1
Instructor: ALSIN, Michael Course: AP Physics Differential Eqn for motion Work, Energy, Power Conservative F Momentum, collisions Center of mass Circular Motion, Torque Rotational Kinematics, Statics, Dynamics Angular Momentum Oscillations, Gravitation SHM U(x) diagrams, Kepler’s Laws (elliptical orbits) Design Observe, Measure Analyze Data, Errors Statics & Dynamics Lab Kinematics, 1D & 2D Mechanics Question Day Date Name (LAST, First):_________________, __________________ Block (circle): 1 2 3 4 5 6 7 8 Date (MM/DD/YY): ___/___/___ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ __/__ 99.A.1 99.A.2 99.A.3 00.A.1 00.A.2 00.A.3 01.A.1 01.A.2 01.A.3 02.A.1 02.A.2 02.A.3 03.A.1 03.A.2 03.A.3 04.A.1 04.A.2 04.A.3 05.A.1 05.A.2 05.A.3 06.A.1 06.A.2 06.A.3 07.A.1 07.A.2 07.A.3 08.A.1 08.A.2 08.A.3 09.A.1 09.A.2 09.A.3 10.A.1 10.A.2 10.A.3 11.A.1 11.A.2 11.A.3 12.A.1 12.A.2 12.A.3 Test Analysis Form C 5/11/2012 10:29 AM 1
1 of 2 2
TABLE OF’ INFORMATION DEVELOPED FOR 2012
CONSTANTS AND CON\ ERSION FAQ WRS
Proton mass,
— 167
10
kg
I
it)
kg
1 electron s di. I eV
kg
Speed of light.
Neutron mass, or
Electron mass, In
As gadr’
=
number. .\
0
*11
l0
,(L
I th mal
-,
Bltzmann’ cnsmnt.
=
l3s
.
nmi.k
lo
.
1 u
=
I
Planck’s constant.
h
=
l
=
Vacuum perIllittis it.
constant.
k
=
1 atmosphere pressure.
PREFIXES
Factor
io
S
nev ton.
A
K
pascal.
joule,
G
mega
NI
cosO
kilo
k
tan6
milli
300
10’
I0
C
ni
-
:n
S m
It)
kg
=
03
.o3 x l0
J.5
=
4 14 ,. 1
1.09> 10
Jon
=
C
Nin
1(1
mol
Hz
N
Pa
J I
124
0(1 x
10
Nnr C
=
4r x 10
T.m A
u,, 4
=
1
10
1 atm
=
1.0
Y
watt,
coulomb,
solt,
ohm,
henrs,
MeV
=
10’ eV.mn
T.m,, A
l0 N/ nf
W
C
V
=
1.0 x 101 Pa
farad.
tesla.
degree Celsius.
elect! on—s nit.
F
‘F
C
e’
H
VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES
giga
10
r
mole.
hertz.
Snibo1
centi
I
ni
ke
Prefix
io-
1
(, —
=
I 4€
p
Magnetic constant. k’
UNIT
SYMBOLS
I0
1 60
-
5.85
Vacuum permeability.
meter.
kiloeram.
oml,
ampere.
kelvin.
<
J K
I unitted at’niic mas unit.
Coulomb’s law
—
=
I nis era1 era itational
en-.tant.
Aceciera’ den due to eras it
,t Earta ‘tiflJ.O,
-
S.’ I i
t nrs eral ca’ constant.
‘
160
Electron charge magnitude,
m
micro
It)
nano
n
10
pim
p
60
90
0
1
4 5
3/2
1
i
2
I
3 5
1 ‘2
4 3
3
L
0
oo
The fo1loing con> entions are used in this exam
1. Unless otherwise stated, the frame of reference of any pr&hlein is
assumed to be inertial.
11. The direction f an> electric current is the direction of tlk u. of positise
charge corn entional current I,
111. For any is lated electric charge, the electric potentid js defined as ze
an infinite dismnce from the charge
-22
at
3
ADVANCED PLACEMENT PHYSICS C EQUATIONS DES’ ELOPED FOR 2012
MECHANICS
a
I
at
c ‘v
,,at
1
a(r
2
r
—t
1)
ma
F
JFdr
3
p
\p
in
*
fr,
<it\
JF.dr
=
1
K
mu
—
=
cit
P
r)
accelerati n
tc ice
1= trequenc y
height
I— rotational mertia
impulse
K= kinetic, energy
spring .a ntant
length
angular mc mentum
In = mass
N = normal lorc
p wer
momentum
I’ = radius or distance
position sector
7’ = period
time
potential energy
velocity or speed
w = work done on a system
5— pasition
coefficient of friction
mgh
A
B
C
d
B
qq,
r
1
4irE
F
q
F
—
F
fr
1
J
B
E.JA
B—
Jr
=
—
—
=
1
n
‘ç’q
r
1
ql
Uj,
icA
d
qq
r
A
=
P
=
Q
1
—
q
R
r
=
t
=
V
v
p
1
—
=
(1
0= angle
torque
(0= angular speed
angular acceleration
phase angle
-
—
ELEC RIC1TY AND MAGNETISM
=
=
=
area
mgnetic field
capautance
distance
electric field
cml
trce
current
current density
inductance
length
number t I I ) ps ci sne
per unit length
number ol charge arriem
per unit solume
posser
charge
point charge
resistance
distance
time
potential or stored energy
electric potential
velocity oi speed
resistisity
magnetic flux
1
dielectric constant
dQ
cit
=
LI
=
=
B*d€=p I
—kg
=
t
—
t
r
—
a’
F
=
a
—
*
mr
L
T
rw
rxp
=
I
=
2,r
I
V
77)
-
—
-
F
pJ
Mu A
JR
IdJxr
fId€XB
—pnl
-
JB.dA
F
2ir—
K
1 ,n
Gm
t((t
r
1
R
P
R
IV
Gm,n
00 +144u1
B
p
4t
RR
Jo>
7I
(0—0)
JB=
R=1
A
F
wJ
11 —mrm
r
—
cos(iut + c
=
=
o
1
1(1
17
t,
=
1
U
=
a’ B
£
L’
11
cit
4
-33
4
ADVANCED PLACEMENT PHYSICS C EQUATIONS DEVELOPEI) FOR 2012
GEOMETRY AND TRIGONOMETR
Rectangle
A area
C = circumference
V solume
5 = surface area
b =base
Ii = height
/ = length
hh
1h
1
—
Circle
yrr
A
C
=
dl
di
—
Triangle
A
CAL CUL1 S
r
2,rr
—
ci’ do
do dr
d
x
ci
dx
)
fi
ax
-
d
(In v)
dx
radius
d
Rectangular Solid
k = iirh
cit
(sini)
ci
Cy finder
V
S
=
—
r2f
2rc
=
+
2
2r
di
7
Jc
Sphere
r cit
1’ —-irr
I—
x
J
S
=
sinO
=
=
ccsr
=
—sinx
1
x
nfl
,
a
—l
mt
sindr
Right Triangle
x
—
J cosxdx
4r
a +b
C
=
—
—cosx
=
a
—
C
b
cosfl
—
9O
12
C
tanO
a
—
-44
5
1999
PHYSICS C
SECTION II, MECHANICS
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the pink
booklet in the spaces provided after each part, NOT in this green insert.
max
ri
Mech 1.
In a laboratory’ experiment, you wish to determine the initial speed of a dart just after it leaves a dart
gun. The dart, of mass m, is fired with the gun very close to a wooden block of mass Mo, which hangs
from a cord of length C and negligible mass, as shown above. Assume the size of the block is negligible
compared to (, and the dart is moving horizontally when it hits the left side of the block at its center and
becomes embedded in it. The block swings up to a maximum angle max from the vertical. Express your
max, and g.
answers to the following in terms of P1, Mo’ 1, 8
(a)
Determine the speed
(b)
The dart and block subsequently swing as a pendulum. Determine the tension in the cord when
it returns to the lowest point of the swing.
(c)
At your lab table you have only the following additional equipment.
Meter stick
Protractor
Spring
0
V
of the dart immediately before it strikes the block.
Stopwatch
5 in of string
Set of known masses
Five more blocks of mass
Without destroying or disassembling any of this equipment, design another practical method
for determining the speed of the dart just after it leaves the gun. Indicate the measurements you
would take. and how the speed could be determined from these measurements.
GOONTOTHENEXTPAGE
5
6
1999 PHYSICS C—MECHANICS
(d)
Mech 2.
The dart is now shot into a block of wood that is fixed in place. The block exerts a force F
on the dart that is proportional to the dart’s velocity V and in the opposite direction, that is
F = —by, where b is a constant. Derive an expression for the distance L that the dart
, and b,
0
penetrates into the block, in terms of m, v
A spherical, nonrotating planet has a radius R and a uniform density P throughout its volume. Suppose
a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above,
in which a small ball of mass in could move freely under the influence of gravity. Let r be the distance
of the ball from the center of the planet.
(a)
Show that the magnitude of the force on the ball at a distance r
planet is given by F = —Cr, where C = irGpm.
(b)
On the axes below, sketch the force F on the ball as a function of distance r from the center
of the planet.
<
R from the center of the
F
0
R
2R
3R
4R
5R
GO ON TO THE NEXT PAGE
6
7
1999 PHYSICS C—MECHANICS
The ball is dropped into the tunnel from rest at point P at the planet’s surface.
(c) Determine the work done by gravity as the ball moves from the surface to the center
of the planet.
(d) Determine the speed of the ball when it reaches the center of the planet.
(e) Fully describe the subsequent motion of the ball from the time it reaches the
center of the planet.
(0
Mech 3.
Write an equation that could be used to calculate the time it takes the ball to move from
point P to the center of the planet. It is not necessary to solve this equation.
As shown above, a uniform disk is mounted to an axle and is free to rotate without friction. A thin
uniform rod is rigidly attached to the disk so that it will rotate with the disk. A block is attached to
the end of the rod. Properties of the disk, rod, and block are as follows.
Disk:
mass
=
3m, radius
Rod:
mass
=
in, length
Block:
mass
=
R, moment of inertia about center
=
mR
‘D
2R. moment of inertia about one end
‘R
=
2,ii
The system is held in equilibrium with the rod at an angle 6
o to the vertical, as shown above, by a
horizontal string of negligible mass with one end attached to the disk and the other to a wall. Express
your answers to the following in terms of m, R, 8, and g.
(a)
Determine the tension in the string.
GO ON TO THE NEXT PAGE
r
7
8
1999 PHYSICS C—MECHANICS
The string is now cut, and the disk-rod-block system is free to rotate.
(b)
Determine the following for the instant immediately after the string is cut.
i. The magnitude of the angular acceleration of the disk
ii. The magnitude of the linear acceleration of the mass at the end of the rod
As the disk rotates, the rod passes the horizontal position shown above.
(c)
Determine the linear speed of the mass at the end of the rod for the instant the rod is in the
horizontal position.
STOP
END OF SECTION II, MECHANICS
8
9
MMMMMMMMMMMMM
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, MECHANICS
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which ase worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part, NOT in this green insert.
Mech 1.
You are conducting an experiment to measure the acceleration due to gravity g at an unknown location. In the
measurement apparatus, a simple pendulum swings past a photogate located at the pendulum’s lowest point, which
records the time t
10 for the pendulum to undergo 10 full oscillations. The pendulum consists of a sphere of mass m
at the end of a string and has a length f. There are four versions of this apparatus, each with a different length. All
four are at the unknown location, and the data shown below are sent to you during the experiment.
tJ()
T
2
T
(cm)
(s)
(s)
(2)
12
7.62
18
8.89
21
10.09
32
12.08
(a) For each pendulum, calculate the period T and the square of the period. Use a reasonable number of
significant figures. Enter these results in the table above.
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AP is a registered trademark of the College Entrance Examination Board.
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-29
_____
10
MMMMMMMMMMMMM
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
(b) On the axes below, plot the square of the period versus the length of the pendulum. Draw a bestfit straight
line for this data.
2
T
1.5
(2)
r
L
1
—
—
I
—
L _1_ J_
—
L _1_
33
_L
—
1
—
L
3_
L _4
4_
—
L
I
_I_ 33_ _L
L
4
I
I_
—
-
r
1.0
F
I
—
—L —4—1—_ L —4— _L _3_I
—
—
—
3—4———
I
—
—
—
_L
43
I
_L
4
I
I
—
L
4
—
3—3— _L —4— I_ —3
3
I
1
_I_
—
L
I
_L
I
—
I
4—3— _L
4
I
_L
I
I
1
—
—I
—
—
L _4_ I
—
—
—
F
—
—
L
—
£
—
_L —3—3—— 3—
•
—
F
—
—
F
—L
1
I
I
-
LI__L_4_I__ L_4____3_I —_L_. -—I—— L —4—1—_ L —4———
0.5
—
—
L _3_ _L _4_ I
r
—
—
—
—I— —L
—
r
—
1
—
I
—
—
T
1
—
—L _4_I_
—
L —1———— 3_I
1
-—
_3_ _L_4_I__ 3_ £____3_I__L_I
I
I
1
—
_I_—L_4— I__L_I____3_I__L
—
—I— —L —4—1—— L —4—_I—
I
I
I
—
3_I— —L
I
I
I
-—
—
3_I— —L —3_I—— L _4____ 3_I
I
I
1
11
——
0
5
10
15
20
25
30
t(cm)
(c) Assuming that each pendulum undergoes small amplitude oscillations, from your fit determine the
experimental value exp of the acceleration due to gravity at this unknown location. Justify your answer.
(d) If the measurement apparatus allows a determination of g that is accurate to within 4%, is your
experimental value in agreement with the value 9.80 m/s
2 ? Justify your answer.
(e) Someone informs you that the experimental apparatus is in fact near Earth’s surface, but that the experiment
has been conducted inside an elevator with a constant acceleration a, Assuming that your experimental
value gexp is exact, determine the magnitude and direction of the elevator’s acceleration.
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AP is a registered trademark of the College Entrance Examination Board.
GO ON TO THE NEXT PAGE.
-310
11
MMMMMMMMMMMMM
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
Mech 2.
A rubber ball of mass in is dropped from a cliff. As the ball falls, it is subject to air drag (a resistive force caused
by the air). The drag force on the bail has magnitude by
. where b is a constant drag coefficient and v is the
2
instantaneous speed of the ball. The drag coefficient b is directly proportional to the cross-sectional area of the
ball and the density of the air and does not depend on the mass of the ball. As the ball falls, its speed approaches
a constant value called the terminal speed.
(a) On the figure below, draw and label all the forces on the ball at some instant before it reaches terminal
speed.
C
(b) State whether the magnitude of the acceleration of the ball of mass m increases, decreases, or remains the
same as the ball approaches terminal speed. Explain.
(c) Write, but do NOT solve, a differential equation for the instantaneous speed v of the ball in terms of time t,
the given quantities, and fundamental constants.
(d) Determine the terminal speed
t
in terms of the given quantities and fundamental constants.
(e) Determine the energy dissipated by the drag force during the fall if the ball is released at height h and
reaches its terminal speed before hitting the ground, in terms of the given quantities and fundamental
constants.
Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.
AP is a registered trademark of the College Entrance Examination Board.
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-411
12
MMMMMMMMMMMMM
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
‘1
Mech 3.
A pulley of radius R
1 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 p slip
on the pulley. Determine the answers to parts (a) and (b) in terms of m, R
, I, and fundamental constants.
1
(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 an acceleration
Determine the following.
i. The tension T
3 in the section of cord supporting the three blocks on the left
ii. The tension T
1 in the section of cord supporting the single block on the right
iii. The rotational inertia 1 of the pulley
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-512
13
MMMMMMMMMMMMM
2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS
, 161
1
2R
(01
(c) The blocks are now removed and the cord is tied into a ioop, which is passed around the original pulley and
a second pulley of radius 2R
1 and rotational inertia I 6I. The axis of the original pulley is attached to a motor
that rotates it at angular speed
which in turn causes the larger pulley to rotate. The loop does not slip on
the pulleys. Determine the following in terms of 1, R
, and w.
1
i. The angular speed w of the larger pulley
ii. The angular momentum L
2 of the larger pulley
iii. The total kinetic energy of the system
STOP
END OF SECTION II, MECHANICS
IF YOU FINISH BEFORE TIME IS CALLED, YOU MAY CHECK YOUR WORK ON SECTION II,
MECHANICS, ONLY. DO NOT TURN TO ANY OTHER TEST MATERIALS.
Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.
AP is a registered trademark of the College Entrance Examination Board.
-613
14
2001 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, MECHANICS
Tirne—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions.
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part. NOT in this green insert.
Motion
Sensor
Force
Sensor
-
11
Mech I.
A motion sensor and a force sensor record the motion of a cart along a track, 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.
0.30
I
I
0.20
40
z
0.10
0
I
I
---
C
> -0.10
—0.20
0.30
30
20
10
—
0
0.32
0.34 0.36
Timet(s)
0.38
0.40
0.30
0.32
0.34 0.36
Time t (s)
0.38
0.40
(a) Determine the cart’s average acceleration between t = 0.33 s and t = 0.37 s.
(b) Determine the magnitude of the change in the cart’s momentum during the collision.
(c) Determine the mass of the cart.
(d) Determine the energy lost in the collision between the force sensor and the cart.
Copyright © 2001 by College Entrance Examination Board, All rights reserved,
Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
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4
14
15
2001 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech 2.
An explorer plans a mission to place a satellite into a circular orbit around the planet Jupiter, which has mass
= 1.90 x 1027 kg aid radius R
7.14 x 10’ m.
(al 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 v
ii.
The period of the orbit is given by 7
=
GM
=
3
R
2
i4
J
(b) The explorer wants the satellite’s orbit to be synchronized with Jupiter’s rotation. This requires an equatorial
orbit whose period equals Jupiter’s rotation period of 9 hr 51 mm = 3.55 x l0 s. Determine the required
orbital radius in meters.
(c) Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differs
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 velocity vector has the correct
direction, but the speed is slightly fter than the correct speed for a circular orbit of that radius.
P
--.-
/
\
/
ii. When the satellite is at the desired altitude over the equator, its velocity vector has the correct
direction, but the speed is slightly slower than the correct speed for a circular orbit of that radius.
P
/
J
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5
15
16
2001 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
H
2L
Experiment A
Mech 3.
A light string that is attached to a large block of mass 4,n 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 of length 2L, with a small block of mass rn attached at each end. The
rotational inertia of the pole and the rod are negligible.
(a) Determine the rotational inertia of the rodand-block apparatus attached to the top of the pole.
(b) Determine the downward acceleration of the large block.
(c) When the large block has descended a distance D, how does the instantaneous total kinetic energy of the
three blocks compare with the value 4mgD ? Check the appropriate space below.
Greater than 4ingD
Equal to 4mgD
Less than 4ingD
Justify your answer.
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Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.
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6
16
17
2001 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
H
Experiment B
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 /. 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.
Greater
—
Equal
Less
Justify your answer.
END OF SECTION II, MECHANICS
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2002 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, MECHANICS
Time—45 minutes
3 Questions
I)irections: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight. Show all our work in the
pink booklet in the spaces provided after each part. NOT in this green insert.
Mech 1.
A crash test car of mass 1,000 kg moving at constant speed of 12 mIs collides completely inelastically with an
object of mass Al at time t = 0. The object was initially at rest. The speed u in rn/s of the car-object system
after the collision is given as a function of time t in seconds by the expression
8
1) =
1 + 5t
(a) Calculate the mass M of the object.
(b) Assuming an initial position of .r = 0, determine an expression for the position of the car-object system after
the collision as a function of time 1.
(c) Determine an expression for the resisting force on the car-object system after the collision as a function of
time t.
(d) Determine the impulse delivered to the car-object system from t
=
0 to t
=
2.0 s.
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19
2002 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
nif4
—m
Bumper
Mech 2.
The cart shown above is made of a block of mass m and four solid rubber tires each of mass ni/4 and radius r.
Each tire
may
he considered to be a disk. (A disk has rotational inertia
—
, where M is the mass and L is
2
ML
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) Detennine 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 Xrn the sprng is
compressed before the cart and bumper come to rest.
(d) Now assume that the bumper has a nonneg1ible mass. After the collision with the bumper, the spring
is compressed to a maximum distance of about 90% of the value of x, in part (c). Give a reasonable
explanation for this decrease.
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2002 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech 3.
An object of mass 0.5 kg experiences a force that is associated with the potential energy function
4.0
U(x) =
where U is injoules and x is in meters.
2.0 + x
——,
(a) On the axes below sketch the graph of Lf(x) versus x.
.
U (.1)
3.0
-
20
-
10
0
1
2
3
4
5
x0Ti)
(b) Determine the force associated with the potential energy function given above.
(c) Suppose that the object is released from rest at the origin. Determine the speed of the particle at x
=
2
m.
In the laboratory, you are given a glider of mass 0.5 kg on an air track. The glider is acted on by the force
determined in part (b). Your goal is to determine experimentally the validity of your theoretical calculation in
part (c).
(d) From the list below, select the additional equipment you will need from the laboratory to do your experiment
by checking the line next to each item. If you need more than one of an item, place the number you need on
the line.
Meterstick
Balance
(e)
Stopwatch
—
Wood block
—
—
Photogate timer
—
String
Spring
Set of objects of different masses
Briefly outline the procedure you will use, being explicit about what measurements you need to make
in order to determine the speed, You may include a labeled diagram of your setup if it will clarify your
procedure
END OF SECTION II, MECHANICS
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2003 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, MECHANICS
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions.
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part, NOT in this green insert.
Mech. 1.
The 100 kg box shown above is being pulled along the x-axis by a student. The box slides across a rough surface,
and its position x varies with time t according to the equation x = 0.5t
3 + 2t, where x is in meters and t is in
seconds.
(a) Determine the speed of the box at time t
=
0.
(b) Detennine the following as functions of time t.
i. The kinetic energy of the box
ii. The net force acting on the box
iii. The power being delivered to the box
(c) Calculate the net work done on the box in the interval t
=
0 to t
=
2 s.
(ci) Indicate below whether the work done on the box by the student in the interval t
than. less than, or equal to the answer in part (c).
Greater than
Less than
=
0 to t
=
2 s would he greater
Equal to
Justify your answer.
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______
22
2003 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
4D
Ii
M
Mech, 2.
An ideal spring is hung from the ceiling and a pan of mass M is suspended from the end of the spring, stretching
it a distance D as shown above. A piece of clay, also of mass A is then dropped from a height H onto the pan
and sticks to it. Express all algebraic answers in terms of the given quantities and fundamental constants.
(a) Determine the speed of the clay at the instant it hits the pan.
(b) Determine the speed of the pan just after the clay strikes it.
(c) Determine the period of the simple harmonic motion that ensues.
(d) Determine the distance the spring is stretched (from its initial unstretched length) at the moment the speed of
the pan is a maximum, Justify your answer.
(e) The clay is now removed from the pan and the pan is returned to equilibrium at the end of the spring. A rubber
ball, also of mass M, is dropped from the same height H onto the pan, and after the collision is caught in midair
before hitting anything else.
Indicate below whether the period of the resulting simple harmonic motion of the pan is greater than, less than,
or the same as it was in part (c).
Greater than
Less than
The same as
Justif your answer.
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___
____________________________________________________________________
23
2003 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
12m
12m
2rn,
4
J
3
/
,
301A
Figure 1
Mech.
Figure 2
3.
Some physics students build a catapult. as shown above. The supporting platform is fixed firmly 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
he 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 1, and begins rotating. There is a mechanism (not shown) that stops the arm 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, recording the following data.
(kg)
F Mass
x(m)
100
18
j
i. The data
are
300
37
500
45
700
48
900
51
plotted on the axes below. Sketch a best-fit curve for these data points.
:
0
-
40—-
1
T
—-—-—
p-——.-
——
—.——
—
30
fl
——--—————1——.——
Hi
—-—,
U
—-—
————
,—
.———.—-—,
200
—---——
—-—-
———
—
—
—
—
—-—
*
.--.*-*
400
600
800
1000
Mass (ku)
ii. Using your best-fit curve, determine the distance x traveled by the projectile if 250 kg is placed in
the counterweight bucket.
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2003 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(b) The students assume that the mass of the rotating aim. 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 x
ur. where v 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?
ii, Derive the equation that describes the gravitational potential energy of the system relative to the ground
when in the position shown in Figure 1, assuming the mass in the counterweight bucket is M.
iii. Derive the equation for the velocity of the projectile as it leaves the cup. as shown in Figure 2.
(cI
i, Complete the theoretical model by writing the relationship for x as a function of the counterweight mass
using the results from (b)i and (b)iii,
ii. Compare the experimental and theoretical values of x for a counterweight bucket mass of 300 kg. Offer a
reason for any difference.
END OF SECTiON Ii, MECHANICS
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25
2004 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C
Section II, MECHANICS
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each, The parts within a question may not have equal weight. Show all your work in the
booklet in the spaces provided after each part, NOT in this green insert.
L
1
in
A
9Q0
C
JR
-—
[.ake
Mech, 1.
A rope of length L is attached to a support at point C. A person of mass in
1 sits on a ledge at position A holding
the other end of the rope so that it is horizontal and taut, as shown above. The person then drops off the ledge
and swings down on the rope toward position B on a lower ledge where an object of mass in
2 is at rest. At
position B the person grabs hold of the object and simultaneously lets go of the rope. The person and object then
land together in the lake at point D, which is a vertical distance L below position B. Air resistance and the mass
of the rope are negligible. Derive expressions for each of the following in terms of nz
. iii,, L, and
1
.
(a) The speed of the person just before the collision with the object
(b) The tension in the rope just before the collision with the object
(c) The speed of the person and object just after the collision
(d> Thc ratio of thu kinetic energ of thu pci son object ssteni butore the collision to the kinetic encrg\ iftur the
col 11 Si Ofl
(e) The total horizontal displacement x of the person from position A until the person and object land in the
water at point D.
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26
2004 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
1)
Mech. 2.
A solid disk of unknown mass and known radius 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 wrapped
around it several times. The block of mass in is released from rest and takes a time t to fall the distance D to
the floor.
(a) Calculate the linear acceleration a of the falling block in terms of the given quantities.
(b) The time t is measured for various heights D and the data are recorded in the following table.
D (an)
t
(s)
0.5
0.68
1
1.02
1.5
1.19
2
1.38
i. What quantities should be graphed in order to best determine the acceleration of the block? Explain
your reasoning.
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2004 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
ii. On the grid below, plot the quantities determined in (b)i., label the axes, and draw the best-fit line to
the data.
———
.1
—
4
-
•_J
L.
.J.,...
.
..,.
.
.
Y-.4
—
4
4
¶
rri-r”i 1P
4.
.4
.4
—
..
..“
.
..
-
J..,
1..4
-1
r
..L
J
.L
.
...L
iii. Use your graph to calculate the magnitude of the acceleration.
(c) Calculate the rotational inertia of the pulley in terms of
in, R, a, and fundamental constants.
(d) The value of acceleration found in (b)iii. along with numerical values for the given quantities and your
answer to (c), can be used to determine the rotational inertia of the pulley. The pulley is removed from its
support and its rotational inertia is found to be greater than this value. Give one explanation for this
discrepancy.
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2004 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
—
I
4 -H r
Pivot
I
Mech. 3.
A uniform rod of mass M and length L is attached to a pivot of negligible friction as shown above. The pivot is
located at a distance
4
from the left end of the rod. 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. Calculate the linear speed of
the bottom end of the rod when the rod passes through the vertical.
TT
Pivot
L
(c) The rod is brought to rest in the vertical position shown above and hangs freely. ft is then displaced slightly
from this position. Calculate the period of oscillation as it swings.
END OF SECTION II, MECHANICS
isit
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2005 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C
Section II. MECHANICS
Tinie—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions.
which are worth 15 points each. The parts within a question may not have equal weight. Show all our work in the
pink booklet in the spaces provided after each part. NOT in this green insert.
Mech. 1.
A ball of mass M is thrown vertically upward with an initial speed of v
. It experiences a force of air resistance
0
given by F = —kv, where k is a positive constant. The positive direction for all vector quantities is upward.
Express all algebraic answers in terms of M. k. v
, and fundamental constants.
0
(a> Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball moves
upward?
increases
decreases
remains the same
Justify your answer.
(b) Write, but do NOT solve, a differential equation for the instantaneous speed u of the ball in terms of time t
as the ball moves upward.
(c) Determine the terminal speed of the ball as it moves downward.
(d) Does it take longer for the ball to rise to its maximum height or to fall from its maximum height back to the
height from which it was thrown?
,,jonger to rise
__jonger to fall
Justify your answer.
(e) On the axes below, sketch a graph of velocity versus time for the upward and downward parts of the ball’s
flight, where t is the time at which the ball returns to the height from which it was thrown.
Velocity
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2005 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech. 2.
A student is given the set of orbital data for some of the moons of Saturn shown below and is asked to use the
data to determine the mass M of Saturn. Assume the orbits of these moons are circular.
Orbital Period. T
( seconds)
Orbital Radius. R
( meters)
8.l4xl0
5
l.85x10
5
l.18x10
8
2.38xl0
1.63 x IO
5
2.95 x 108
2.37 x i0
5
3.77 x 10
(a) Write an algebraic expression for the gravitational force between Saturn and one of its moons.
(bI Use your expression from part (a) and the assumption of circular orbits to derive an equation for the orbital
period T of a moon as a function of its orbital radius R.
(c) Which quantities should be graphed to yield a straight line whose slope could be used to determine Saturn’s
mass?
(d) Complete the data table by calculating the two quantities to be graphed. Label the top of each column,
including units.
(e) Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers to indicate
the scale.
.LJ.JL
3
———-—
—r— —i——r-—
._J
LL1
LLJ.
I
—I4——
—-i——trr—-——r
I
LJLLJ. L_L[
:4.
44
I
I
I
—-1—--I———
I
——-—
I
—1—-H——
I
H
3
1
-—3--——i--—H--—--I—f
I
._—t——
—
I
LJ_J.L
1
—
-•
I
JLL1
I
LLLJ.i.J_L... .J
I
44--t—
4I—-H --4— 4
—-1r—r
.-
3
—.4-1--
—m——
T1—r——,r—r —i———
JLLL.
L
‘
3
I
L_ILLI_
.
—H——I—-——I—— 34-—-4- —H—4—-1—.—
-r—1i
L.
r—r—r—
—
LL.
-_-
---
—-r—r—j- —-i—
—
LJL.
—4—H—-— H—
-—t—
-
*
JLJ.
——-H—H—.4—
—
—H-—4—
-——r—- —,—-rr—r—
L_i_L__ 1_JJ__L
3
1
---
3
--
-
—
—
-
-
—
-
(f) Using the graph, calculate a value for the mass of Saturn,
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2005 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
P
Al,
Before Collision
After collision
TOP VIEWS
Mech. 3.
A system consists of a ball of mass M
7 and a uniform rod of mass M 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 (0 as shown above
,
M
d
2. The rod strikes the ball, which is initially at rest.
left. The rotational inertia of the rod about point P is 1
As a result of this collision, the rod is stopped and the ball moves in the direction shown above right. Express all
answers in terms of M
, M
1
, (0. d, and fundamental constants.
2
(a) Derive an expression for the angular momentum of the rod about point P before the collision.
(b) Derive an expression for the speed v of the ball after the collision.
(c) Assuming that this collision is elastic, calculate the numerical value of the ratio 1
M
/
M
T
Ix
•
Al
Bekre Collision
(d) A new ball with the same mass 1
Al as the rod is now placed a distance x from the pivot, as shown above.
Again assuming the collision is elastic, for what value of x will the rod stop moving after hitting the bail?
END OF SECTION 11, MECHANICS
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2006 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Time—45 minutes
3
Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part, NOT in this green insert.
Block. MB
=
0.50 kg
Slab. M
=
3.0 k
4.0 m/s
Mech I.
A small block of mass MB = 0.50 kg is placed on a long slab of mass M
3.0 kg as shown above. Initially, the
slab is at rest and the block has a speed v
0 of 4.0 mIs to the right. The coefficient of kinetic friction between the
block and the slab is 0.20, and there is no friction between the slab and the horizontal surface on which it moves.
(a) On the dots below that represent the block and the slab, draw and label vectors to represent the forces acting on
each as the block slides on the slab.
Block
Slab
.
.
At some moment later, before the block reaches the right end of the slab, both the block and the slab attain identical
speeds Vf.
(b) Calculate v
.
1
(c) Calculate the distance the slab has traveled at the moment it reaches v
(d) Calculate the work done by
friction
on the slab from the beginning of its motion until it reaches u
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32
33
2006 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech 2.
A nonlinear spring is compressed various distances x, and the force F required to compress it is measured for each
distance. The data are shown in the table below.
x (m)
F (N)
0.05
4
0.10
17
0.15
38
0.20
68
0.25
106
Assume that the magnitude of the force applied by the spring is of the form F(x)
=
.
2
Ax
(a) Which quantities should be graphed in order to yield a straight line whose slope could be used to calculate a
numerical value for A?
(b) Calculate values for any of the quantities identified in (a) that are not given in the data, and record these values
in the table above. Label the top of the column, including units.
(c) On the axes below, plot the quantities you indicated in (a) Label the axes with the variables and appropriate
numbers to indicate the scale.
.
r
L
(d) Using your graph, calculate A.
The spring is then placed horizontally on the floor. One end of the spring is fixed to a wall. A cart of mass 0,50 kg
moves on the floor with negligible friction and collides hea&on with the free end of the spring, compressing it a
maximum distance of 0.10 rn.
te) Calculate the work done by the cart in compressing the spring 0.10 in from its equilibrium leneth.
(f) Calculate the speed of the cart just before it strikes the spring.
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34
2006 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
lvi. R
Mech 3.
A thin hoop of mass lvi. radius R. and rotational inertia MR
2 is released from rest from the top of the ramp of length
L above. The ramp makes an angle 8 with respect to a horizontal tabletop to which the ramp is fixed. The table is a
height H above the floor. 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 constants.
(a) Derive an expression for the acceleration of the center of mass of the hoop as it rolls down the ramp.
(b) Derive an expression for the speed of the center of mass of the hoop when it reaches the bottom of the ramp.
(c) Derive an expression for the horizontal distance from the edge of the table to where the hoop lands on the floor.
(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 edge of the table to where the disk lands on the floor compare with the distance determined in part (c)
for the hoop?
Less than
The same as
Greater than
Briefly justify your response.
END OF EXAM
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2007 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions.
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part. NOT in this green insert.
Mech.
1.
A block of mass in
is
in terms of in. F
1
0
pulled along a rough horizontal surface by a constant applied force of magnitude F1 that acts at
an angle 8 to the horizontal, as indicated above. The acceleration of the block is a
. Express all algebraic answers
1
.
. and fundamental constants.
1
a
(a) On the figure below, draw and label a free-body diagram showing all the forces on the block.
El
(b) Derive an expression for the normal force exerted by the surface on the block.
(c) Derive an expression for the coefficient of kinetic friction u between the block and the surface.
(d) On the axes below, sketch graphs of the speed v and displacement x of the block as functions of time t if the
block started from rest at x = 0 and t = 0.
x
U
U
U
(c) If the applied force is large enough. the block will lose contact with the surface. Derive an expression tbr the
magnitude of the greatest acceleration a
0 that the block can have and still maintain contact with the ground.
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36
2007 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech.
2.
In March 1999 the Mars Global Surveyor (GS) entered its final orbit about Mars, sending data back to Earth.
Assume a circular orbit with a period of 1.18 x 102 minutes = 7.08 x i0
3 s and orbital speed of 3.40 x
rn/s.
The mass of the GS is 930 kg and the radius of Mars is 3.43
X
106 m
(a) Calculate the radius of the GS orbit.
(b’ Calculate the mass of Mars.
(c) Calculate the total mechanical energy of the GS in this orbit.
(d) If the GS was to he placed in a lower circular orbit (closer to the surface of Mars), would the new orbital period
of the GS be greater than or less than the given period?
than
than
Justify your answer.
(e) In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars at 3.71 x l0 m
above the surface and its furthest distance at 4.36 x 10 rn above the surface. If the speed of the GS at
closest approach is 3.40 x i0
3 rn/s calculate the speed at the furthest point of the orbit.
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37
2007 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Spring
Mech.
3.
The apparatus above is used to study conservation of mechanical energy. A spring of force constant 40 N/rn is held
horizontal over a horizontal air track, with one end attached to the air track. A light string is attached to the other
end of the spring and connects it to a glider of mass m The glider is pulled to stretch the spring an amount x from
equilibrium and then released. Before reaching the photogate, the glider attains its maximum speed and the string
becomes slack. The photogate measures the time t that it takes the small block on top of the glider to pass through.
Information about the distance x and the speed v of the glider as it passes through the photogate are given below.
Extension of the Spring
x (m)
Speed of Glider
v (m/s)
Extension Squared
2 (m2)
x
Speed Squared
1
030xl0
0.47
0.09 x 102
0.22
2
0.60 x 10
0.87
0.36 x 10_2
0.76
3
0.90 x 10-’
1.3
0.81 x 10-2
1.7
4
1.2 x l0’
1.6
1.4 x 102
2.6
5
1.5 x i0
2.2
2.3 x l0_2
4.8
Trial
2 (m2/s2)
v
(a) Assuming no energy is lost, write the equation for conservation of mechanical energy that would apply to this
situation.
(b) On the grid below, plot v
2 versus x
. Label the axes, including units and scale.
2
I
I!
44 4
I
4-_4----4----
I
I
4.4
I
I
—
—r—4——-r
I
——--—-t
——-—r
-—_4__—
LL___
b
L11
I
———i-—I
—I
1
—
I
rr1T
T
r
—
—-------f-_
-_
4__4
:
::---j
—-j---j.--
.—÷_
---•
—---
•----•--
JJII
Ii
JJJL
I
I
...i.
----
LLI
4
J.JJ
.
T
4-—4--,
;i
11
:-
J..JL. LI
.
-
.-—-..
-----
..
I-.i
Li £I.
—•
L
.
.
rr
-.
—
.
I-.
.
liii
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37
38
2007 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(c)
i. Draw a best-fit straight line through the data.
ii. Use the best-fit line to obtain the mass in of the glider.
(d) The track is now tilted at an angle U as shown below. When the spring is unstretched, the center of the glider is
a height h above the photogate. The experiment is repeated with a variety of values of x.
i. Assuming no energy is lost, write the new equation for conservation of mechanical energy that would
apply to this situation.
ii. Will the graph of v2 versus x
2 for this new experiment be a straight line?
Yes
No
Justify your answer.
END OF EXAM
\‘lslt
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39
2008 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions.
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part, NOT in this green insert.
Mech. 1.
A skier of mass M is skiing down a frictionless hill that makes an angle 0 with the horizontal, as shown in the
diagram. The skier starts from rest at time t 0 and is subject to a velocity-dependent drag force due to air
resistance of the form F = —by. where v is the velocity of the skier and b is a positive constant. Express all
algebraic answers in terms of M, h, 6, and fundamental constants.
(a) On the dot below that represents the skier, draw a free-body diagram indicating and labeling all of the forces
that act on the skier while the skier descends the hill.
(b) Write a differential equation that can be used to solve for the velocity of the skier as a function of time.
(C)
Determine an expression for the terminal velocity
T
of the skier.
(d) Solve the differential equation in part (b) to determine the velocity of the skier as a function of time,
wiiallourstes.
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2008 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(e) On the axes below, sketch a graph of the acceleration a of the skier as a function of time r, and indicate the
initial value of a. Take downhill as positive.
a
0
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2008 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Hinge
Mech. 2.
The horizontal uniform 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 frictionless 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 of 30 with the rod. A spring scale of negligible mass measures the
tension in the cord. A 0.50 kg block is also attached to the right end of the rod.
(a) On the diagram below, draw and label vectors to represent all the forces acting on the rod. Show each force
vector originating at its point of application.
(b) Calculate the reading on the spring scale.
(c) The rotational inertia of a rod about its center is
, where M is the mass of the rod and L is its length.
2
ML
Calculate the rotational inertia of the rod-block system about the hinge.
(d) If the cord that supports the rod is cut near the end of the rod, calculate the initial angular acceleration of the
rod-block system about the hinge.
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42
2008 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech. 3.
In an experiment to determine the spring constant of an elastic cord of length 0.60 m, a student hangs the cord
from a rod as represented above and then attaches a variety of weights to the cord. For each weight, the student
allows the weight to hang in equilibrium and then measures the entire length of the cord. The data are recorded
in the table below:
ght(N)0l0152OT
0.6O037L
(a) Use the data to plot a graph of weight versus length on the axes below. Sketch a best-fit straight line through
the data.
30 -—rr—-———rr—rrr—r-—rrrr-—r
I
ttt
2c ---±--------l
H------ --,----H-------,
I
—
10
5
I
I
I
I
I
I
I
—
—
I
I
_-..,-—.,-—-1_—_l_- i,
I
I
Il
I
()
0.5
I
1.0
l.cnuth (in)
1.5
2.0
(b) Use the best-fit line you sketched in part (a) to determine an experimental value for the spring constant k of
the cord.
Visit
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43
2008 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
The student now attaches an object of unknown mass rn to the cord and holds the object adjacent to the point at
which the top of the cord is tied to the rod, as represented above. When the object is released from rest, it falls
1.5 in before stopping and turning around. Assume that air resistance is negligible.
(c) Calculate the value of the unknown mass
(d)
i.
in
of the object.
Calculate how far down the object has fallen at the moment it attains its maximum speed.
ii. Explain why this is the point at which the object has its maximum speed.
iii. Calculate the maximum speed of the object.
END OF EXAM
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-943
44
2009 APu PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in this
booklet in the spaces provided after each part, NOT in the green insert.
Mech. 1.
A 3.0 kg object is
the x-axis in a region where its potential energy as a function of x is given
as U(x) = 4.0x
. where U is injoules and x is in meters. When the object passes the point x = —0.50 m. its
2
velocity is +2.0 mIs. All forces acting on the object are conservative.
moving along
(a) Calculate the total mechanical energy of the object.
(b) Calculate the x-coordinate of any points at which the object has zero kinetic energy.
(c) Calculate the magnitude of the momentum of the object at x
=
0.60 m.
(d) Calculate the magnitude of the acceleration of the object as it passes .v
VisO
=
0.60 m.
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44
45
2009 AP® PHYSICS C: MECHANICS FREERESPONSE QUESTIONS
(e) On the axes below, sketch graphs of the object’s position x versus time t and kinetic energy K versus time
Assume that x = 0 at time i = 0. The two graphs should cover the same time interval and use the same
scale on the horizontal axes.
x
—
0
—
0
\‘sit
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46
2009 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
ThTT
WI’
Mech. 2.
You are given a long, thin, rectangular bar of known mass Al and length C with a pivot attached to one end.
The bar has a nonuniform mass density, and the center of mass is located a known distance x from the end with
the pivot. YOU are to determine the rotational inertia
of the bar about the pivot by suspending the bar from the
pivot, as shown above, and allowing it to swing. Express all algebraic answers in terms of I,,. the given
quantities, and fundamental constants.
(a)
i.
By applying the appropriate equation of motion to the bar, write the differential equation for the angle 0
the bar makes with the vertical.
ii.
By applying the small-angle approximation to your differential equation, calculate the period of the bar’s
motion.
(h) Describe the experimental procedure you would use to make the additional measurements needed to
determine ‘b’ Include how you would use your measurements to obtain ‘b and how you would minimize
experimental error.
(c) Now suppose that you were not given the location of the center of mass of the bar. Describe an experimental
procedure that you could use to determine it. including the equipment that you would need.
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47
2009 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
T
(I
Mech. 3,
A block of mass M/2 rests on a frictionless horizontal table, as shown above. It is connected to one end of a
string that passes over a massless pulley and has another block of mass M/2 hanging from its other end. The
apparatus is released from rest.
(a) Derive an expression for the speed v of the hanging block as a function of the distance d it descends.
Now the block and pulley system is replaced by a uniform rope of length L and mass M, with one end of the
rope hanging slightly over the edge of the frictionless table. The rope is released from rest, and at some time later
there is a length y of rope hanging over the edge, as shown below, Express your answers to parts (b). (c), and (d)
in terms of y. L, M, and fundamental constants.
(b) Determine an expression for the force of gravity on the hanging part of the rope as a function of y.
(c) Dens e an expression for the work done by gravity on the rope as a function of y, assuming y is initially zero.
(d) Derive an expression for the speed Vr of the rope as a function of y.
e) The hanging block and the right end of the rope are each allowed to fall a distance L (the length of the rope).
The string is long enough that the sliding block does not hit the pulley. Indicate whether v
5 from part (a) or
v from part (d) is greater after the block and the end of the rope have traveled this distance.
is
greater.
14 is greater.
The speeds are equal.
Justify your answer.
END OF EXAM
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48
2010 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part, NOT in this green insert,
—i
Mech. 1.
Students are to conduct an experiment to investigate the relationship between the terminal speed of a stack of
falling paper coffee filters and its mass. Their procedure involves stacking a number of coffee filters, like the
one shown in the figure above, and dropping the stack from rest. The students change the number of filters in the
stack to vary the mass in while keeping the shape of the stack the same. As a stack of coffee filters falls, there
is an air resistance (drag) force acting on the filters.
= Cu
, where
2
(a) The students suspect that the drag force FD is proportional to the square of the speed v:
C is a constant. Using this relationship, derive an expression relating the terminal speed VT to the mass in.
The students conduct the experiment and obtain the following data.
Mass of the stack of filters in (kg)
Terminal speed,
VT
(m/s)
1 12 x i0
204 x i0
0.51
0.62
296>< i0
0.82
4 18 x 10
0.92
5 10 x l0
1.06
(b)
(i) Assuming the functional relationship for the drag force above, use the grid below to plot a linear graph
as a function of in to verify the relationship. Use the empty boxes in the data table, as appropriate,
to record any calculated values you are graphing. Label the vertical axis as appropriate, and place
numbers on both axes.
:
:
: : : :r: : : : : : : : : : : : : : ::c:
in
(kg)
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49
2010 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(ii) Use your graph to calculate C.
A particular stack of filters with mass m is dropped from rest and reaches a speed very close to terminal speed
by the time it has fallen a vertical distance Y.
(c)
(i) Sketch an approximate graph of speed versus time from the time the filters are released up to the time
on
t = T that the filters have fallen the distance Y. Indicate time t = T and terminal speed v =
the graph.
1)
(ii) Suppose you had a graph like the one sketched in (c)(i) that had a numerical scale on each axis.
Describe how you could use the graph to approximate the distance Y.
(d) Determine an expression for the approximate amount of mechanical energy dissipated, E. due to air
resistance during the time the stack falls a distance y, where v > Y. Express your answer in terms of v.
in,
and fundamental constants.
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50
2010 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
N©ic: Figure not drawn to scale.
Mech. 2.
A bowling ball of mass 6.0 kg is released from rest from the top of a slanted roof that is 4.0 m long and angled
300, as shown above. The ball rolls along the roof without slipping.
The rotational inertia of a sphere of
at
mass M and radius R about its center of mass is MR2.
(a) On the figure below, draw and label the forces (not components) acting on the ball at their points of
application as it rolls along the roof.
(b) Calculate the force due to friction acting on the ball as it rolls along the roof. If you need to draw anything
other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add
anything to the figure in part (a).
(c) Calculate the linear speed of the center of mass of the ball when it reaches the bottom edge of the roof.
(d) A wagon containing a box is at rest on the ground below the roof so that the ball falls a vertical distance of
3.0 m and lands and sticks in the center of the box. The total mass of the wagon and the box is 12 kg.
Calculate the horizontal speed of the wagon immediately after the ball lands in it.
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2010 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech. 3.
A skier of mass in will be pulled up a hill by a rope. as shown above, The magnitude of the acceleration of the
skier as a function of time t can be modeled by the equations
In
a= amslnT
(0< r<T)
(r
=0
T),
where amax and Tare constants. The hill is inclined at an angle U above the horizontal, and friction between the
skis and the snow is negligible. Express your answers in terms of given quantities and fundamental constants.
(a) Derive an expression for the velocity of the skier as a function of time during the acceleration, Assume the
skier starts from rest.
(b) Derive an expression for the work done by the net force on the skier from rest until terminal speed is reached.
(c) Determine the magnitude of the force exerted by the rope on the skier at terminal speed.
(d) Derive an expression for the total impulse imparted to the skier during the acceleration.
(e) Suppose that the magnitude of the acceleration is instead modeled as a
=
e
1
a
,ntT
for all
t >
0. where
am and T are the same as in the original model. On the axes below, sketch the graphs of the force exerted
by the rope on the skier for the two models, from
new model F
.
2
F
t
=
0 to a time ( > T. Label the original model I and the
iiig sin8
T
END OF EXAM
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2011 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Tinie—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions.
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part, NOT in this green insert.
I
Launching
Device
Mech.
Uxr.
I
—-
Projectile
1.
A projectile is fired horizontally from a launching device, exiting with a speed v While the projectile is in the
launching device, the impulse imparted to it is J,,, and the average force on it is f. Assume the force becomes
.
zero just as the projectile reaches the end of the launching device. Express your answers to parts (a) and (b) in terms
of vi., Jr,, F, and fundamental constants, as appropriate.
(a) Determine an expression for the time required for the projectile to travel the length of the launching device.
(b) Determine an expression for the mass of the projectile.
The projectile is fired horizontally into a block of wood that is clamped to a tabletop so that it cannot move. The
projectile travels a distance d into the block before it stops. Express all algebraic answers to the following in terms
of d and the given quantities previously indicated, as appropriate.
(c) Derive an expression for the work done in stopping the projectile.
(d) Derive an expression for the average force f exerted on the projectile as it comes to rest in the block.
Now a new projectile and block are used, identical to the first ones, but the block is
clamped to the table. The
projectile is again fired into the block of wood and travels a new distance d into the block while the block slides
across the table a short distance D, Assume the following: the projectile enters the block with speed v., the average
force
between the projectile and the block has the same value as determined in part (d), the average force of
friction between the table and the block is
and the collision is instantaneous so the frictional force is negligible
during the collision.
(e) Derive an expression for d,, in terms of d, D,
f
.
and fr, as appropriate.
D Derive an expression for d,, in terms of d. the mass
,
,t,
of the projectile. and the mass M of the block.
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53
2011 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Passenger
I
E
D
3R
Mech. 2.
An amusement park ride features a passenger compartment of mass M that is released from rest at point A, as
shown in the figure above, and moves along a track to point E. The compartment is in free fall between points A
and B, which are a distance of 3R/4 apart, then moves along the circular arc of radius R between points B and D.
Assume the track is frictionless from point A to point D and the dimensions of the passenger compartment are
negligible compared to R.
(a) On the dot below that represents the passenger compartment, draw and label the forces (not components) that
act on the passenger compartment when it is at point C. which is at an angle 8 from point B.
(h) In terms of 9 and the magnitudes of the forces drawn in part (a), determine an expression for the magnitude of
the centripetal force acting on the compartment at point C. If you need to draw anything besides what you have
shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figure in part (a).
(c) Derive an expression for the speed VD of the passenger compartment as it reaches point D in terms of M, R.
and fundamental constants, as appropriate.
A force acts on the compartment between points D and E and brings it to rest at point E.
(d) If the compartment is brought to rest by friction, calculate the numerical value of the coefficient of friction i’
between the compartment and the track.
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2011 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(e) Now consider the case in which there is no friction between the compartment and the track, but instead the
compartment is brought to rest by a braking force —kv, where k is a constant and v is the velocity of the
compartment. Express all algebraic answers to the following in terms of M, R, k, UD, and fundamental
constants, as appropriate.
i. Write, but do NOT solve, the differential equation for v(t).
ii. Solve the differential equation you wrote in parti.
iii. On the axes below, sketch a graph of the magnitude of the acceleration of the compartment as a function of
time. On the axes, explicitly label any intercepts, asymptotes. maxima, or minima with numerical values or
algebraic expressions, as appropriate.
Magnitude of
Acceleration
0
Time
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54
55
2011 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
\\
ivlech.
0
3.
The torsion pendulum shown above consists of a disk of rotational inertia I suspended by a tiexible rod attached to
a rigid support. When the disk is twisted through a small angle 8, the twisted rod exerts a restoring torque r that is
proportional to the angular displacement: r = —/38 where ,8 is a constant. The motion of a torsion pendulum is
analogous to the motion of a mass oscillating on a spring.
,
(a) In terms of the quantities given above, write but do NOT solve the differential equation that could he used to
determine the angular displacement 8 of the torsion pendulum as a function of time t.
(b) Using the analogy to a mass oscillating on a spring, determine the period of the torsion pendulum in terms of the
given quantities and fundamental constants, as appropriate.
To determine the torsion constant /3 of the rod, disks of different, known values of rotational inertia are attached to
the rod, and the data below are obtained from the resulting oscillations.
Rotational Inertia I of Disk(kg.m
)
2
Average Time for Ten Oscillations (s)
Period T (s)
0.025
0.036
0.049
0.064
0.081
22.4
26.8
29.5
33.3
35.9
2.24
2.68
2.95
3.33
3.59
2
T
(2)
5.0
7.2
8.7
11.1
12.9
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-855
56
2011 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(c) On the graph below, plot the data points. Draw a straight line that best represents the data,
14.0
zz:
12.0
10.()
-
Lt
-‘
4-
L
1
6.0
:H
4.0
fJ
-*
f
--
-H
-
2.0
0
z
0.02
zzz
zz
0.04
0.06
I of Disk 2
(kg.rn
)
0.08
0.10
(d) Determine the equation for your line.
(e) Calculate the torsion constant /3 of the rod from your line.
(f) What is the physical significance of the intercept of your line with the vertical axis?
END OF EXAM
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57
2012 AP” PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTft)N H
Time—45 minutes
3 Questions
Directions: Answer all threa que’aions. [ha suggested time is ab nit 15 minutes tor answ acing ach d the questir ns,
hnih are w rth 15 points each The parts within a questic n ina not have equal weight Sh w all y ut w ik ii. tais
booklet in the spaces pr vided after each part
hquilibrium Postam
Spring Constici I.
iass—O
Mech, 1.
jpimentl. A block of mass 030 kg is placed on a frictionless table and is attached to one end ol a horizontal
spring of spring con’tant k, as shown abos e. The other end of the spring is attached to a fixed wall. Ihe block ic
set into cscillatory motion by stretching the spring and releasing the block from rest at time t = 0. A motion
detector is used to record the position of the block as it oscillates. The resulting graph of selocity e versus time t
is shown helms, ‘I he positis e direction for all quantities is to the right.
(i2
0
C
flnie its)
a) Determine the equation for u
t;.
including numerical s alues for all constants.
b) Given that the equilibrium position is at a
for all constants.
0, determine the equation for x(t), including numerical values
(c) Calculate the salue ci k
i
0h
II g B
21
rI
i
ilc iiord
V
11
g
B
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58
2012 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
laced on a wueh surface, as shev n beie The bli
Esperiinent. The bluck and spring rrincenient 5
se
compresed
Is
a
dislnce
J
and released from rest.
that
the
sprine
is displaced
Equl ihr.un l’i’sit un
d On the dots below that reptesent the block, draw and label the torce (not coinp nentc) that at n the bk ek
d 2 and the block ic moving in the direction indicated hek w
when the spring is cmpressed a distance a
each dot
—
S
Away from
the equilibrium position
Toward
the equilibrium position
te) Draw a sketch of v sersus t in this case, Assume that there is a neeligible change in the period and that the
positive direction is still to the right.
0
0.0
0.5
1 .0
,_l,inie os
‘
Vi it
c
d g P
1 .5
1 Ih C ikge Bo.id.
CiiL h’ rd. ri;
d r ‘n V f
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58
59
2012 APe PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech.
2.
u are to per1 rtn an experiment ins estigating the censors ation c f mechaniLal energy ns Is in a
tiausft rmatIon Ir rti initial gras itati nal p ental energy t tianslatu nal kinetic energs
u are gisen the equipment listed hel iW, all the sUpp( rts required to hold the equipment, and a lab table
On the list belcw, inlicate each piece ii equipment you isould use by checking the line next ti eah item
hiects f different niasses
Irack
Meterstick
Set
Cart
Elecer mc balance
Lmghtsseight Ii sstriuit n policy
Stung
Stipwatch
Ii
b) Outline a procedure for performing the experiment. Include a diagram of your experimental setup. Labe the
equipment in your diagram. Also include a description of the measurements you would make and a syiribi 1
for each measurement,
ci Gis e a detailed account of the calculatw nc of gras itational potential energy and translati mal kinetic enerey
h th heft re and after the transformation, in terms of the quantities measured in part rh).
(di Aftem your first trial, your calculatk ns show that the energy increased during the experiment. Assuming you
made no mathematical errorc, gise a reasonable explanation for this result.
e On all other trials, your calculations show that the energy decreased during the experiment. Acsuming y )u
made no mathematical errors, gis e a reas unable phy sical explanation for the fact that the average energy y lu
determined decreased, Include references to consersatise and nonconsersatise forces, as appropriate.
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60
2012 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Slidina Only
Slfthng and luinin’ I nannp with
F
F
\‘
\hding
—
S
I
Nlnictin
Mach.
I iction with ( clticien!
i
I
A ring t mass M, radius R, and r jtational inertia MR i initially sliding on a fricti mless surface at c ntant
val city a tithe right, as ch wn above At time
0 it encountei a surface with cool ilcient of friull n i and
beams sliding and r tatmg. Alter traveling a distanco L, the ring begins ft lung with )ut sliding Express all
answ cr5 tC the follow inn in terms of M. R, a ,p, and fundamental constants, as appn priate
(a) Starting from Newton’s second law in either translational or rotti nal form, as appropriate. derive a
differential equation that can he used to s lye for the magnitude of the following as the ring is cliding and
rotating.
i,
ii,
The linear velocity v of the ring as a function of time
The angular velocity
oi
of the ring as a function of time
(b) Deny e an expression for the magnitude of the following as the ring is sliding and rotating
i.
The linear velocity a of the ring as a function of time
ii. The angular velocity
of the ring as a function of time
(c) Derive an expression for the time it takes the ring to tray el the distance L
d) Derive an expression for the magnitude of the velocity of the ring immediately alter it hac traveled the
distance L.
(e
Derive an expression for the distance L.
STOP
END OF EXAM
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61
2013 AP PHYSICS C MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions.
which are worth 15 points each. The parts within a question may nor have equal weight. Show all your work in this
booklet in the spaces provided after each part.
I.i
a S.’i
Il
:1
\n lhIk
I
II
Figure 1
Motion Sensor
Re I1eci r
IIU I
lOin
o5 In
1.5w
Figuic 2
Mech
I.
A student places a 0.40 kg glider on an air track of negligible friction and holds it so that it touches an
uncompressed ideal spring, as shown in Figure 1 above. The student then pushes the glider back to compress the
spring by 0.25 m, as shown in Figure 2. At time t 0, the student releases the glider, and a motion sensor begins
recording the velocity of the reflector at the front of the glider as a function of time. The data points are shown in
the table below. At time t = 0.79 s, the glider loses contact with the spring.
[jme(s)
\eloji (mIs)
\
0
025
050
07S
l00
ISo
200
0
02S
04
01S
() 50)
049
051
isit
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-.561
62
2013 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(a) On the axes below, plot the data points for velocity u as a function of time r for the glider, and dras a
smooth cune that best fits the data. Be sure to label an appropriate scale on the sertical axis,
I
I
I
I
I
2(3
10
(b) The student sishes to use the data to plot position x as a function of time t for the glider.
i. Describe a method the student could use to do this.
ii. On the axes below, sketch the position x as a function of time i for the glider. Explicitly label any
intercepts. asymptotes, maxima. or minima with numerical values or algebraic expressions. as
appropriate.
i g
I •()
(c) Calculate the time at which the glider makes contact with the bumper at the far right.
(d) Calculate the force constant of the spring.
te The experiment is run again. but this time the glider is attached to the spring rather than simply being pushed
against it.
i. Determine the amplitude of the resulting periodic motion.
ii. Calculate the period of oscillation of the resulting periodic motion.
iJ3
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63
2013 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
1]
I
Me,c.h 2.
A box of mass in initially at rest is acted upon by a constant applied force of magnitude F as shown in the
figure above. The friction between the box and the horizontal surface can he assumed to he negligible. hut the
box is subject to a drag force of magnitude kv where u is the speed of the box and k is a positive constant.
Express all your answers in terms of the given quantities and fundamental constants, as appropriate.
.
(a) The dot below represents the box. Draw and label the forces (not components) that act on the box.
(b) Write, but do not solve, a differential equation that could be used to determine the speed v of the box as a
function of time t. If you need to draw anything other than what you have shown in part (a) to assist in your
solution. use the space below. Do NOT add anything to the figure in part (a).
(c) Determine the magnitude of the terminal velocity of the box.
(d Use the differential equation from part (b) to derive the equation for the speed u of the box as a function of
time r. Assume that v = 0 at time t = 0.
(e) On the axes below, sketch a graph of the speed u of the box as a function of time t. Explicitly label any
intercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.
(1
\
st
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64
2013 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
I
‘\ote: I igule not dran to ik.
Mech 3.
A disk of mass M = 2,0 kg and radius R 0,10 m is supported by a rope of negligible mass, as shown abose. The
rope is attached to the ceiling at one end and passes under the disk. The other end of the rope is pulled upward
with a force F
1 The rotational inertia of the disk around its center is MR
/2.
2
4 necessary to hold the disk at rest,
(a) Calculate the magnitude of the force F
At time t = 0, the force FA is increased to 12 N, causing the disk to accelerate upward. The rope does not slip on
the disk as the disk rotates.
(b) Calculate the linear acceleration of the disk.
(c) Calculate the angular speed of the disk at t = 3.0 s.
(d) Calculate the increase in total mechanical energy of the disk from t
0 to t
=
3.0 s.
(e) The disk is replaced by a hoop of the same mass and radius. Indicate whether the linear acceleration of the
hoop is greater than, less than, or the same as the linear acceleration of the disk,
—
Greater than
—
Less than
—
The same as
Justify your answer.
STOP
END OF EXAM
2013 iheC OIIe,L BOMd.
S t th C oIkg. Board on thc WLb: iwwcollegboird.org.
-864