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2007 Semi-Final Exam

INSTRUCTIONS

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Work Part A first. You have 90 minutes to complete all four problems. After you have completed Part A, you may take a break. Then work Part B. You have 90 minutes to complete both problems. Show all your work. Partial credit will be given. Start each question on a new sheet of paper. Be sure to put your name in the upper right-

hand corner of each page, along with the question number and the page number/total pagesfor this problem. For example,

Doe, JamieA1 1/3

A hand-held calculator may be used. Its memory must be cleared of data and programs. Youmay use only the basic functions found on a simple scientific calculator. Calculators may notbe shared. Cell phones, PDAs, or cameras may not be used during the exam or while theexam papers are present. You may not use any tables, books, or collections of formulas.

Questions with the same point value are not necessarily of the same difficulty. Do not discuss the contents of this exam with anyone until after March 27th. Good luck!

Possibly Useful Information - (Use for both part A and for part B)Gravitational field at the Earths surface g = 9.8 N/kgNewtons gravitational constant G = 6.67 x 10-11 Nm2/kg2

Coulombs constant k = 1/4 = 8.99 x 109Nm2/C2

Biot-Savart constant km = /4 = 10-7 Tm/A

Speed of light in a vacuum c = 3.00 x 108 m/sBoltzmanns constant kB = 1.38 x 10

-23 J/KAvogadros number NA = 6.02 x 10

23 (mol)-1

Ideal gas constant R = NAkB = 8.31 J/(molK)Stefan-Boltzmann constant = 5.67 x 10-8 J/(sm2K4)

Elementary charge e = 1.602 x 10-19

C1 electron volt 1 eV = 1.602 x 10-19 JPlancks constant h = 6.63 x 10-34 Js = 4.14 x 10-15 eVsElectron mass m = 9.109 x 10-31 kg = 0.511 MeV/c2

Binomial expansion (1 + x)n 1 + nx for |x|


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Copyright 2007, American Association of Physics Teachers

Semi-Final Exam

Part A

A1. A group of 12 resistors is arranged along the edges of a cube as shown in the diagram below.

The vertices of the cube are labeled a-h.

a b

c d

f

gh

e

a. (13 pts) The resistance between each pair of vertices is as follows:

Rab = Rac = Rae = 3.0

Rcg = Ref= Rbd= 8.0

Rcd= Rbf= Reg = 12.0

Rdh = Rfh = Rgh = 1.0

What is the equivalent resistance between points a and h?

b. (12 pts) The three 12.0 resistors are replaced by identical capacitors. Ccd= Cbf= Ceg =15.0 F. A 12.0 V battery is attached across points a and h and the circuit is allowed to

operate for a long period of time. What is the charge (Qcd, Qbf, Qeg) on each capacitor

after this long period of time?


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,

A2. A simple gun can be made from a uniform cylinder of lengthL0 and inside radius rc. One

end of the cylinder is sealed with a moveable plunger and the other end is plugged with acylindrical cork bullet. The bullet is held in place by friction with the walls of the cylinder.

The pressure outside the cylinder is atmospheric pressure, . The bullet will just start to slide

out of the cylinder if the pressure inside the cylinder exceeds .

0P

crP

a. There are two ways to launch the bullet: either by heating the gas inside the cylinder and

keeping the plunger fixed, or by suddenly pushing the plunger into the cylinder. In either case,assume that an ideal monatomic gas is inside the cylinder, and that originally the gas is at

temperature , the pressure inside the cylinder is , and the length of the cylinder is0T 0P 0.L

(8 pts) i. Assume that we launch the bullet by heating the gas without moving the plunger.

Find the minimum temperature of the gas necessary to launch the bullet. Express

your answer in terms of any or all of the variables: and .0 0 0, , , ,cr T L P crP

(8 pts) ii. Assume, instead that we launch the bullet by pushing in the plunger, and that we

do so quickly enough so that no heat is transferred into or out of the gas. Find the

length of the gas column inside the cylinder when the bullet just starts to move.

Express your answer in terms of any or all of the variables: and .0 0 0, , ,cr T L P crP

b. (9 pts) It is necessary to squeeze the bullet to get it into the cylinder in the first place. Thebullet normally has a radius that is slightly larger than the inside radius of the cylinder;

, is small compared to . The bullet has a length The walls of the cylinder

apply a pressure to the cork bullet. When a pressure is applied to the bullet along a givendirection, the bullets dimensions in that direction change by

br

b cr r r = cr 0.h L

P

x P

x E

=

for a constantEknown as Youngs modulus. You may assume that compression along one

direction does not cause expansion in any other direction. (This is true if the so-called Poissonratio is close to zero, which is the case for cork.)

If the coefficient of static friction between the cork and the cylinder is , find an expression

for . Express your answer in terms of any or all of the variables:crP 0 , , , , ,P h E r and .cr


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A3. A volume fV of fluid with uniform charge density is sprayed into a room, forming

spherical drops. As they float around the room, the drops may break apart into smaller dropsor coalesce into larger ones. Suppose that all of the drops have radiusR. Ignore inter-drop

forces and assume that 3.fV R

(10 pts) a. Calculate the electrostatic potential energy of a single drop. (Hint: suppose the

sphere has radius r. How much work is required to increase the radius by dr?).

(4 pts) b. What is the total electrostatic energy of the drops?

Your answer to (b) should indicate that the total energy increases with R. In the absence ofsurface tension, then, the fluid would break apart into infinitesimally small drops. Suppose,

however, that the fluid has a surface tension . (This value is the potential energy per unit

surface area, and is positive.)

(4 pts) c. What is the total energy of the drops due to surface tension?

(7 pts) d. What is the equilibrium radius of the drops?


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A4. A nonlinear circuit element can be made out of a parallel plate capacitor and small balls,

each of mass m, that can move between the plates. The balls collide inelastically with the

plates, dissipate all kinetic energy as thermal energy, and immediately release the charge theyare carrying to the plate. Almost instantaneously, the balls then pick up a small charge of

magnitude q from the plate; the balls are then repelled directly toward the other plate under

electrostatic forces only. Another collision happens, kinetic energy is dissipated, the balls giveup the charge, collect a new charge, and the cycle repeats. There are n0 balls per unit surface

area of the plate. The capacitor has a capacitance C. The separation dbetween the plates is

much larger than the radius rof the balls. A battery is connected to the plates in order to

maintain a constant potential difference V. Neglect edge effects and assume that magneticforces and gravitational forces may be ignored.

(5) a. Determine the time it takes for one ball to travel between the plates in terms of any or

all of the following variables: m, q, d, and V.

(5) b. Calculate the kinetic energy dissipated as thermal energy when one ball collides

inelastically with a plate surface in terms of any or all of the following variables: m,

q, d, and V.

(5) c. Derive an expression for the current between the plates in terms of the permittivity offree space, 0 , and any or all of the following variables: m, q, n0, C, and V.

(5) d. Derive an expression for the effective resistance of the device in terms of 0 , and any

or all of the following variables: m, q, n0, C, and V.

(5) e. Calculate the rate at which the kinetic energy of the balls is converted into thermal

energy in terms of 0 , and any or all of the following variables: m, q, n0, C, and V.


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Semi-Final Exam

Part B

B1. A certain mechanical oscillator can be modeled as an ideal massless spring connected to amoveable plate on an incline. The spring has spring constant k, the plate has mass m, and theincline makes an angle with the horizontal. When the system is operating correctly, the

plate oscillates between pointsA andB in the figure, located a distanceL apart. When the platereaches pointA it has zero kinetic energy, but then trips a small lever that instantaneously loads

a block of massMonto the plate. The block and plate then move down the incline to pointB,where the force from the spring stops the plate. At this point, the block falls through a hole inthe incline, allowing the plate to move back up under the force of the spring. Upon returningto pointA it collects another block, and the cycle repeats. Both the plate and the block have acoefficient of friction with the incline for both kinetic and static friction. It is reasonable that

the motion in either direction is simple harmonic in nature.

(10 pts) a. Let c be the critical value of the coefficient of friction where the block will just

start to slide under the force of gravity on an incline (without the spring acting on

it). Then let2

c = . Find in terms ofg, the acceleration of free fall, and any

or all of the following variables: andM.


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(14 pts) b. In order for this system to work correctly, it is necessary to have the correct ratiobetween the mass of the block and the mass of the plate. These masses are chosenso that the downward moving block and plate just stop at pointB while the

upward moving plate just stops at pointA. Find the ratio .M

Rm

=

(13 pts) c. The system delivers blocks to pointB with period , until the blocks run out.

After that, the plate alone oscillates with a periodT

0T

. Find the ratio 0T

T.

(13 pts) d. The plate only oscillates a few times after delivering the last block. At whatdistance up the incline, measured from pointB, does the plate come to apermanent stop?


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e

B2. A model of the magnetic properties of materials is based upon small magnetic momentsgenerated by each atom in the material. One source of this magnetic moment is the magneticfield generated by the electron in its orbit around the nucleus. For simplicity, we will assume

that each atom consists of a single electron of charge e and mass , a single proton of charge

+e and mass , and that the electron orbits in a circular orbit of radiusR about the

proton.

em

pm m

a. Magnetic Moments.

Assume that the electron orbits in thex-y plane.

(3 pts) i. Calculate the net electrostatic force on the electron from the proton. Express your

answer in terms of any or all of the following parameters: e, , R, and the

permittivity of free space,

em ,pm

0 , where

0

1

4 k

= .

(kis the Coulombs Law constant).

(5 pts) ii. Determine the angular velocity 0 of the electron around the proton in terms of

any or all of the following parameters: e, ,R, andem 0 .

(8 pts) iii. Derive an expression for the magnitude of the magnetic field eB due to the orbital

motion of the electron at a distance from thex-y plane along the axis oforbital rotation of the electron. Express your answer in terms of any or all of the

following parameters: e, ,R,

z R

em 0 , and the permeability of free space,z 0 .

(4 pts) iv. A small bar magnet has a magnetic field far from the magnet given by

0

3,

2

mB

z

=

wherez is the distance from the magnet on the axis connecting the north andsouth poles, m is the magnetic dipole moment, and 0 is the permeability of free

space. Assuming that an electron orbiting a proton acts like a small bar magnet,find the dipole moment m for an electron orbiting an atom in terms of any or all of

the following parameters: e, ,R, andem 0 .


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,

b. Diamagnetism.

We model a diamagnetic substance to have all atoms oriented so that the electron orbits are inthex-y plane, exactly half of which are clockwise and half counterclockwise when viewedfrom the positivez axis looking toward the origin. Some substances are predominantlydiamagnetic.

(3 pts) i. Calculate the total magnetic moment of a diamagnetic substance withNatoms.Write your answer in terms of any or all of the following parameters:

and, , ,ee m R N 0.

(6 pts) ii. An external magnetic field 0 0 B B z=G

is applied to the substance. Assume that the

introduction of the external field doesnt change the fact that the electron movesin a circular orbit of radiusR. Determine , the change in angular velocity of

the electron, for both the clockwise and counterclockwise orbits. Throughout this

entire problem you can assume that 0 . Write your answer in terms of

and, ee m , 0B only.

(6 pts) iii. Assume that the external field is turned on at a constant rate in a time interval t .

That is to say, when 0t=

the external field is zero and when the externalfield is

t=

t0.BG

Determine the induced emfEexperienced by the electron. Write your

answer in terms of any or all of the following parameters: , , , ,ee m R N 0B , 0 , and

0.

(6 pts) iv. Verify that the change in the kinetic energy of the electron satisfies E.

This justifies our assumption in (ii) thatR does not change.

K e =

(6 pts) v. Determine the change in the total magnetic moment m for theNatoms when the

external field is applied, writing your answer in terms of 0, , , ,ee m R N and 0.B

(3 pts) vi. Suppose that the uniform magnetic field used in the previous parts of this problemis replaced with a bar magnet. Would the diamagnetic substance be attracted orrepelled by the bar magnet? How does your answer show this?


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2008 Semifinal Exam 1

AAPT UNITED STATES PHYSICS TEAM

AIP 2008

Semifinal Exam

DO NOT DISTRIBUTE THIS PAGE

Important Instructions for the Exam Supervisor

This examination consists of three parts.

Part A has four questions and is allowed 90 minutes.

Part B has two questions and is allowed 90 minutes.

Part C has one question and is allowed 20 minutes. The answer for Part C will not be used for teamselection, but will be used for special recognition from the Optical Society of America.

The first page that follows is a cover sheet. Examinees may keep the cover sheet for all three parts ofthe exam.

The three parts are then identified by the center header on each page. Examinees are only allowed todo one part at a time, and may not work on other parts, even if they have time remaining.

Allow 90 minutes to complete Part A. Do not let students look at Part B or Part C. Collect the answersto Part A before allowing the examinee to begin Part B. Examinees are allowed a 10 to 15 minutesbreak between parts A and B.

Allow 90 minutes to complete Part B. Do not let students look at Part C or go back to Part A. Collectthe answers to part B before allowing the examinee to begin Part C. Examinees are allowed a 10 to 15minutes break between Parts B and C.

Allow 20 minutes to complete Part C. This part is optional; scores on Part C will not be used to selectthe US Team. Examinees may not go back to Part A or B.

Ideally the test supervisor will divide the question paper into 4 parts: the cover sheet (page 2), PartA (pages 3-7), Part B (pages 8-10), and Part C (page 11). Examinees should be provided the partsindividually, although they may keep the cover sheet.

The supervisor mustcollect all examination questions, including the cover sheet, at the end of the exam,as well as any scratch paper used by the examinees. Examinees may not take the exam questions. Theexamination questions may be returned to the students after March 31, 2008.

Examinees are allowed calculators, but they may not use symbolic math, programming, or graphicfeatures of these calculators. Calculators may not be shared and their memory must be cleared of dataand programs. Cell phones, PDAs or cameras may not be used during the exam or while the exampapers are present. Examinees may not use any tables, books, or collections of formulas.

Please provide the examinees with graph paper for Part A.

Copyright c2008 American Association of Physics Teachers


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2008 Semifinal Exam Cover Sheet 2


AIP 2008

Semifinal Exam

INSTRUCTIONS


Work Part A first. You have 90 minutes to complete all four problems. Each question is worth 25points. Do not look at Parts B or C during this time.

After you have completed Part A you may take a break.

Then work Part B. You have 90 minutes to complete both problems. Each question is worth 50 points.

Do not look at Parts A or C during this time. Show all your work. Partial credit will be given. Do not write on the back of any page. Do not write

anything that you wish graded on the question sheets.

Start each question on a new sheet of paper. Put your school ID number, your name, the questionnumber and the page number/total pages for this problem, in the upper right hand corner of eachpage. For example,

School ID #

Doe, Jamie

A1 - 1/3

A hand-held calculator may be used. Its memory must be cleared of data and programs. You may useonly the basic functions found on a simple scientific calculator. Calculators may not be shared. Cellphones, PDAs or cameras may not be used during the exam or while the exam papers are present.You may not use any tables, books, or collections of formulas.

Questions with the same point value are not necessarily of the same difficulty.

Part C is an optional part of the test. You will be given 20 additional minutes to complete Part C.Your score on Part C will not affect the selection for the US Team, but can be used for special prizesand recognition to be awarded by the Optical Society of America.

In order to maintain exam security, do not communicate any information about thequestions (or their answers/solutions) on this contest until after March 31, 2008.

Possibly Useful Information. You may use this sheet for all three parts of the exam.g = 9.8 N/kg G = 6.67 1011 N m2/kg2

k = 1/40 = 8.99 109 N m2/C2 km = 0/4 = 107 T m/Ac = 3.00 108 m/s kB = 1.38 10

23 J/KNA = 6.02 1023 (mol)1 R = NAkB = 8.31 J/(mol K) = 5.67 108 J/(s m2 K4) e = 1.602 1019 C1eV = 1.602 1019 J h = 6.63 1034 J s = 4.14 1015 eV sme = 9.109 10

31 kg = 0.511 MeV/c2 (1 + x)n 1 + nx for |x| 1sin 1

63 for || 1 cos 1 1

22 for || 1



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2008 Semifinal Exam Part A 3

Part A

Question A1

Four square metal plates of area A are arranged at an even spacing d as shown in the diagram. (Assumethat A d2.)

Plate 1

Plate 2

Plate 3

Plate 4d

d

d

Plates 1 and 4 are first connected to a voltage source of magnitude V0, with plate 1 positive; plates 2 and3 are then connected together with a wire. The wire is subsequently removed. Finally, the voltage sourceattached between plates 1 and 4 is replaced with a wire. The steps are summarized in the diagrams below.

Step 1 Step 2 Step 3

Find the resulting potential difference V12 between plates 1 and 2; like wise find V23 and V34, definedsimilarly.

Assume, in each case, that a positive potential difference means that the top plate is at a higher potentialthan the bottom plate.



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Question A2

A simple heat engine consists of a moveable piston in a cylinder filled with an ideal monatomic gas. Initiallythe gas in the cylinder is at a pressure P0 and volume V0. The gas is slowly heated at constant volume. Oncethe pressure reaches 32P0 the piston is released, allowing the gas to expand so that no heat either entersor escapes the gas as the piston moves. Once the pressure has returned to P0 the outside of the cylinder iscooled back to the original temperature, keeping the pressure constant. For the monatomic ideal gas youshould assume that the molar heat capacity at constant volume is given by CV = 32R, where R is the idealgas constant. You may express your answers in fractional form or as decimals. If you choose decimals, keepthree significant figures in your calculations. The diagram below is not necessarily drawn to scale.

Pressure

P

0

0

V0

32P

VolumeVmax

a. Let Vmax be the maximum volume achieved by the gas during the cycle. What is Vmax in terms of

V0? If you are unable to solve this part of the problem, you may express your answers to the remainingparts in terms of Vmax without further loss of points.

b. In terms ofP0 and V0 determine the heat added to the gas during a complete cycle.

c. In terms ofP0 and V0 determine the heat removed from the gas during a complete cycle.

d. What is the efficiency of this cycle?



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Question A3

A certain planet of radius R is composed of a uniform material that, through radioactive decay, generates anet power P. This results in a temperature differential between the inside and outside of the planet as heatis transfered from the interior to the surface.

The rate of heat transfer is governed by the thermal conductivity. The thermal conductivity of a ma-terial is a measure of how quickly heat flows through that material in response to a temperature gradient.Specifically, consider a thin slab of material of area A and thickness x where one surface is hotter than theother by an amount T. Suppose that an amount of heat Q flows through the slab in a time t. Thethermal conductivity k of the material is then

k =Q

t

1

A

x

T.

It is found that k is approximately constant for many materials; assume that it is constant for the planet.For the following assume that the planet is in a steady state; temperature might depend on position, but

does not depend on time.

a. Find an expression for the temperature of the surface of the planet assuming blackbody radiation, anemissivity of 1, and no radiation incident on the planet surface. You may express your answer in termsof any of the above variables and the Stephan-Boltzmann constant .

b. Find an expression for the temperature difference between the surface of the planet and the center ofthe planet. You may express your answer in terms of any of the above variables; you do not need toanswer part (a) to be able to answer this part.



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Question A4

A tape recorder playing a single tone of frequency f0 is dropped from rest at a height h. You stand directlyunderneath the tape recorder and measure the frequency observed as a function of time. Here t = 0 s is thetime at which the tape recorder was dropped.

t (s) f (Hz)

2.0 5814.0 6196.0 6658.0 723

10.0 801

The acceleration due to gravity is g = 9.80 m/s2 and the speed of sound in air is vsnd = 340 m/s. Ignoreair resistance. You might need to use the Doppler shift formula for co-linear motion of sources and observersin still air,

f = f0vsnd vobsvsnd vsrc

where f0 is the emitted frequency as determined by the source, f is the frequency as detected by the observer,and vsnd, vsrc, and vobs are the speed of sound in air, the speed of the source, and the speed of the observer.

The positive and negative signs are dependent upon the relative directions of the motions of the source andthe observer.

a. Determine the frequency measured on the ground at time t, in terms of f0, g, h, and vsnd. Consideronly the case where the falling tape recorder doesnt exceed the speed of sound vsnd.

b. Verify graphically that your result is consistent with the provided data.

c. What (numerically) is the frequency played by the tape recorder?

d. From what height h was the tape recorder dropped?



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STOP: Do Not Continue to Part BIf there is still time remaining for Part A, you should review your work for

Part A, but do not continue to Part B until instructed by your exam

supervisor.



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2008 Semifinal Exam Part B 8

Part B

Question B1

A platform is attached to the ground by an ideal spring of constant k; both the spring and the platformhave negligible mass; assume that your mass is mp. Sitting on the platform is a rather large lump of clay ofmass mc = rmp, with r some positive constant that measures the ratio mc/mp. You then gently step ontothe platform, and the platform settles down to a new equilibrium position, a vertical distance D below theoriginal position. Throughout the problem assume that you never lose contact with the platform.

D

h

a. You then slowly pick up the lump of clay and hold it a height h above the platform. Upon releasingthe clay you and the platform will oscillate up and down; you notice that the clay strikes the platformafter the platform has completed exactly one oscillation. Determine the numerical value of the ratioh/D.

b. Assume the resulting collision between the clay and the platform is completely inelastic. Find theratio of the amplitude of the oscillation of the platform after the collision ( Af) to the amplitude of theoscillations of the platform before the collision (Ai). Determine Af/Ai in terms of the mass ratio r andany necessary numerical constants.

c. Sketch a graph of the position of the platform as a function of time, with t = 0 corresponding to themoment when the clay is dropped. Show one complete oscillation after the clay has collided with theplatform. It is not necessary to use graph paper.

d. The above experiment is only possible if the mass ratio r is less than some critical value rc. Otherwise,despite the clay having been dropped from the height determined in part (a), the oscillating platformwill hit the clay before the platform has completed one full oscillation. On your graph in part (c)sketch the position of the clay as a function of time relative to the position of the platform for themass ratio r = rc.



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Question B2

Consider a parallel plate capacitor with the plates vertical. The plates of the capacitor are rigidly supportedin place. The distance between the plates is d. The plates have height h and area A d2. Assumethroughout this problem that the force of air resistance may be neglected; however, the force of gravitycannot be neglected. Neglect any edge effects as well as any magnetic effects.

h

/2

h/2

L

h

d

RigidSupport

String

a. A small metal ball with a mass M and a charge q is suspended from a string of length L that is tiedto a rigid support. When the capacitor is not charged, the metal ball is located at the center of thecapacitor at a distance d/2 from both plates and at a height h/2 above the bottom edge of the plates.If instead a constant potential difference V0 is applied across the plates, the string will make an angle0 to the vertical when the metal ball is in equilibrium.

i. Determine 0 in terms of the given quantities and fundamental constants.

ii. The metal ball is then lifted until it makes an angle to the vertical where is only slightlygreater than 0. The metal ball is then released from rest. Show that the resulting motion issimple harmonic motion and find the period of the oscillations in terms of the given quantitiesand fundamental constants.

iii. When the ball is at rest in the equilibrium position 0, the string is cut. What is the maximumvalue for V0 so that the ball will not hit one of the plates before exiting? Express your answer in

terms of the given quantities and fundamental constants.

b. Suppose instead that the ball of mass M and charge q is released from rest at a point halfway betweenthe plates at a time t = 0. Now, an AC potential difference V(t) = V0 sin t is also placed across thecapacitor. The ball may hit one of the plates before it falls (under the influence of gravity) out of theregion between the plates. If V0 is sufficiently large, this will only occur for some range of angularfrequencies min < < max. You may assume that min

g/h and max

g/h. Making these

assumptions, find expressions for min and max in terms of the given quantities and/or fundamentalconstants.

c. Assume that the region between the plates is not quite a vacuum, but instead humid air with a uniformresistivity . Ignore any effects because of the motion of the ball, and assume that the humid air doesntchange the capacitance of the original system.

i. Determine the resistance between the plates.

ii. If the plates are originally charged to a constant potential source V0, and then the potential isremoved, how much time is required for the potential difference between the plates to decrease toa value of V0/e, where ln e = 1?

iii. If the plates are instead connected to an AC potential source so that the potential differenceacross the plates is V0 sin t, determine the amplitude I0 of the alternating current through thepotential source.



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STOP: Do Not Continue to Part CIf there is still time remaining for Part B, you should review your work for

Part B, but do not continue to Part C until instructed by your exam

supervisor. You may not return to Part A



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2008 Semifinal Exam Part C 11

Optical Society of America Bonus Question

Researchers have developed a lens made of liquid. The spherical lens consists of a droplet of transparentliquid resting on an electrically controllable surface. When the voltage of the surface is changed, the dropletitself changes shape; it either tries to ball-up more strongly or it becomes flatter. The figure below is asketch of the liquid lens and several parameters that describe it, including the thickness of the lens ( t), the

radius of curvature of the top surface (R) and the contact angle (), which represents the angle between theflat surface beneath the droplet and the tangent to the curved surface at the point of contact.

RR

t

a. When a certain voltage is applied, both the contact angle and lens thickness increase (and the lensbecomes more curved). In this case, is the liquid attracted or repelled by the surface?

b. Express the contact angle as a function ofR and t.

c. The total volume of the liquid lens is an important parameter because as the liquid lens changes shape,its volume is conserved. Calculate the volume of the lens as a function of R and t.

d. Use your result to part (b) to eliminate the variable t from your expression for the volume and findV(R, ).

e. By changing the voltage on the control surface, the contact angle, , can be changed, which in turnchanges the focal length of the lens, f. The lensmakers formula can be used to calculate the focallength and is given by

1f

= (nliquid nair)

1R1

1R2

,

where nliquid and nair are the refractive indices of the liquid in the lens and air around it, and R1 andR2 are the radii of curvature of the two surfaces of the lens. In figure 1, R1 is the curved face and R2is the flat face. Use the lensmakers formula to calculate the focal length of the lens in terms of thetotal volume of the liquid, the contact angle, and the relevant refractive indices.

Sidenote: liquid lenses are interesting because they are electrically controllable, variable focus lensesthat can be very compact. People are working on putting them into cell phone cameras for ultra-compact zoom lenses. For more information on this type of liquid lens, see T. Krupenkin, S. Yang,and P. Mach, Tunable liquid microlens, Appl. Phys. Lett. 82, 316-318 (2003).



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AIP 2009

Semifinal Exam



This examination consists of two parts.



The first page that follows is a cover sheet. Examinees may keep the cover sheet for bothparts of the exam.

The parts are then identified by the center header on each page. Examinees are only allowedto do one part at a time, and may not work on other parts, even if they have time remaining.

Allow 90 minutes to complete Part A. Do not let students look at Part B. Collect the answersto Part A before allowing the examinee to begin Part B. Examinees are allowed a 10 to 15minutes break between parts A and B.

Allow 90 minutes to complete Part B. Do not let students go back to Part A.

Ideally the test supervisor will divide the question paper into 3 parts: the cover sheet (page2), Part A (pages 3-4), and Part B (pages 6-7). Examinees should be provided parts A andB individually, although they may keep the cover sheet.

The supervisor must collect all examination questions, including the cover sheet, at the endof the exam, as well as any scratch paper used by the examinees. Examinees may not takethe exam questions. The examination questions may be returned to the students after March31, 2009.

Examinees are allowed calculators, but they may not use symbolic math, programming, orgraphic features of these calculators. Calculators may not be shared and their memory mustbe cleared of data and programs. Cell phones, PDAs or cameras may not be used duringthe exam or while the exam papers are present. Examinees may not use any tables, books,or collections of formulas.




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AIP 2009

Semifinal Exam

INSTRUCTIONS


Work Part A first. You have 90 minutes to complete all four problems. Each question isworth 25 points. Do not look at Part B during this time.


Then work Part B. You have 90 minutes to complete both problems. Each question is worth

50 points. Do not look at Part A during this time.

Show all your work. Partial credit will be given. Do not write on the back of any page. Donot write anything that you wish graded on the question sheets.

Start each question on a new sheet of paper. Put your school ID number, your name, thequestion number and the page number/total pages for this problem, in the upper right handcorner of each page. For example,

School ID #

Doe, Jamie

A1 - 1/3

A hand-held calculator may be used. Its memory must be cleared of data and programs. Youmay use only the basic functions found on a simple scientific calculator. Calculators may notbe shared. Cell phones, PDAs or cameras may not be used during the exam or while theexam papers are present. You may not use any tables, books, or collections of formulas.


In order to maintain exam security, do not communicate any information aboutthe questions (or their answers/solutions) on this contest until after March 31,2009.

Possibly Useful Information. You may use this sheet for both parts of the exam.

g = 9.8 N/kg G = 6.67 10

11 N m2/kg2k = 1/40 = 8.99 10

9 N m2/C2 km = 0/4 = 107 T m/A

c = 3.00 108 m/s kB = 1.38 1023 J/K

NA = 6.02 1023 (mol)1 R = NAkB = 8.31 J/(mol K)

= 5.67 108 J/(s m2 K4) e = 1.602 1019 C1eV = 1.602 1019 J h = 6.63 1034 J s = 4.14 1015 eV sme = 9.109 10


63 for || 1 cos 1 1

22 for || 1



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Part A

Question A1

A hollow cylinder has length l, radius r, and thickness d, where l r d, and is made of amaterial with resistivity . A time-varying current I flows through the cylinder in the tangential

direction. Assume the current is always uniformly distributed along the length of the cylinder. Thecylinder is fixed so that it cannot move; assume that there are no externally generated magneticfields during the time considered for the problems below.

l

r

I

a. What is the magnetic field strength B inside the cylinder in terms ofI, the dimensions of thecylinder, and fundamental constants?

b. Relate the emfE developed along the circumference of the cylinder to the rate of change ofthe current dI

dt, the dimensions of the cylinder, and fundamental constants.

c. Relate E to the current I, the resistivity , and the dimensions of the cylinder.

d. The current at t = 0 is I0. What is the current I(t) for t > 0?

Question A2

A mixture of32P and 35S (two beta emitters widely used in biochemical research) is placed next toa detector and allowed to decay, resulting in the data below. The detector has equal sensitivity tothe beta particles emitted by each isotope, and both isotopes decay into stable daughters.

You should analyze the data graphically. Error estimates are not required.

Day Activity Day Activity Day Activity

0 64557 40 12441 200 11215 51714 60 6385 250 673

10 41444 80 3855 300 46720 27020 100 273430 18003 150 1626

a. Determine the half-life of each isotope. 35S has a significantly longer half-life than 32P.

b. Determine the ratio of the number of 32P atoms to the number of 35S atoms in the originalsample.



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Question A3

Two stars, each of mass M and separated by a distance d, orbit about their center of mass. Aplanetoid of mass m (m M) moves along the axis of this system perpendicular to the orbitalplane.

zd

axis perpendicular to plane of orbit

Let Tp be the period of simple harmonic motion for the planetoid for small displacements fromthe center of mass along the z-axis, and let Ts be the period of motion for the two stars. Determine

the ratio Tp/Ts.This problem was adapted from a problem by French in Newtonian Mechanics.

Question A4

A potato gun fires a potato horizontally down a half-open cylinder of cross-sectional area A. Whenthe gun is fired, the potato slug is at rest, the volume between the end of the cylinder and thepotato is V0, and the pressure of the gas in this volume is P0. The atmospheric pressure is Patm,where P0 > Patm. The gas in the cylinder is diatomic; this means that Cv = 5R/2 and Cp = 7R/2.The potato moves down the cylinder quickly enough that no heat is transferred to the gas. Frictionbetween the potato and the barrel is negligible and no gas leaks around the potato.

Lclosed end open end

potato

The parameters P0, Patm, V0, and A are fixed, but the overall length L of the barrel may bevaried.

a. What is the maximum kinetic energy Emax with which the potato can exit the barrel? Express

your answer in terms of P0, Patm, and V0.

b. What is the length L in this case? Express your answer in terms of P0, Patm, V0, and A.



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STOP: Do Not Continue to Part B

If there is still time remaining for Part A, you should review your work for


supervisor.



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Part B

Question B1

A bowling ball and a golf ball are dropped together onto a flat surface from a height h. The bowlingball is much more massive than the golf ball, and both have radii much less than h. The bowling

ball collides with the surface and immediately thereafter with the golf ball; the balls are droppedso that all motion is vertical before the second collision, and the golf ball hits the bowling ball atan angle from its uppermost point, as shown in the diagram. All collisions are perfectly elastic,and there is no surface friction between the bowling ball and the golf ball.

h

l

After the collision the golf ball travels in the absence of air resistance and lands a distance laway. The height h is fixed, but may be varied. What is the maximum possible value of l, andat what angle is it achieved?

You may present your results as decimals, but remember that you are not allowed to usegraphical or algebraic functions of your calculator.



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Question B2

An electric dipole consists of two charges of equal magnitude q and opposite sign, held rigidly apartby a distance d. The dipole moment is defined by p = qd.

Now consider two identical, oppositely oriented electric dipoles, separated by a distance r, asshown in the diagram.

r

A B

dd

a. It is convenient when considering the interaction between the dipoles to choose the zero ofpotential energy such that the potential energy is zero when the dipoles are very far apart

from each other. Using this convention, write an exact expression for the potential energy ofthis arrangement in terms of q, d, r, and fundamental constants.

b. Assume that d r. Give an approximation of your expression for the potential energyto lowest order in d. Rewrite this approximation in terms of only p, r, and fundamentalconstants.

c. What is the force (magnitude and direction) exerted on one dipole by the other? Continueto make the assumption that d r, and again express your result in terms of only p, r, andfundamental constants.

d. What is the electric field near dipole B produced by dipole A? Continue to make the assump-

tion that d r and express your result in terms of only p, r, and fundamental constants.



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AIP 2010

Semifinal Exam








Allow 90 minutes to complete Part A. Do not let students look at Part B. Collect the answersto Part A before allowing the examinee to begin Part B. Examinees are allowed a 10 to 15minutes break between parts A and B.



The supervisor must collect all examination questions, including the cover sheet, at the endof the exam, as well as any scratch paper used by the examinees. Examinees may not takethe exam questions. The examination questions may be returned to the students after March31, 2010.




29/48



AIP 2010

Semifinal Exam

INSTRUCTIONS




Then work Part B. You have 90 minutes to complete both problems. Each question is worth

50 points. Do not look at Part A during this time.


Start each question on a new sheet of paper. Put your school ID number, your name, thequestion number and the page number/total pages for this problem, in the upper right handcorner of each page. For example,

School ID #

Doe, Jamie

A1 - 1/3

A hand-held calculator may be used. Its memory must be cleared of data and programs. Youmay use only the basic functions found on a simple scientific calculator. Calculators may notbe shared. Cell phones, PDAs or cameras may not be used during the exam or while theexam papers are present. You may not use any tables, books, or collections of formulas.


In order to maintain exam security, do not communicate any information aboutthe questions (or their answers/solutions) on this contest until after March 31,2010.


g = 9.8 N/kg G = 6.67 1011

N m2

/kg2

k = 1/40 = 8.99 109 N m2/C2 km = 0/4 = 10

7 T m/Ac = 3.00 108 m/s kB = 1.38 10

23 J/KNA = 6.02 10

23 (mol)1 R = NAkB = 8.31 J/(mol K) = 5.67 108 J/(s m2 K4) e = 1.602 1019 C1eV = 1.602 1019 J h = 6.63 1034 J s = 4.14 1015 eV sme = 9.109 1031 kg = 0.511 MeV/c2 (1 + x)

n 1 + nx for |x| 1sin 1

63 for || 1 cos 1 1

22 for || 1



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Part A

Question A1

An object of mass m is sitting at the northernmost edge of a stationary merry-go-round of radiusR. The merry-go-round begins rotating clockwise (as seen from above) with constant angular

acceleration of . The coefficient of static friction between the object and the merry-go-round iss.

a. Derive an expression for the magnitude of the objects velocity at the instant when it slidesoff the merry-go-round in terms of s, R, , and any necessary fundamental constants.

b. For this problem assume that s = 0.5, = 0.2 rad/s2, and R = 4 m. At what angle, asmeasured clockwise from north, is the direction of the objects velocity at the instant whenit slides off the merry-go-round? Report your answer to the nearest degree in the range 0 to360.

Question A2

A spherical shell of inner radius a and outer radius b is made of a material of resistivity andnegligible dielectric activity. A single point charge q0 is located at the center of the shell. At timet = 0 all of the material of the shell is electrically neutral, including both the inner and outersurfaces. What is the total charge on the outer surface of the shell as a function of time for t > 0?Ignore any effects due to magnetism or radiation; do not assume that b a is small.

Question A3

A cylindrical pipe contains a movable piston that traps 2.00 mols of air. Originally, the air is atone atmosphere of pressure, a volume V0, and at a temperature of T0 = 298 K. First (process A)the air in the cylinder is compressed at constant temperature to a volume of 1

4V0. Then (process

B) the air is allowed to expand adiabatically to a volume of V = 15.0 L. After this (process C)this piston is withdrawn allowing the gas to expand to the original volume V0 while maintaining aconstant temperature. Finally (process D) while maintaining a fixed volume, the gas is allowed toreturn to the original temperature T0. Assume air is a diatomic ideal gas, no air flows into, or outof, the pipe at any time, and that the temperature outside the remains constant always. Possiblyuseful information: Cp =

72

R, Cv =52

R, 1 atm = 1.01 105 Pa.

a. Draw a P-V diagram of the whole process.

b. How much work is done on the trapped air during process A?

c. What is the temperature of the air at the end of process B?



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Question A4

The energy radiated by the Sun is generated primarily by the fusion of hydrogen into helium-4. Instars the size of the Sun, the primary mechanism by which fusion takes place is the proton-protonchain. The chain begins with the following reactions:

2 p X1 + e+

+ X2 (0.42 MeV) (A4-1)

p + X1 X3 + (5.49 MeV) (A4-2)

The amounts listed in parentheses are the total kinetic energy carried by the products, includinggamma rays. p is a proton, e+ is a positron, is a gamma ray, and X1, X2, and X3 are particlesfor you to identify.

The density of electrons in the Suns core is sufficient that the positron is annihilated almostimmediately, releasing an energy x:

e+ + e 2 (x) (A4-3)

Subsequently, two major processes occur simultaneously. The pp I branch is the single reaction

2 X3 4He + 2 X4 (y), (A4-4)

which releases an energy y. The pp II branch consists of three reactions:

X3 +4He X5 + (A4-5)

X5 + e X6 + X7 (z) (A4-6)

X6 + X4 24He (A4-7)

where z is the energy released in step A4-6.

a. Identify X1 through X7. X2 and X7 are neutral particles of negligible mass. It is useful toknow that the first few elements, in order of atomic number, are H, He, Li, Be, B, C, N, O.

b. The mass of the electron is 0.51 MeV/c2, the mass of the proton is 938.27 MeV/c2, andthe mass of the helium-4 nucleus is 3 727.38 MeV/c2. Find the energy released during theproduction of one helium-4 nucleus, including the kinetic energy of all products and all energycarried by gamma rays.

c. Find the unknown energies x and y above.

d. Step (A4-6) does not proceed as follows because there is insufficient energy.

X5 X6 + e+

+ X7

What constraint does this fact place on z?

e. In which of the reaction steps is the energy carried by any given product the same every timethe step occurs? Assume that the kinetic energy carried in by the reactants in each step isnegligible, and that the products are in the ground state.



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STOP: Do Not Continue to Part B

If there is still time remaining for Part A, you should review your work for


supervisor.



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Part B

Question B1

A thin plank of mass M and length L rotates about a pivot at its center. A block of mass m Mslides on the top of the plank. The system moves without friction. Initially, the plank makes an

angle 0 with the horizontal, the block is at the upper end of the plank, and the system is at rest.Throughout the problem you may assume that 1, and that the physical dimensions of theblock are much, much smaller than the length of the plank.

x

L/2

Let x be the displacement of the block along the plank, as measured from the pivot, and let be the angle between the plank and the horizontal. You may assume that centripetal accelerationof the block is negligible compared with the linear acceleration of the block up and down the plank.

a. For a certain value of 0, x = k throughout the motion, where k is a constant. What is thisvalue of0? Express your answer in terms of M, m, and any fundamental constants that yourequire.

b. Given that 0 takes this special value, what is the period of oscillation of the system? Expressyour answer in terms of M, m, and any fundamental constants that you require.

c. Determine the maximum value of the ratio between the centripetal acceleration of the blockand the linear acceleration of the block along the plank, writing your answer in terms of mand M, therefore justifying our approximation.



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Question B2

These three parts can be answered independently.

a. One pair of ends of two long, parallel wires are connected by a resistor, R = 0.25 , and afuse that will break instantaneously if 5 amperes of current pass through it. The other pair

of ends are unconnected. A conducting rod of mass m is free to slide along the wires underthe influence of gravity. The wires are separated by 30 cm, and the rod starts out 10 cmfrom the resistor and fuse. The whole system is placed in a uniform, constant magnetic fieldof B = 1.2 T as shown in the figure. The resistance of the rod and the wires is negligible.When the rod is released is falls under the influence of gravity, but never loses contact withthe long parallel wires.

Resistor Fuse

Sliding Rod

The magnetic field isdirected into the page

i. What is the smallest mass needed to break the fuse?

ii. How fast is the mass moving when the fuse breaks?

b. A fuse is composed of a cylindrical wire with length L and radius r L. The resistivity(not resistance!) of the fuse is small, and given by f. Assume that a uniform current I flowsthrough the fuse. Write your answers below in terms of L, r, f, I, and any fundamentalconstants.

i. What is the magnitude and direction of the electric field on the surface of the fuse wire?

ii. What is the magnitude and direction of the magnetic field on the surface of the fusewire?

iii. The Poynting vector, S is a measure of the rate of electromagnetic energy flow througha unit surface area; the vector gives the direction of the energy flow. Since S = 1

0

E B,

where 0 is the permeability of free space and and E and B are the electric and magnetic

field vectors, find the magnitude and direction of the Poynting vector associated withthe current in the fuse wire.

c. A fuse will break when it reaches its melting point. We know from modern physics that ahot object will radiate energy (approximately) according to the black body law P = AT4,where T is the temperature in Kelvin, A the surface area, and is the Stefan-Boltzmannconstant. If Tf = 500 K is the melting point of the metal for the fuse wire, with resistivityf = 120 n m, and If = 5 A is the desired breaking current, what should be the radius ofthe wire r?



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42/48



AIP 2012

Semifinal Exam








Allow 90 minutes to complete Part A. Do not let students look at Part B. Collect the answersto Part A before allowing the examinee to begin Part B. Examinees are allowed a 10 to 15minute break between parts A and B.



The supervisor must collect all examination questions, including the cover sheet, at the endof the exam, as well as any scratch paper used by the examinees. Examinees may not takethe exam questions. The examination questions may be returned to the students after April1, 2012.





43/48



AIP 2012

Semifinal Exam

INSTRUCTIONS




Then work Part B. You have 90 minutes to complete both problems. Each question is worth50 points. Do not look at Part A during this time.


Start each question on a new sheet of paper. Put your AAPT ID number, your name, thequestion number and the page number/total pages for this problem, in the upper right handcorner of each page. For example,

AAPT ID #

Doe, Jamie

A1 - 1/3 A hand-held calculator may be used. Its memory must be cleared of data and programs. You

may use only the basic functions found on a simple scientific calculator. Calculators may notbe shared. Cell phones, PDAs or cameras may not be used during the exam or while theexam papers are present. You may not use any tables, books, or collections of formulas.


In order to maintain exam security, do not communicate any information aboutthe questions (or their answers/solutions) on this contest until after April 1, 2012.


g = 9.8 N/kg G = 6.67 1011

N m2

/kg2

k = 1/40 = 8.99 109 N m2/C2 km = 0/4 = 10

7 T m/Ac = 3.00 108 m/s kB = 1.38 10

23 J/KNA = 6.02 10

23 (mol)1 R = NAkB = 8.31 J/(mol K) = 5.67 108 J/(s m2 K4) e = 1.602 1019 C1eV = 1.602 1019 J h = 6.63 1034 J s = 4.14 1015 eV sme = 9.109 10


63 for || 1 cos 1 1

22 for || 1



44/48


Part A

Question A1

A newly discovered subatomic particle, the S meson, has a mass M. When at rest, it lives forexactly = 3 108 seconds before decaying into two identical particles called P mesons (peons?)

that each have a mass of M.

a. In a reference frame where the S meson is at rest, determine

i. the kinetic energy,

ii. the momentum, and

iii. the velocity

of each P meson particle in terms of M, , the speed of light c, and any numerical constants.

b. In a reference frame where the S meson travels 9 meters between creation and decay, determine

i. the velocity andii. kinetic energy of the S meson.

Write the answers in terms of M, the speed of light c, and any numerical constants.

Question A2

An ideal (but not necessarily perfect monatomic) gas undergoes the following cycle.

The gas starts at pressure P0, volume V0 and temperature T0.

The gas is heated at constant volume to a pressure P0, where > 1.

The gas is then allowed to expand adiabatically (no heat is transferred to or from the gas) topressure P0

The gas is cooled at constant pressure back to the original state.

The adiabatic constant is defined in terms of the specific heat at constant pressure Cp and thespecific heat at constant volume Cv by the ratio = Cp/Cv.

a. Determine the efficiency of this cycle in terms of and the adiabatic constant . As areminder, efficiency is defined as the ratio of work out divided by heat in.

b. A lab worker makes measurements of the temperature and pressure of the gas during the

adiabatic process. The results, in terms of T0 and P0 are

Pressure units ofP0 1.21 1.41 1.59 1.73 2.14

Temperature units ofT0 2.11 2.21 2.28 2.34 2.49

Plot an appropriate graph from this data that can be used to determine the adiabatic constant.

c. What is for this gas?



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Question A3

This problem inspired by the 2008 Guangdong Province Physics Olympiad

Two infinitely long concentric hollow cylinders have radii a and 4a. Both cylinders are insulators;the inner cylinder has a uniformly distributed charge per length of + ; the outer cylinder has auniformly distributed charge per length of.

An infinitely long dielectric cylinder with permittivity = 0, where is the dielectric constant,has a inner radius 2a and outer radius 3a is also concentric with the insulating cylinders. Thedielectric cylinder is rotating about its axis with an angular velocity c/a, where c is the speedof light. Assume that the permeability of the dielectric cylinder and the space between the cylindersis that of free space, 0.

a. Determine the electric field for all regions.

b. Determine the magnetic field for all regions.

Question A4

Two masses m separated by a distance l are given initial velocities v0 as shown in the diagram.The masses interact only through universal gravitation.

l

v0

v0

a. Under what conditions will the masses eventually collide?

b. Under what conditions will the masses follow circular orbits of diameter l?

c. Under what conditions will the masses follow closed orbits?

d. What is the minimum distance achieved between the masses along their path?



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Part B

Question B1

A particle of mass m moves under a force similar to that of an ideal spring, except that the forcerepels the particle from the origin:

F = +m2

x

In simple harmonic motion, the position of the particle as a function of time can be written

x(t) = A cos t + B sin t

Likewise, in the present case we have

x(t) = A f1(t) + B f2(t)

for some appropriate functions f1 and f2.

a. f1(t) and f2(t) can be chosen to have the form ert. What are the two appropriate values of

r?

b. Suppose that the particle begins at position x(0) = x0 and with velocity v(0) = 0. What isx(t)?

c. A second, identical particle begins at position x(0) = 0 with velocity v(0) = v0. The secondparticle becomes closer and closer to the first particle as time goes on. What is v0?



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Question B2

For this problem, assume the existence of a hypothetical particle known as a magnetic monopole.Such a particle would have a magnetic charge qm, and in analogy to an electrically chargedparticle would produce a radially directed magnetic field of magnitude

B =04

qmr2

and be subject to a force (in the absence of electric fields)

F = qmB

A magnetic monopole of mass m and magnetic charge qm is constrained to move on a vertical,nonmagnetic, insulating, frictionless U-shaped track. At the bottom of the track is a wire loopwhose radius b is much smaller than the width of the U of the track. The section of track nearthe loop can thus be approximated as a long straight line. The wire that makes up the loop hasradius a b and resistivity . The monopole is released from rest a height H above the bottom ofthe track.

Ignore the self-inductance of the loop, and assume that the monopole passes through the loopmany times before coming to a rest.

a. Suppose the monopole is a distance x from the center of the loop. What is the magnetic fluxB through the loop?

b. Suppose in addition that the monopole is traveling at a velocity v. What is the emfE in theloop?

c. Find the change in speed v of the monopole on one trip through the loop.

d. How many times does the monopole pass through the loop before coming to a rest?

e. Alternate Approach: You may, instead, opt to find the above answers to within a dimen-sionless multiplicative constant (like 2

3or 2). If you only do this approach, you will be able

to earn up to 60% of the possible score for each part of this question.

You might want to make use of the integral

1

(1 + u2)3du =

3

8

or the integral 0

sin4 d =3

8

provas - olimpiada americana

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