylinder

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APPhysics C Rotational Kinematics and Dynamics C Rotational Kinematics and Dynamics OBJECTIVES:...

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Transcript of APPhysics C Rotational Kinematics and Dynamics C Rotational Kinematics and Dynamics OBJECTIVES:...