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 Body
Review: 4.5 - 4.7 Angular Kinematics
II. APPROXIMATE
SCHEDULE
W
10/23 Rotational Kinematics & Rotational Dynamics: Rotational KE and Rotational Inertia
Th
10/24 Rotational Inertia of Extended Objects
F
10/25 Work on book problems (pg 378: 3ab, 7, 11, 13,25,29, 35b,c, 37a,b)
M
T
W
Th
F
10/28
10/29
10/30
10/31
11/01
Rotation and Torque
Homework quiz on book problems/Independent
Combined Rotation and Translation,
Independcnt practice
No School
M
T
W
Tb
F
I 1/04
I1/0S
11/06
11/07
11/08
Angular Momentum
Homework Quiz: 04,83,89,01,00
Indepcndent Practice
Homework Quiz:, 08, 03, 06, 86, 02
Independent Practice
M
T
W
11/11 Homework Quiz: 82, 92, 87, OS
11/12 Review
11/13 Rotational Motion Test (toO pt)
practice
III. ZERO HOUR LABS
F 10/25
Rolling Motion Lab
F 1111
NO SCHOOL
F 11/8
Spin Torquc Lab
IV. ASSIGNI\IENTS:
Ch. 12 Rotational Book problems pg 378: 3ab. 7, 11. 13.25.29,
Homework Quiz: 04. 83, 89, 0 I. 00
Homework Quiz:. 08, 03, 06, 86. 02
Homework Quiz: 82.92,87,05
DUE DATES:
Mon 11/4
Fri 11115
35b.c. 37a.b
DUE DATES:
T 10/29
T 11/05
Th 11/07
M 11111
Rotational Motion AP problems
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2004M2. A solid disk of unknown mass and known mdius R is used as a pulley in a lab experiment, as shown above. A
small block of mass m is attached to a string, the other end of which is attached to the pulley and wmpped around it
several 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)..
b. The time
measured for various heights D and the data are recorded in the following table.
k.....•..
D (In)
I (5)
I is
-'20
\
0.68
t
1,02
1/ r;l{
1.19
II
1.38
I,qoCf
~
1.
V)...
What quantities should be graphed in order to best determine the accelemtion of the block? Explain your reasoning.
/J.})
11.
o (C{(p l--
0.5
V
(i-
S
On the grid below, ~ot the quantities determined in (b)
DCM)
...•.
~.,.I
..
"'_.j
&_'"
~nddmw the best-fit line to the data.
I~ ••, ...••.••..•••••
•.•.
.~~L:L~~-~L~
_~t~i:t:~~_
..J~ML~t:l::'-'r'.'
--;".
1-- :
+ _.- ... ~ ... ' I.--; ..• -l ..~
H"
3
,.
i.: label theaxes,
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..... ,..
.... ~._.J_...
.: ..
I.Y
, , ,
.1 .• _!
.,4
/
-=-
'd.
I~IV\-I
Calculate the rotational inertia of tIle pulley in tc.-:.msofm,~,
(.UC.0
s L-and fundamental constants._
T""
5 - YYla.-") - T 4') 1= -;~q _~
The value of acceleration found in (b)iii, along with numerical values for the given
used to determine the rotational inertia of the pulley. The pulley is removed from
found to be greater than.this value. Give one explanation fa this .discrcpanc~
-{'t\.R....
t.,.A.)a4
elL
0-0
~-tv."''1
J/;~;l~
_.
o calculate the magnitude of the acceleration.
S(0""j0Q...r;; = 1:' 0\.
•• _"'
•
".,.
~. __ ••. ....L ••. J •••.••.•.•••••.
I
I,
""~p, .•..•••••.•.
, •.
4
,/LO-dAV S
-I.?"
quantities and your answer to (c), can be
its support and its rotational inertia is
(_ L
~
.tf
~
~
YIt<.
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a.
1983M2. A uniform solid cylinder of mass m, and radius R is mounted on frictionless bearings about a fixed axis
through O. The moment of inertia of the cylinder about the axis is I = Y2m,R'. A block of mass m,. suspended by a
cord wrapped around the cylinder as shown above, is released at time t = O.
On the diagram below draw and identify all of the forces acting on the cylinder and on the block.
b.
In temlS of fit. 012. R. and g, determine each of the following.
i. The acceleration of the block
0'-: T 0<
T j1 co J.:[rn /l. ~
T ':'
(J(
d- «h...
~ ='~~o. +
11.
The tension in the cord
rn, ,"1-')1"
t-=Y¥'I,
-
•
:t1Y1~C(
. ~
rn I tz.1111.
iii. The angular momentum orthe disk as a function of time
l.
ivOf
~M
l
e- ,!:m
?;/
... \2'"
...
-
1- Cj
LJ
\
o
-.:
J-t¥\2.-~ \
yYI. -t / lYi
r<-
-+ 2.#\ 1- )
e.
t'Y\l-CJ
0v ~ <At
b
-Page 6 01'30
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B
4M
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 3M
that is free to move on a frictionless horizontal surface, as shown above. The pulley is a disk with frictionless
bearings, having a radius R and moment of inertia 3MR'. Block C of mass 4M is on top of block B. The surface
between blocks Band C is NOT frictionless. Shortly after the system is released from rest, block A moves with a
downward acceleration a, and the two blocks on the table move relative to each other.
In terms of M, g, and a, determine the
3.
tension T...in the vertical section of the cord
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,)
1
\q.lo-10 - )AI<. (3'1.2.) ~ Co
- -M,,-l?''1,'l.) " - 3,("
d.
acceleration
of block C
'F"\( :::'mOl-
e: 6\-)'~c\. 2 ~W CA
Q::
o,~In/s?
-
Page II 01'30
Rotational
Motion AP problems
m
m
radius rF
Ellpcrimenl A
2001M3. A light string that is attached to a large block of mass 4m passes over a pulley with negligible rotational inertia
and is wrapped around a vertical pole of radius r, as shown in Experiment A above. The system is released from rest, and as
the block descends the string unwinds and the vertical pole with its attached apparatus rotates. The apparatus consists of a
horizontal rod oflength 2L, with asmall block of mass m attadied at each end. The rotational inertia of the pole and the rod
are 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.
rr
. I roJ. -=-
b. Determine the downward acceleration of the large block.
~
l'
T-t ifrvla.:J -= t.fm
1E0L'Zo, -+ 717"Cl -=1r1et-
vL=o
J l//Yl~
-
r"2-
:.J
L."l.C\
~
c.
+- 2rJ..ik-
I .1.-
c;;(
C\
Tr
1
:: ;;t1Y\ L'l.- -0. (Ll.--'l.r1.-}"=''lt:3".'l-
~(
1---
•..
When the large block has descendecra distance D, how does the in~tantaneous total kinetic-energyof
compare with the value 4mgp? Check the appropriate space below and justify your answer.
Greater than 4mgD
....
Equal to 4mgD
Less than 4mgD
~.d)
\or ,....
r0.:: (L1.-'~r~)
zgr
~
_
:29(2L
t e three blocks
.
0 ~
The system is now reset. The string is rewound around the pole to bring the large block back to its original location. The
small blocks are detached from the rod and then suspended from each end of the rod, using strings of length I. The system is
again released from rest so that as the large block descends and the apparatus rotates, the small blocks swing outward, as
shown in Experiment B above. This time the 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 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
_._
cao;."
Less than before
+\l
~-{Y\...L
~
Rotational Motion AP problems
~
2000M3. A pulley of radius R, and rotational inertia I, is mounted on an axle with negligible friction. A light cord passing
over the pulley has two blocks of mass m attached to either end, as shown above. Assume that the cord does not slip on the
pulley. 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 on
the left accelerate downward with an acceleration g/3 . Determine the following.
0. ~
i. The tensio~ ..TJ in the section of co~ supporting the thre~blocks on the left
_
Vis
3~
- 1$ -::
~rn( (jIg)
iii. The rotational inertia II oCthe pulley
~ry =:- I
eft
'I
C{
J
~I
T3
" 310"1(3 - rY"\
9 -
-1\ rei
2nr1~
~h'I~e,-~~~)Se(
=.2l " 2L
(,l"
2
!rJ fr\ f-,
~,
\
c.
-
~
2 rY"\ f2.
~(Y\(2.:--
2.
I
-
J
;":r J
The blocks are now removed and the cord is tied into a loop, which is passed around the original pulley and a second
pulley of radius 2R1 nnd rotational inertia 1611• The axis of the original pulley is attached to a motor that rotates it at
angular speed 00], which in tum causes the larger pulley to rotate. The loop does not slip on the pulleys. Determine the
following in terms of II. Rio and COl_
i. The angular speed
of the larger pulley
"'2
V I :::. V'l..
.f)!
()J11-,-:=: W .•.~r~\
ii. The angular momentum
L2 of tile larger pulley
:: {CD:I ,
1-z.- tuJ£.
( \
(!If)
Z
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Rotational
Motion AP problems
2.0 kg./
Hinge
0.50 kg
0.60m---1
2008M2. The horizontal unifonn rod shown above has length 0.60 m and mass 2.0 kg. The left end of the rod is
attached to a vertical support by a fiictionless hinge that allows the rod to swing up or down. The right end of the
rod is supported by a cord that makes an angle onoo with the rod. A spring scale of negligible mass measures
the 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 force
vector originating at its point of application.
(b) Calculate
the reading on the spring scale.
'F1"(.(,,~-;: (l,}C9.B)C3)
JI-::O
5. <&1
Sio1yf
tYI~ l\ I
1Y1~
t
+
.'\
S)[q)?)l'~J
eEl
d .'1 C{
0-
d- 9, Y Ai :(
(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
-;:tt
n'll.. 'Z- + '((1
-3 [2.)( .G)2 + (.5X.foyZ.
+- mIt.
~ 2.
-+
L
f{\ II 2.
2. <.j
-,
-\-
•
1 IS -=CO-, l{-2-~5-,.,,-1...-J
lrf\ L'- 4- ((\'0 \.."1-
3
(d) If the cord that supports the rod is cut
rod-block system about the hinge.
nc<'lI'
the md of the rod, calculate the initial angular acceleration of the
-£!1~:E-~
C~)(q,'G)~:,)+-Cts )C'l.cc.)(,<e)
S. ~ t c? .'1Y -=. 'i L<:><-
==
.4?.- 0(
d.-t (Ac{/<;'
Rotational Motion AP problems
"
.•...'--:
\.
,0
J",\
A
---'------
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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 be
loaded with various masses greater than 10 kg is located at the other end of the arm. The arm is released from the
horizontal position, shown in Figure I, and begins rotating. There is a mechanism (not shown) that stops the ann in the
vertical 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 resulting
distance x traveled by the 10 kg projectile, recordin the followin data.
Mass (k)
100 300 500 700
x(m)
18
37
45
48
51
I. The data are plotted on the axes below. Sketch a best-fit curve for these data points .
:'ri)
40
-. 30
S
", 20
III
=r=r=
-_._- --- '-'1"
__
__
".
.._---
..
__--_•..•... _. .
-'-'."-'T
,
'T
.-~-_.-
--
I
I
- ,__
,
(.00
200
-- --- -
--" '.'
._ ..
..~-'
_.~-
..
....
-
,I
"-""
•• _n'
()
,.
--".,
1-
... ,
'-
-1-------
..'_._._- -----_ . -
.._. __ ._ .......
,J
!)flO
- ..._-
.._ ..•..•. _._ ..
--I------r
1000
M••• (k~)
II.
b.
C.
Using your best-fit curve, detemline the distance x traveled by the projectile if250 kg is placed in the counterweight
bucket.
The students assume Ihat the mass of the rotating arm, the cup, and the counterweight bucket can be neglected. With this
assumption, 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 the
position 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.
I.
Complete the theoretical model by writing the relationship for x as a function of the counterweight mass using (he
results 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 for
any difference.
Rotational Motion AP problems
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2006M3. A thin hoop of mass M, radius R, and rotational inertia MR' is released from rest from the top of the ramp of
leugth L above. The ramp makes an angle 0 with respect to a horizontal tabletop to which the ramp is fixed. The table is a
heightH abovc the 110or.Assume that the hoop rolls without slipping down the ramp and across the table. Express all
algebraic
answers
in terms of given quantities and fundamental
r-
constants.
I
a.
Derive an expression for the acceleration ofll1C center of mass of the hoop as it rolls down the ramp.
C\.CM
£'
-=. d- I'
J
95\'1\ &"' 6l
b.
/'I
LACM
n.
W
1-
\Ie~1.
~i~L~I-&~ -!-z.,~'\/ ""," t- J. II"~ ~~
.
'2.
"
I'"
aJ LS,./\ e ~ V
t1i"L'
.a..
V -=- I~gsI (\ V
Denve
an expression
ts~";V Y.
for the horizontal
distance
I ~
::.TL.8{'i~(J-1-~-.d.
_
""'O'\'\ -
•
?-
Derive an expression for the spced of the center of maOf.-ft 1e oop w en
rY'<3h z Y-z-.I'Y\V 1.- +- '/"2- T
c.
ry-=I"""
fY\g~ lAG :=c. t'YIOCYV'
yY\I051",G -,tYI,ac .•-=- M,a I"t\
~
'J
1
~
"$I,,,e
J'rz
=. m~C\ 01.
j'f-"«If2. ~
I/C/:" f
~a.
~ ~
, _ 0' I
-
f'I11,'t,,,,,,
t-
)
2-
!'
J
oS
reac es the boltom of the ramp.
");;l (~IVI
1.-
L.
'"
If.,;:)
5"";1\9
from the edge of the table to where the hoop lands on the floor.
H ~ 11. g{~ -t '-1~
TJiI-LSI'r16
Suppose that the hoop is now replaced by a disk having the same mass M and radius R. How will the distance from the edge
of the table to where the disk lands on the floor cOlnpare with the distance detenilined iii. part (c) for the hoop?
Less than
BricOy justify
__
your response.
Thc some as
~ter
than
I ~ L." J. noop.
.5:> mffL "rfi\
~
cJ- h4 t/.-n"
SD
y'~
JeJD ~(
y M'f\j1
t--V)
M~
~f~
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1986M2. An inclined plane makes an angle of8 with the horizontal, as shown above. A solid sphere of radius R
and mass M is initially at rest in the position shown, such that the lowest point of the sphere is a vertical height It
above the base of the plane. The sphere is released and rolls down the plane without slipping. The moment of inertia
of 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.
I.
Determine the following for the sphere when it is on the plane.
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 is
released
c.
from the same location as the solid sphere, rolls down the incline without slipping.
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 of
the solid sphere at the bottom orthe plane. Justify your answer
Page 8 of30
Rotational Motion AP problems
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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 of
the disk.) The cart is released from rest and rolls without slipping from the top of an inclined plane of height h. Express all
algebraic 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 mass
attached to an ideal spring, which has a spring constant k.
Determine
the distance
Xm
the spring is compressed
before
the 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 compressed
to 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
2m
21-i
,p
I
I
I
I
I
I
1982M3. A system consists nftwo small disks, of masses m and 2m, attached to a rod of negligible mass
3/ as shown above. The rod is free to turn about a vertical axis through point P. The two disks rest on a
horizontal surface; the coefficient of friction between the disks and the surface is ~. At time t = 0, the rod
initial counterclockwise angular velocity 00, about P. The system is gradually brought to rest by friction.
expressions for the following quantities in terms of ~ m, /, g. and roo
a. 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
oflength
rough
has an
Develop
all Rot FR
.l:'
Axis
•M
3M ~I'------
2£
.,
Bug ~>---
1992M2. Two identical spheres, each of mass M and negligible radius, arc fastened to opposite ends of a rod of
negligible mass and length 21. This system is initially at rest with the rod horizontal, as shown above, and is free to
rotate about a frictionless, horizontal axis through the center of the rod and perpendicular to the plane of the page. A
bug, of mass 3M, lands gently on the sphere on the left. Assume that the size of the bug is small compared to the
length 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- :: 3 rrg
b.
e
Determine the angular acceleration of the rod-spheres-bug system immediately after the bug lands.
<f '/ To<..
3JrSt ~ t(~ 1.--
-ll-
7.
Z
:;)fYI~ £ML
~ :lfr1 e-
0(
~D(
~',
,,
,,
,,
I
\'
\
\
Axis
,,
,,
,
,
--
',,- M
The rod-sphcres-bug
system swings about the axis.
3M
\
\
I
!
At the instant that the rod is vCI1ical, as shown above,
determine each of the following.
e.
TheMq~r
~ee~gt~e
'-lrf\9f' +
d.
g
Su
J
J W 1-
.y(~l-:;::~~ -;;.X
The angular momentum
+--k
of the system
L-<I'w ~
~ 5'rf\ Q}- Iit.
e.
The magnitude
and direction
of~e
lorce that must be exerted on the bug by the sphere to keep the bug from
being thrown off the sphere
Page 14 of30
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all Rot FR
l
Befort
Collision
I
I
After
Collision
I
I
3kg
I
/
/
/
/
/'
/
/
-•
"g
~~-------
10 mfs
,
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, when
it makes a glancing collision with the lower end of a bar that was hanging vertically at rest before the collision. For
the system consisting of the object and bar, linear momentum is not conserved in this collision, but kinetic energy is
conserved. The bar, which has a length I of 1.2 meters and a mass m of 3.0 kilograms, is pivoted about the upper
end. 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. The
moment 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
Rotational Motion AP problems
All •• Collisi""
TO? VIEWS
2005M3. A system consists of a ball of mass M, and a unifonn rod of mass MI and length d. The rod is attached to a
horizontal frictionless table by a pivot at point P and initially rotates at an angular speed Cll, as shown above left. The
.!. Mid'
. The rod strikes the ball, which is initially at rest. As a result of this
3
collision, the rod is stopped and the ball moves in the direction shown above rig~. Express all answers in tenns of MI, M"
d, and fundamental constants.
- rotational inertia of the rod about point P is
W,
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 ~ :;
l'
of the ball after the collision.
-j, YY\ .&
L.('
ZtAj -:- 1112
/
vi
/
c. Assuming that this collision is elastic, calculate the numerical value of the ratio MJ
~ YY\,
/
M]
j~1
3:--Vl.-
--/YIv - duJ
Bdhrc: Colli,.;on
7;111
d. A new ball wIth the same mass j\1j as the rod
the C~10:/Js;c;;;r
3 Y'
I
J-;;
w:a~
I
Ii
d2w -
~l'i.!
'1--.,1
IS
i ~ ~V t (t t<t.')lJ
now placed a distance x from the pivot, as shown above. Again assuming
the rod stop moving after hitting the ball?
~
X-;;
r3
\j ?
o~
---
~d~'~1~4
::--'i-~ f-I
1-:;.
I
= ~o,\,.')l.3,
]...
fjIIt Advanced
I.
rf1Ldi iJPhysics Rotational
Motion Tcst
+0
I
Name:
A turntable that is initially at rest is sct in motion with
a constant angular acceleration a. What is the angular
velocity oCthe turntable after it has made one compkte
1--.£ - •••..--
revolution?
(A)
.ha
•
(11) .J2JW
(0) 2a
_
(C)
5.
(E)4JW
magnitude
g
g
(A)
A
0
•
l..Pivot
M
3 0
Point
n
A light rigid rod with masses attached to its ends is
pivoled about a horizontal axis as shown above. When
rekased Cromrest in a horizontal orientation, the rod
begins to rotate with an angular acceleration of
M
.J4JW
Questions 2 - 3
10 kg
2£----<
r
C
~ ..
D
71
(11) 51
(C)
g
5g
41
(0) 71
(11)B
(C) C
2F
(D) D
,.
F
(E) E
F
6.
3.
The sphere-rod combination can be pivoted about an
axis that is perpendicular to the plane oCthe page and
that passes through onc oCthe live lettered points.
Through which point should the axis pass Corthc
moment of inertia oCthe sphere-rod combination about
this axis to be greatest?
(A) A
4.
(B) 11
(C) C
I
E
Which oCthe live lettcred points represents the center
oCmass oCthe sphere-rod combination'!
(A) A
(E)
85kg
A 5-kilogram sphere is connected to a IO-kilogram
sphere by a rigid rod oCnegligible mass, as shown
above.
2.
g
(D) D
A system of two wheels fixed to each other is free to
rotate about a frictionless axis through the common
eemer oCthe wheels and perpendicular to the page.
Four forces are exerted tangentially to the rims of the
wheels, as shown above. The magnitude of the net
torque on the system aboutlhe axis is
(A) zero
(11) foR
(C) 2FR (0) 5FR (E) 14FR
(E) E
A uniConn stick has length L. The moment of inertia
about the center of the stick is I". A particle of mass M
is attached to one end of the stick. The moment oC
7. An ice skater is spinning about a vertical axis with
inertia oCthe combined system aboutlhe center oCthe
anns Cullvextended. ICthe arn1S are pulled in closer to
stick is
the body,~in which oCthe following ways are the
3
,
angular momentum and kinetic energy oCthe skater
I
,
(C) 10 +-AIL(A) 10 +}.,,\lL"
(Il) lo+-MLafleetcd'!
4
4
2
Kinetic Energv
Angular Momentum
-)
5
,
(1
(D) 10 +elf!."
+-AJLIncreases
(A)
Increascs
4
Remains Constant
(11) Incrcases
Increases
(C) Rcmains Constant
Remains Constant
(D) Remains Constant
Remains Constant
(E) Dcereases
: 'n
2009 Advanced
Physics Rotational
Name:
Motion Test
_
y
Questions 8-10
A cylinder rotates with constant angular acceleration
about a fixed axis. The cvlinder's moment of inertia
about the axis is 4 kg m': At time t = 0 the cylinder is
at rest. At time t = 2 seconds its angular velocity is
1 radian per second.
8.
What is the angular acceleration of the cylinder
between t = 0 and t = 2 seconds?
(A) 0.5 radian/s'
(13) I radian/s2
(C) 2 radian/s'
(D) 4 radian/s'
9,
(E) 5 radian/s'
----------~,---(O,a)
v
n1
I
--Or<t--------~,,..O--~.f
12. A particle of mass m moves with a constant speed v
along the dashed line y = a. When the x-coordinate of
the particle is x"' the magnitude of the angular
momentum of the particle with respect to the origin of
the system is
(A) zero
(B) mm
(e) mvx.
(D)
r",)"o'
+ ,,'
What is the angular momentum of the cylinder at time
t = 2 seconds?
(B) 2 kg m'/s
(C) 3 kg m'/s
(A) I kg m'/s
(D) 4 kg m'/5
(E) It cannot be detennined without
knowing the radius of the cylinder.
10, What is the kinetic energy of the cylinder at time t = 2
seconds?
(13) 2 J
(A) 1 J
(C) 3 J
(D) 4 J
(E) cannot be detennined without knowing the radius
of the cylinder
11. A bowling ball of mass M and radius R. whose
moment of inertia about its center is (2/5)MR', rolls
without slipping along a level surface at speed v. The
maximum vertical height to which it can roll if it
ascends an incline is
(A) ~
5g
(D)
(13)
o '
~I"
5g
(C) ~
2g
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