ALSIN, Michael Name (LAST, First): , Physics Block 1 2 3 4 ... Physics/AP Tests/AP Physics … · Without destroying or disassembling any ofthis equipment, design another practical

Instructor: ALSIN, Michael Name (LAST, First):_________________, __________________ Course: AP Physics Block (circle): 1 2 3 4 5 6 7 8 Date (MM/DD/YY): ___/___/___

Test Analysis Form C 5/11/2012 10:29 AM 1 of 2

Mechanics Lab Date

Day

Question

Kinem

atics, 1D & 2D

Statics & Dynam

ics

Differential Eqn for motion

Work, Energy, Power

Conservative F

Momen

tum, collisions

Center of mass

Circular M

otion, Torque

Rotational Kinem

atics,

Statics, D

ynamics

Angular Momentum

Oscillations, Gravitation

SHM

U(x) diagrams,

Kepler’s Laws (elliptical orbits)

Design

Observe, M

easure

Analyze Data, Errors

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1

1

TABLE OF’ INFORMATION DEVELOPED FOR 2012

CONSTANTS AND CON\ ERSION FAQ WRS

Proton mass, — 167 10 kg Electron charge magnitude, 160 < I 0 C

Neutron mass, or I 0 it) kg 1 electron s di. I eV — 1 60 1

Electron mass, In = * 1 1 l0 ‘ kg Speed of light. = 300 10’ ni

- I nis era1 era itationalAs gadr’ number. .\ ,(L I th mal (, — I 0 - :nen-.tant.-, Aceciera’ den due to eras itt nrs eral ca’ constant. S.’ I i nmi.k . - S m,t Earta ‘tiflJ.O,

Bltzmann’ cnsmnt. = l3s . lo J K

I unitted at’niic mas unit. 1 u = I It) kg = 03 MeV =

Planck’s constant. h = .o3 x l0 J.5 = 4 14 ,. 1

l = 1.09> 10 Jon = 124 10’ eV.mn

Vacuum perIllittis it. 5.85 1(1 C Nin

Coulomb’s law constant. k = I 4€ = 0(1 x 10 Nnr C

Vacuum permeability. p = 4r x 10 T.m A

Magnetic constant. k’ u,, 4 = 1 10 T.m,, A

1 atmosphere pressure. 1 atm = 1.0 Y l0 N/ nf = 1.0 x 101 Pa

meter. ni mole. mol watt, W farad. Fkiloeram. ke hertz. Hz coulomb, C tesla. ‘F

UNITSYMBOLS

oml, S nev ton. N solt, V degree Celsius. C

ampere. A pascal. Pa ohm, elect! on—s nit. e’

kelvin. K joule, J I henrs, H

PREFIXES VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES

Factor Prefix Snibo1 60 90

giga G 0 1

io mega NI cosO 1 3/2 4 5 2 3 5 1 ‘2 L 0

kilo k tan6 i I 4 3 3 oo

io- centiThe fo1loing con> entions are used in this exam

10 milli m 1. Unless otherwise stated, the frame of reference of any pr&hlein is

I r micro assumed to be inertial.11. The direction f an> electric current is the direction of tlk u. of positise

It) nano ncharge corn entional current I,

10 pim p 111. For any is lated electric charge, the electric potentid js defined as ze at

an infinite dismnce from the charge

-2-

2

2

ADVANCED PLACEMENT PHYSICS C EQUATIONS DES’ ELOPED FOR 2012

ma

F

3 JFdr \p

p * in

fr, <it\

= JF.dr

1K

—

mu

=

cit

P -

— mgh

LI=

accelerati ntc icetrequenc yheightrotational mertiaimpulsekinetic, energyspring .a ntantlengthangular mc mentummassnormal lorcp wermomentumradius or distanceposition sectorperiodtimepotential energyvelocity or speedwork done on a systempasitioncoefficient of friction

angletorqueangular speedangular accelerationphase angle

77)T = 2,r

7 2ir—I

Gm1,nr

Gm,n

1 qq,4irE r

FFq

E.JA

B—Jr

1 ‘ç’qr

Uj, ql 1 qqr

A areaB mgnetic fieldC capautanced distanceB — electric fieldF cml

fr trce1 = currentJ — current densityB — inductance

= lengthn — number t I I ) ps ci sne

per unit lengthA = number ol charge arriem

per unit solumeP = posser

Q — chargeq point chargeR resistancer = distancet = time

= potential or stored energy

V = electric potentialv = velocity oi speedp = resistisity

magnetic flux

dielectric constant

B*d€=p I

p IdJxrJB=

4t

MECHANICS

at

c ‘v 1 ,,at

1) —t r2a(r r)

ELEC RIC1TY AND MAGNETISM

aI1=

I—

K=

In =

N =

I’ =

7’ =

w =

5—

0=

(0=

= —kg

= 1

a = cos(iut + c

= = IwJ

icA

d

(1

1 1 1

= dQcit

= =

R=1A

F- pJ

I — Mu A

V - JR

RR

1R R

P IV

U1 = a’ B

t — r a’ F

t — t,17 1(1

— * mr

r11 —mrm

o —

rw

L rxp Jo>

K

(0—0) t((t

00 +144u1

F fId€XB

B —pnl

- JB.dA

F

£ L’11cit

4

-3-

3

3

ADVANCED PLACEMENT PHYSICS C EQUATIONS DEVELOPEI) FOR 2012

GEOMETRY AND TRIGONOMETR

Rectangle A area

hh C = circumferenceV — solume

Triangle5 = surface area

A 11h b =base— Ii = height

Circle / = length

A yrrr — radius

C = 2,rr

Rectangular Solid

k = iirh

Cy finder

V = r2f

S — 2rc + 2r2

Sphere

1’ —-irr

S = 4r

Right Triangle

a + b =

asinO = —

C

bcosfl —

C

atanO —

CALCUL1 S

dl ci’ dodi do dr

dx ) ax

cifi - C

dx

d(In v)

dx x

d(sini) = ccsr

cit

ci= —sinx

= 1x , a —l

nfl

Jc7di —

r citI— mt

J x

J cosxdx =

sindr — —cosx

9O12

-4-

4

4

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 pinkbooklet in the spaces provided after each part, NOT in this green insert.

max

riMech 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 hangsfrom a cord of length C and negligible mass, as shown above. Assume the size of the block is negligiblecompared to (, and the dart is moving horizontally when it hits the left side of the block at its center andbecomes embedded in it. The block swings up to a maximum angle max from the vertical. Express your

answers to the following in terms of P1, Mo’ 1, 8max, and g.

(a) Determine the speed V0 of the dart immediately before it strikes the block.

(b) The dart and block subsequently swing as a pendulum. Determine the tension in the cord whenit returns to the lowest point of the swing.

(c) At your lab table you have only the following additional equipment.

Meter stick Stopwatch Set of known massesProtractor 5 in of string Five more blocks of massSpring

Without destroying or disassembling any of this equipment, design another practical methodfor determining the speed of the dart just after it leaves the gun. Indicate the measurements youwould take. and how the speed could be determined from these measurements.

GOONTOTHENEXTPAGE

5

5

1999 PHYSICS C—MECHANICS

(d) The dart is now shot into a block of wood that is fixed in place. The block exerts a force Fon the dart that is proportional to the dart’s velocity V and in the opposite direction, that isF = —by, where b is a constant. Derive an expression for the distance L that the dartpenetrates into the block, in terms of m, v0, and b,

Mech 2. A spherical, nonrotating planet has a radius R and a uniform density P throughout its volume. Supposea 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 distanceof the ball from the center of the planet.

(a) Show that the magnitude of the force on the ball at a distance r < R from the center of the

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 centerof the planet.

F

R 2R 3R 4R 5R0

GO ON TO THE NEXT PAGE

6

6


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 centerof 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 thecenter of the planet.

(0 Write an equation that could be used to calculate the time it takes the ball to move frompoint P to the center of the planet. It is not necessary to solve this equation.

Mech 3. As shown above, a uniform disk is mounted to an axle and is free to rotate without friction. A thinuniform rod is rigidly attached to the disk so that it will rotate with the disk. A block is attached tothe end of the rod. Properties of the disk, rod, and block are as follows.

Disk: mass = 3m, radius R, moment of inertia about center ‘D mR

Rod: mass = in, length = 2R. moment of inertia about one end ‘R =

Block: mass = 2,ii

The system is held in equilibrium with the rod at an angle 6o to the vertical, as shown above, by ahorizontal string of negligible mass with one end attached to the disk and the other to a wall. Expressyour answers to the following in terms of m, R, 8, and g.

(a) Determine the tension in the string.

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7

7


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 thehorizontal position.

STOPEND OF SECTION II, MECHANICS

8

8

MMMMMMMMMMMMM2000 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 thepink 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 themeasurement apparatus, a simple pendulum swings past a photogate located at the pendulum’s lowest point, which

records the time t10 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 T2

(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 ofsignificant figures. Enter these results in the table above.

Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.AP is a registered trademark of the College Entrance Examination Board.

GO ON TO THE NEXT PAGE.-2-

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9


(b) On the axes below, plot the square of the period versus the length of the pendulum. Draw a bestfit straightline for this data.

T2 (2)

r

L — I — L _1_ J_ — L _1_ 33 _L —

L 3_ L _4 4_ — L I _I_ 33_ _L —

r

L _3_ _L _4_ I — — L 4 I_ — 3—4——— —

I I

r —

— I — — L _4_ I — — L — £ — _L —3—3—— 3—

F — — — F — —

LI__L_4_I__ L_4____3_I —_L_.

F —

_3_ _L_4_I__ 3_ £____3_I__L_-

_I_—L_4— I__L_I____3_I__L —

5 10

— 1

— 1

- — I — —L —4—1—_ L —4— _L _3_I — —L — 3—3— _L —4— I_ —3 I 3 1

— 1I I I I

- I — _L 43 _L 4 I — L 4 _I_ — L I _L — 4—3— _L 4 I _L II I

— 1

•

— F — r 1 — T — 1

-—I—— L —4—1—_ L —4——— — —I— —L — — I — —L _4_I_ — L —1———— 3_I

-— 1

I I I

— 1

—I— —L —4—1—— L —4—_I— — 3_I— —L — 3_I— —L —3_I—— L _4____ 3_II I I

I I I I I-— 1

_____

——

20

(c) Assuming that each pendulum undergoes small amplitude oscillations, from your fit determine theexperimental 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/s2 ? Justify your answer.

(e) Someone informs you that the experimental apparatus is in fact near Earth’s surface, but that the experimenthas been conducted inside an elevator with a constant acceleration a, Assuming that your experimentalvalue gexp is exact, determine the magnitude and direction of the elevator’s acceleration.


GO ON TO THE NEXT PAGE.

1.5

1.0

0.5

11

0 15 25 30 t(cm)

-3-

10

10


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 causedby the air). The drag force on the bail has magnitude by2. where b is a constant drag coefficient and v is theinstantaneous speed of the ball. The drag coefficient b is directly proportional to the cross-sectional area of theball and the density of the air and does not depend on the mass of the ball. As the ball falls, its speed approachesa 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 terminalspeed.

C

(b) State whether the magnitude of the acceleration of the ball of mass m increases, decreases, or remains thesame 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 andreaches its terminal speed before hitting the ground, in terms of the given quantities and fundamentalconstants.



11

11

MMMMMMMMMMMMM

Mech 3.

2000 AP® PHYSICS C FREE-RESPONSE QUESTIONS

A pulley of radius R1 and rotational inertia I is mounted on an axle with negligible friction. A light cord passingover the pulley has two blocks of mass m attached to either end, as shown above. Assume that the cord does p slip

on the pulley. Determine the answers to parts (a) and (b) in terms of m, R1, I, and fundamental constants.

(a) Determine the tension T in the cord.

(b) One block is now removed from the right and hung on the left. When the system is released from rest, the

three blocks on the left accelerate downward with an acceleration Determine the following.

i. The tension T3 in the section of cord supporting the three blocks on the left

ii. The tension T1 in the section of cord supporting the single block on the right

iii. The rotational inertia 1 of the pulley

Copyright © 2000 College Entrance Examination Board and Educational Testing Service. All rights reserved.AP is a registered trademark of the College Entrance Examination Board,


‘1

-5-

12

12


2R1, 161

(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 2R1 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, R1, and w.

i. The angular speed w of the larger pulley

ii. The angular momentum L2 of the larger pulley

iii. The total kinetic energy of the system

STOPEND 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.


-6-

13

13

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 thepink booklet in the spaces provided after each part. NOT in this green insert.

Mech I.

MotionSensor

-

ForceSensor

11

A motion sensor and a force sensor record the motion of a cart along a track, as shown above. The cart is given apush so that it moves toward the force sensor and then collides with it. The two sensors record the values shownin the following graphs.

(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.


>

0.30

0.20

0.10

0 ---

-0.10

z

C

40

30

20

10

0

—

0.30—0.20

0.30 0.32 0.34Timet(s)

I II I

0.36 0.38 0.40 0.32 0.34 0.36Time t (s)

0.38 0.40

4

14

14


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.

GMi. The orbital speed of the planned satellite is given by v =

i42R3ii. The period of the orbit is given by 7 =

J

(b) The explorer wants the satellite’s orbit to be synchronized with Jupiter’s rotation. This requires an equatorialorbit whose period equals Jupiter’s rotation period of 9 hr 51 mm = 3.55 x l0 s. Determine the requiredorbital radius in meters.

(c) Suppose that the injection of the satellite into orbit is less than perfect. For an injection velocity that differsfrom the desired value in each of the following ways, sketch the resulting orbit on the figure. (J is the centerof Jupiter, the dashed circle is the desired orbit, and P is the injection point.) Also, describe the resultingorbit qualitatively but specifically.

i. When the satellite is at the desired altitude over the equator, its velocity vector has the correctdirection, 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 correctdirection, but the speed is slightly slower than the correct speed for a circular orbit of that radius.

P

/

J

Copyright © 2001 by College Entrance Examination Board. All rights reserved.Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

GO ON TO THE NEXT PAGE.5

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15

Mech 3.


H 2L

A light string that is attached to a large block of mass 4,n passes over a pulley with negligible rotational inertiaand is wrapped around a vertical pole of radius r, as shown in Experiment A above. The system is released fromrest, and as the block descends the string unwinds and the vertical pole with its attached apparatus rotates. Theapparatus consists of a horizontal rod of length 2L, with a small block of mass rn attached at each end. Therotational 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 thethree blocks compare with the value 4mgD ? Check the appropriate space below.

Greater than 4ingD Equal to 4mgD Less than 4ingD

Justify your answer.

Copyright @ 2001 1w College Entrance Examination Board. All rights reserved,Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.


Experiment A

6

16

16


The system is now reset. The string is rewound around the pole to bring the large block back to its originallocation. 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 theblock 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 thethree blocks compare to that in part (c) ? Check the appropriate space below.

— Equal

END OF SECTION II, MECHANICS

Copyright © 2001 by College Entrance Examination Board. All rights reserved.Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.


H

Experiment B


Greater Less

7

17

17


PHYSICS CSection 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 thepink 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 anobject of mass Al at time t = 0. The object was initially at rest. The speed u in rn/s of the car-object systemafter the collision is given as a function of time t in seconds by the expression

81) =

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 afterthe 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 oftime t.

(d) Determine the impulse delivered to the car-object system from t = 0 to t = 2.0 s.

Copyright © 2002 by College Entrance Examination Board, Al] rights reserved.Advanced Placement Program and AP are registered trademarks of the College Entrance Examination Board.

S GO ON TO THE NEXT PAGE.

18

18

Mech 2.


nif4 —m

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 — ML2, 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) 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 negligiblemass attached to an ideal spring, which has a spring constant k, Determine the distance Xrn the sprng iscompressed 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 springis compressed to a maximum distance of about 90% of the value of x, in part (c). Give a reasonableexplanation for this decrease.

Copyright © 2002 h College Entrance Examination Board All rights reserved,Advanced Placement Program and AP are registered trademarks of the College Entrance Examtnaton Board.

Bumper

6 GO ON TO THE NEXT PAGE.

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Mech 3.

An object of mass 0.5 kg experiences a force that is associated with the potential energy function4.0U(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

x0Ti)0 1 2 3 4 5

(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 forcedetermined in part (b). Your goal is to determine experimentally the validity of your theoretical calculation inpart (c).

(d) From the list below, select the additional equipment you will need from the laboratory to do your experimentby checking the line next to each item. If you need more than one of an item, place the number you need onthe line.

Meterstick Stopwatch — Photogate timer — String Spring

Balance — Wood block — Set of objects of different masses

(e) Briefly outline the procedure you will use, being explicit about what measurements you need to makein order to determine the speed, You may include a labeled diagram of your setup if it will clarify yourprocedure


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7 GO ON TO THE NEXT PAGE.

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20




Time—45 minutes3 Questions

Copyright © 2003 by College Entrance Exandnation Board. All nghts reserved.Available to AP professionals at apcentral.collegeboardcorn and to

students and parents at ssww.collegeboard.com/apstudents.


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 thepink 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 = 0.5t3 + 2t, where x is in meters and t is inseconds.

(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 = 0 to t = 2 s would he greaterthan. less than, or equal to the answer in part (c).

Greater than Less than Equal to

5

21

21


Mech, 2.

4D

Ii

An ideal spring is hung from the ceiling and a pan of mass M is suspended from the end of the spring, stretchingit a distance D as shown above. A piece of clay, also of mass A is then dropped from a height H onto the panand 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 ofthe 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 rubberball, also of mass M, is dropped from the same height H onto the pan, and after the collision is caught in midairbefore 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).

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students and parents at vww.collegchoardcons!apswdents.


M

Greater than

Justif your answer.

Less than

______

The same as

6

22

22

Mech. 3.


Figure 1 Figure 2

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 tohe loaded with various masses greater than 10 kg is located at the other end of the arm. The arm is released from thehorizontal position, shown in Figure 1, and begins rotating. There is a mechanism (not shown) that stops the arm inthe 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 theresulting distance x traveled by the 10 kg projectile, recording the following data.

F Mass (kg) 100 300 500 700 900

j 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.

0 -:40—- T

—-—-— 1—— p-——.-

30

—.——

—

fl ——--—————1——.—— —---—— ——— — — — — —-— *

ii. Using your best-fit curve, determine the distance x traveled by the projectile if 250 kg is placed inthe counterweight bucket.

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12m

12m 2rn,

301A

___

3J,4/

Hi —-— ———— —-—- .--.*-*

—-—, ,— .———.—-—,

U 200 400 600 800 1000Mass (ku)

____________________________________________________________________

7

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23


(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 therelationship x ur. where v is the horizontal velocity of the projectile as it leaves the cup and i is the timeafter 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 groundwhen 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 massusing 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 areason for any difference.

END OF SECTiON Ii, MECHANICS

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Time—45 minutes3 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 thebooklet in the spaces provided after each part, NOT in this green insert.

Lin1 C

A 9Q0

JR

-—

[.ake

Mech, 1.

A rope of length L is attached to a support at point C. A person of mass in1 sits on a ledge at position A holdingthe other end of the rope so that it is horizontal and taut, as shown above. The person then drops off the ledgeand swings down on the rope toward position B on a lower ledge where an object of mass in2 is at rest. Atposition B the person grabs hold of the object and simultaneously lets go of the rope. The person and object thenland together in the lake at point D, which is a vertical distance L below position B. Air resistance and the massof the rope are negligible. Derive expressions for each of the following in terms of nz1. iii,, L, and .

(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 thecol 11 Si Ofl

(e) The total horizontal displacement x of the person from position A until the person and object land in thewater at point D.

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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 wrappedaround it several times. The block of mass in is released from rest and takes a time t to fall the distance D tothe 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? Explainyour reasoning.

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1)

6

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26


ii. On the grid below, plot the quantities determined in (b)i., label the axes, and draw the best-fit line tothe data.

——— .1 —- 4

•_J L. . .J.,... ..,.

..

Y-- rri-r”i 1P— .4 4 4. .4 .4 —

4 ¶..

..“.

..

- -1 r.

J.., 1..4 ..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 youranswer to (c), can be used to determine the rotational inertia of the pulley. The pulley is removed from itssupport and its rotational inertia is found to be greater than this value. Give one explanation for thisdiscrepancy.

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27

27


— 4 -H r Pivot

I 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 ofthe bottom end of the rod when the rod passes through the vertical.

TT

PivotL

(c) The rod is brought to rest in the vertical position shown above and hangs freely. ft is then displaced slightlyfrom this position. Calculate the period of oscillation as it swings.


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2005 AP PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS

PHYSICS CSection 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 thepink 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 v0. It experiences a force of air resistancegiven 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. v0, and fundamental constants.

(a> Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball movesupward?

increases decreases remains the same


(b) Write, but do NOT solve, a differential equation for the instantaneous speed u of the ball in terms of time tas 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 theheight from which it was thrown?

,,jonger to rise __jonger to fall


(e) On the axes below, sketch a graph of velocity versus time for the upward and downward parts of the ball’sflight, where t is the time at which the ball returns to the height from which it was thrown.

Velocity

.....,

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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 thedata to determine the mass M of Saturn. Assume the orbits of these moons are circular.

Orbital Period. T Orbital Radius. R( seconds) ( meters)

8.l4xl0 l.85x105

l.18x105 2.38xl08

1.63 x IO5 2.95 x 108

2.37 x i05 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 orbitalperiod 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’smass?

(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 indicatethe scale.

.LJ.JL ._J LL1 LLJ. LJLLJ.L_L[3 I———-— —I4—— 44 :4.

I—r— —i——r-— —-i——trr—-——r

I I I

I I—-1—--I——— ——-— —1—-H—— H

I3 1 3 1 —

-—3--——i--—H--—--I—f -•

I I._—t—— — I

LJ_J.L JLL1 LLLJ.i.J_L... .JI I

44--t— 4I—-H--4— 4 —.4-1--

.- —-1r—r —m—— T1—r——,r—r —i———L JLLL.

‘

3 I — - *

L_ILLI_ LJL. JLJ.

. —H——I—-——I—— 34-—-4- —H—4—-1—.— —4—H—-— H— ——-H—H—.4— — —H-—4—

-r—1- r—r—r— — --- —-r—r—j- —-i— -—t— -——r—- —,—-rr—r—i L. LL. L_i_L__ 1_JJ__L

3 1-_- ---

3

--

- — — - - — -

(f) Using the graph, calculate a value for the mass of Saturn,Copyright © 2005 by College Entrance Examination Board. All rights reserved.

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Before Collision After collision

Mech. 3.

TOP VIEWS

A system consists of a ball of mass M7 and a uniform rod of mass M and length d. The rod is attached to ahorizontal frictionless table by a pivot at point P and initially rotates at an angular speed (0 , as shown above

left. The rotational inertia of the rod about point P is M1d2. The rod strikes the ball, which is initially at rest.

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 M1, M2, (0. d, and fundamental constants.

(a) Derive an expression for the angular momentum of the rod about point P before the collision.

(b) Derive an expression for the speed v of the ball after the collision.

(c) Assuming that this collision is elastic, calculate the numerical value of the ratio M1/M

Bekre Collision

(d) A new ball with the same mass Al1 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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P

Al,

TIx

•Al

7

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31


PHYSICS C: MECHANICS

SECTION IITime—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 thepink 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 v0 of 4.0 mIs to the right. The coefficient of kinetic friction between theblock 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 oneach 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 identicalspeeds Vf.

(b) Calculate v1.

(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


Mech 2.

A nonlinear spring is compressed various distances x, and the force F required to compress it is measured for eachdistance. 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) = Ax2.

(a) Which quantities should be graphed in order to yield a straight line whose slope could be used to calculate anumerical value for A?

(b) Calculate values for any of the quantities identified in (a) that are not given in the data, and record these valuesin 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 appropriatenumbers to indicate the scale.

L

r

(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 kgmoves on the floor with negligible friction and collides hea&on with the free end of the spring, compressing it amaximum 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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33

Mech 3.


A thin hoop of mass lvi. radius R. and rotational inertia MR2 is released from rest from the top of the ramp of lengthL above. The ramp makes an angle 8 with respect to a horizontal tabletop to which the ramp is fixed. The table is aheight H above the floor. Assume that the hoop rolls without slipping down the ramp and across the table. Expressall 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 distancefrom 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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lvi. R

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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 thepink booklet in the spaces provided after each part. NOT in this green insert.

Mech. 1.

A block of mass in is 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 a1. Express all algebraic answers

in terms of in. F1 0 . a1. and fundamental constants.

(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 theblock started from rest at x = 0 and t = 0.

U

U

x

U

(c) If the applied force is large enough. the block will lose contact with the surface. Derive an expression tbr themagnitude of the greatest acceleration a0 that the block can have and still maintain contact with the ground.

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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 i03 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 periodof the GS be greater than or less than the given period?

________Greater

than

__________Less

than


(e) In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars at 3.71 x l0 mabove the surface and its furthest distance at 4.36 x 10 rn above the surface. If the speed of the GS atclosest approach is 3.40 x i03 rn/s calculate the speed at the furthest point of the orbit.

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GO ON TO ThE NEXT PAGE.-6-

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Mech. 3.

The apparatus above is used to study conservation of mechanical energy. A spring of force constant 40 N/rn is heldhorizontal over a horizontal air track, with one end attached to the air track. A light string is attached to the otherend of the spring and connects it to a glider of mass m The glider is pulled to stretch the spring an amount x fromequilibrium and then released. Before reaching the photogate, the glider attains its maximum speed and the stringbecomes 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.

TrialExtension of the Spring Speed of Glider Extension Squared Speed Squared

x (m) v (m/s) x2 (m2) v2 (m2/s2)

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

(a) Assuming no energy is lost, write the equation for conservation of mechanical energy that would apply to thissituation.

(b) On the grid below, plot v2 versus x2. Label the axes, including units and scale.

I I! I4-_4----4---- 44 4 4.4

I I I I

—

—r—4——-r ——-—r -—_4__—I I

1——--—-t ———i-—I b —

—- LL___ L11I I

rr1T T r—

-_ —-------f-_ 4__4

: ::-

---j

—-----_-----l

.—÷_ —-j---j.--

---•—--- •----•-- —• -----

JJII LIi .

.. I

I-.i ---- ...i. 4 ;i :-Li£I. JJJL LLI J.JJ 11 J..JL. LI

I . ..

-

T rr.-—-.. 4-—4--, — -.

.

I-. . liii

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Spring

-7-.

37

37


(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 isa 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 wouldapply to this situation.

ii. Will the graph of v2 versus x2 for this new experiment be a straight line?

____

Yes

____

No


END OF EXAM

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PHYSICS C: MECHANICSSECTION II

Time—45 minutes

3 Questions


Mech. 1.

A skier of mass M is skiing down a frictionless hill that makes an angle 0 with the horizontal, as shown in thediagram. The skier starts from rest at time t 0 and is subject to a velocity-dependent drag force due to airresistance of the form F = —by. where v is the velocity of the skier and b is a positive constant. Express allalgebraic 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 forcesthat 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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39

39


(e) On the axes below, sketch a graph of the acceleration a of the skier as a function of time r, and indicate theinitial value of a. Take downhill as positive.

a

0

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40

40


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 attachedto a vertical support by a frictionless hinge that allows the rod to swing up or down. The right end of the rod issupported by a cord that makes an angle of 30 with the rod. A spring scale of negligible mass measures thetension 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 forcevector 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 ML2,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.

(d) If the cord that supports the rod is cut near the end of the rod, calculate the initial angular acceleration of therod-block system about the hinge.

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Hinge

-7-

41

41


Mech. 3.

In an experiment to determine the spring constant of an elastic cord of length 0.60 m, a student hangs the cordfrom a rod as represented above and then attaches a variety of weights to the cord. For each weight, the studentallows the weight to hang in equilibrium and then measures the entire length of the cord. The data are recordedin 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 throughthe data.

30 -—rr—-———rr—rrr—r-—rrrr-—rI I

ttt2c ---±--------- H------ --,----H-------,

— Il

I I I

10 ——

I II I I

5 _-..,-—.,-—-1_—_l_- i,

I IIl

I I

() 0.5 1.0 1.5 2.0l.cnuth (in)

(b) Use the best-fit line you sketched in part (a) to determine an experimental value for the spring constant k ofthe cord.

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42

42


The student now attaches an object of unknown mass rn to the cord and holds the object adjacent to the point atwhich the top of the cord is tied to the rod, as represented above. When the object is released from rest, it falls1.5 in before stopping and turning around. Assume that air resistance is negligible.

(c) Calculate the value of the unknown mass in of the object.

(d) i. 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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-9-

43

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2009 APu PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS



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 thisbooklet in the spaces provided after each part, NOT in the green insert.

Mech. 1.

A 3.0 kg object is moving along the x-axis in a region where its potential energy as a function of x is givenas U(x) = 4.0x2. where U is injoules and x is in meters. When the object passes the point x = —0.50 m. itsvelocity is +2.0 mIs. All forces acting on the object are conservative.

(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 = 0.60 m.

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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 t.Assume that x = 0 at time i = 0. The two graphs should cover the same time interval and use the samescale on the horizontal axes.

x

—

0

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V OO9 The College Board. All rights reserved.\‘sit the College Board on the Web: www.collcgeboard.com.


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ThTTWI’

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 withthe 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 0the bar makes with the vertical.

ii. By applying the small-angle approximation to your differential equation, calculate the period of the bar’smotion.

(h) Describe the experimental procedure you would use to make the additional measurements needed todetermine ‘b’ Include how you would use your measurements to obtain ‘b and how you would minimizeexperimental error.

(c) Now suppose that you were not given the location of the center of mass of the bar. Describe an experimentalprocedure that you could use to determine it. including the equipment that you would need.

2009 The College Board. All rights reserved,Visit the College Board on the Web: www.collegeboard.com.


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Mech. 3,


T(I

A block of mass M/2 rests on a frictionless horizontal table, as shown above. It is connected to one end of astring that passes over a massless pulley and has another block of mass M/2 hanging from its other end. Theapparatus 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 therope hanging slightly over the edge of the frictionless table. The rope is released from rest, and at some time laterthere 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 v5 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.


END OF EXAM

The speeds are equal.

2009 The (Zoliege Board. All rights reer’ ed.Visit the College Board on the Web: www.collegehoard.som.

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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 thepink booklet in the spaces provided after each part, NOT in this green insert,

—iMech. 1.

Students are to conduct an experiment to investigate the relationship between the terminal speed of a stack offalling paper coffee filters and its mass. Their procedure involves stacking a number of coffee filters, like theone shown in the figure above, and dropping the stack from rest. The students change the number of filters in thestack to vary the mass in while keeping the shape of the stack the same. As a stack of coffee filters falls, thereis an air resistance (drag) force acting on the filters.

(a) The students suspect that the drag force FD is proportional to the square of the speed v: = Cu2,where

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) 1 12 x i0 204 x i0 296>< i0 4 18 x 10 5 10 x l0

Terminal speed, VT (m/s) 0.51 0.62 0.82 0.92 1.06

(i) Assuming the functional relationship for the drag force above, use the grid below to plot a linear graphas 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 placenumbers on both axes.

: : : : : :r: : : : : : : : : : : : : : ::c:

in (kg)

0 2010 The College Beard.Visit the College Board on the Web: www.collegeboardcern.


(b)

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(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 speedby 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

t = T that the filters have fallen the distance Y. Indicate time t = T and terminal speed v = on

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 airresistance during the time the stack falls a distance y, where v > Y. Express your answer in terms of v. in,

and fundamental constants.

2010 The College Board.Visit the College Board on the Web: www.collegeboard.orn.


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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 angledat 300, as shown above. The ball rolls along the roof without slipping. The rotational inertia of a sphere of

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 ofapplication 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 anythingother than what you have shown in part (a) to assist in your solution, use the space below. Do NOT addanything 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 of3.0 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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N©ic: Figure not drawn to scale.

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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 theskier as a function of time t can be modeled by the equations

Ina= amslnT (0< r<T)

=0 (r T),

where amax and Tare constants. The hill is inclined at an angle U above the horizontal, and friction between theskis 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 theskier 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 = a1e ,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 t = 0 to a time ( > T. Label the original model I and the

new model F2.

iiig sin8

F

T

END OF EXAM

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SECTION II

Tinie—45 minutes

3 Questions


I I Uxr.

Launching—- Projectile

Device

Mech. 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 termsof 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. Theprojectile travels a distance d into the block before it stops. Express all algebraic answers to the following in termsof 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. Theprojectile 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 negligibleduring 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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GO ON TOTHE NEXT PAGE.

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Passenger

Mech. 2.An amusement park ride features a passenger compartment of mass M that is released from rest at point A, asshown in the figure above, and moves along a track to point E. The compartment is in free fall between points Aand 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 arenegligible compared to R.

(a) On the dot below that represents the passenger compartment, draw and label the forces (not components) thatact 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 ofthe centripetal force acting on the compartment at point C. If you need to draw anything besides what you haveshown 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.

@‘ .O1 1 The College Board.Visit the College Board on the Weh: wwwcollegehoardorg.

I

D E

3R


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(e) Now consider the case in which there is no friction between the compartment and the track, but instead thecompartment is brought to rest by a braking force —kv, where k is a constant and v is the velocity of thecompartment. 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 oftime. On the axes, explicitly label any intercepts, asymptotes. maxima, or minima with numerical values oralgebraic expressions, as appropriate.

Magnitude ofAcceleration

0 Time

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ivlech. 3.


The torsion pendulum shown above consists of a disk of rotational inertia I suspended by a tiexible rod attached toa rigid support. When the disk is twisted through a small angle 8, the twisted rod exerts a restoring torque r that isproportional to the angular displacement: r = —/38 , where ,8 is a constant. The motion of a torsion pendulum isanalogous 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 todetermine 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 thegiven 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 tothe rod, and the data below are obtained from the resulting oscillations.

Rotational Inertia I of Disk(kg.m2) Average Time for Ten Oscillations (s) Period T (s) T2 (2)

0.025 22.4 2.24 5.00.036 26.8 2.68 7.20.049 29.5 2.95 8.70.064 33.3 3.33 11.10.081 35.9 3.59 12.9

2011 The College Board.Visit the College Board on the Web: ww.co1iegehoard.oro.


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(c) On the graph below, plot the data points. Draw a straight line that best represents the data,

(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

O H ‘1 he College Board.v hit the C ollege Board on the Web: swcu1kgcboard.org.

14.0

12.0

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6.0

4.0

2.0

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I of Disk (kg.rn2)

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2012 AP” PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS


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

Mech, 1.

Spring Constici I.

hquilibrium Postam

iass—O

jpimentl. A block of mass 030 kg is placed on a frictionless table and is attached to one end ol a horizontalspring of spring con’tant k, as shown abos e. The other end of the spring is attached to a fixed wall. Ihe block icset into cscillatory motion by stretching the spring and releasing the block from rest at time t = 0. A motiondetector 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.

0C

a) Determine the equation for u t;. including numerical s alues for all constants.

b) Given that the equilibrium position is at a 0, determine the equation for x(t), including numerical valuesfor all constants.

(c) Calculate the salue ci k

21 i ilc iiordi 0 h II g B r I V 11 g B


(i2

flnie its)

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Esperiinent. The bluck and spring rrincenient 5 laced on a wueh surface, as shev n beie The bliis displaced se that the sprine Is compresed a dislnce J and released from rest.

d On the dots below that reptesent the block, draw and label the torce (not coinp nentc) that at n the bk ekwhen the spring is cmpressed a distance a — d 2 and the block ic moving in the direction indicated hek w

each dot

S

Towardthe equilibrium position

Away fromthe 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 thepositive direction is still to the right.

0.0 0.5 1 .0 1 .5,_l,inie os

Equl ihr.un l’i’sit un

0

‘ 1 Ih C ikge Bo.id.Vi it c d g P d r ‘n V f CiiL h’ rd. ri;


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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 atiausft 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 tableOn the list belcw, inlicate each piece ii equipment you isould use by checking the line next ti eah item

Irack Meterstick Set Ii hiects f different niasses

Cart Elecer mc balance Lmghtsseight Ii sstriuit n policy

Stung Stipwatch

b) Outline a procedure for performing the experiment. Include a diagram of your experimental setup. Labe theequipment in your diagram. Also include a description of the measurements you would make and a syiribi 1for each measurement,

ci Gis e a detailed account of the calculatw nc of gras itational potential energy and translati mal kinetic enereyh 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 youmade 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 )umade no mathematical errors, gis e a reas unable phy sical explanation for the fact that the average energy y ludetermined decreased, Include references to consersatise and nonconsersatise forces, as appropriate.


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Slidina Only Slfthng and luinin’ I nannp with \‘ \hding

—F

FS

A ring t mass M, radius R, and r jtational inertia MR i initially sliding on a fricti mless surface at c ntantval 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 allansw 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 adifferential equation that can he used to s lye for the magnitude of the following as the ring is cliding androtating.

i, The linear velocity v of the ring as a function of time

ii, 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 thedistance L.

(e Derive an expression for the distance L.

STOP

END OF EXAM

Mach. I

Nlnictin

I

I iction with ( clticien! i

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2013 AP PHYSICS C MECHANICS FREE-RESPONSE QUESTIONS


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.

a S.’i I.i

Il

:1

\n lhIk

I II

Motion Sensor Re I1eci r

Figure 1

I I U I

Mech I.

A student places a 0.40 kg glider on an air track of negligible friction and holds it so that it touches anuncompressed ideal spring, as shown in Figure 1 above. The student then pushes the glider back to compress thespring by 0.25 m, as shown in Figure 2. At time t 0, the student releases the glider, and a motion sensor beginsrecording the velocity of the reflector at the front of the glider as a function of time. The data points are shown inthe table below. At time t = 0.79 s, the glider loses contact with the spring.

[jme(s) 0 025 050 07S l00 ISo 200

\eloji (mIs) 0 02S 04 01S () 50) 049 051

2013 the Coltee Board.\ isit the College Board on the Web: www.eollegehoard.org.


o5 In

Figuic 2

lOin 1.5w

-.5-

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(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

10 2(3

(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 anyintercepts. asymptotes, maxima. or minima with numerical values or algebraic expressions. asappropriate.

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 pushedagainst it.

i. Determine the amplitude of the resulting periodic motion.

ii. Calculate the period of oscillation of the resulting periodic motion.

iJ3 I he College Board.Visit the College Board on the 5\ eh: w%oIkgehoardrg.


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1]

IMe,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 thebox 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 yoursolution. 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 anyintercepts, asymptotes, maxima, or minima with numerical values or algebraic expressions, as appropriate.

(1

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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. Therope 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 F1 The rotational inertia of the disk around its center is MR2/2.

(a) Calculate the magnitude of the force F4 necessary to hold the disk at rest,

At time t = 0, the force FA is increased to 12 N, causing the disk to accelerate upward. The rope does not slip onthe 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 thehoop is greater than, less than, or the same as the linear acceleration of the disk,

— Greater than — Less than — The same as


STOP

END OF EXAM

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