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2014 USA Physics Olympiad Exam 1 USA Physics Olympiad Exam DO NOT DISTRIBUTE THIS PAGE Important Instructions for the Exam Supervisor This examination consists of two parts. Part A has four questions and is allowed 90 minutes. Part B has two questions and is allowed 90 minutes. The rst page that follows is a cover sheet. Examinees may keep the ...

2014 USA Physics Olympiad Exam Cover Sheet 2

AAPT UNITED STATES PHYSICS TEAM

USA Physics Olympiad 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 Each question is

worth 25 points Do not look at Part B 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 Part A 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 AAPT ID number your name the

question number and the page number total pages for this problem in the upper right hand

corner of each page For example

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 not

be shared Cell phones PDA s 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

In order to maintain exam security do not communicate any information about

the questions or their answers solutions on this contest until after April 15

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 kg2

k 1 4 0 8 99 109 N m2 C2 km 0 4 10 7 T m A

c 3 00 108 m s kB 1 38 10 23 J K

NA 6 02 1023 mol 1 R NA kB 8 31 J mol K

5 67 10 8 J s m2 K4 e 1 602 10 19 C

1eV 1 602 10 19 J h 6 63 10 34 J s 4 14 10 15 eV s

me 9 109 10 31 kg 0 511 MeV c2 1 x n 1 nx for x 1

sin 16 3 for 1 cos 1 21 2 for 1

Copyright 2014

c American Association of Physics Teachers

2014 USA Physics Olympiad Exam Part A 3

Question A1

Inspired by http www wired com wiredscience 2012 04 a leaning motorcycle on a vertical wall

A unicyclist of total height h goes around a circular track of radius R while leaning inward at

an angle to the vertical The acceleration due to gravity is g

a Suppose h R What angular velocity must the unicyclist sustain

Work in the rotating frame where four forces act on the unicyclist a normal and frictional

force at the point of contact gravity downwards at the center of mass and a fictitious

centrifugal force

If h R all parts of the unicyclist are at a distance of approximately R from the center of the

circle so the centripetal acceleration of every part of the unicyclist is 2 R The centrifugal

force can then be taken to act at the center of mass for purposes of computing the torque

If the center of mass is a distance l from the point of contact the torque about the point of

contact is

m 2 Rl cos mgl sin

Since the unicyclist is stationary in this frame 0

m 2 Rl cos mgl sin 0

b Now model the unicyclist as a uniform rod of length h where h is less than R but not

negligible This refined model introduces a correction to the previous result What is the new

expression for the angular velocity Assume that the rod remains in the plane formed by

the vertical and radial directions and that R is measured from the center of the circle to the

point of contact at the ground

The centripetal acceleration now varies meaningfully along the length of the unicyclist In

the rotating frame the torque about the point of contact is given by

where r is the distance from the center of the circle z is the height above the ground and

dm is a mass element Because the mass of the unicyclist is uniformly distributed along a

length h the mass element dm can be written as m h ds for a length element ds and we have

c 2 R s sin s cos ds

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c American Association of Physics Teachers

2014 USA Physics Olympiad Exam Part A 4

c m h cos sin

Gravity continues to act at the center of mass a distance h2 from the point of contact and

in the opposite direction

Again the total torque is zero so

m h cos sin mg sin 0

Question A2

A room air conditioner is modeled as a heat engine run in reverse an amount of heat QL is

absorbed from the room at a temperature TL into cooling coils containing a working gas this gas is

compressed adiabatically to a temperature TH the gas is compressed isothermally in a coil outside

the house giving off an amount of heat QH the gas expands adiabatically back to a temperature

TL and the cycle repeats An amount of energy W is input into the system every cycle through

an electric pump This model describes the air conditioner with the best possible efficiency

heating coil

valve cooling coil

Assume that the outside air temperature is TH and the inside air temperature is TL The

air conditioner unit consumes electric power P Assume that the air is sufficiently dry so that no

condensation of water occurs in the cooling coils of the air conditioner Water boils at 373 K and

freezes at 273 K at normal atmospheric pressure

a Derive an expression for the maximum rate at which heat is removed from the room in terms

of the air temperatures TH TL and the power consumed by the air conditioner P Your

derivation must refer to the entropy changes that occur in a Carnot cycle in order to receive

full marks for this part

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c American Association of Physics Teachers

2014 USA Physics Olympiad Exam Part A 5

From Carnot cycles and by entropy conservation we have

Also by energy conservation

So heat is removed at a rate QL t But

QL QH W QL W

Rearrange and divide by time

b The room is insulated but heat still passes into the room at a rate R k T where T is

the temperature difference between the inside and the outside of the room and k is a constant

Find the coldest possible temperature of the room in terms of TH k and P

k T 2 P TH P T

which is a quadratic that can be solved as

P P 2 4P kTH

but only the positive root has physical significance Writing x P k

T 1 4TH x 1

That s the amount the room is colder than the outside so

TL TH 1 4TH x 1

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2014 USA Physics Olympiad Exam Part A 6

c A typical room has a value of k 173 W C If the outside temperature is 40 C what

minimum power should the air conditioner have to get the inside temperature down to 25 C

Don t forget to convert to Kelvin

From above

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c American Association of Physics Teachers

2014 USA Physics Olympiad Exam Part A 7

Question A3

When studying problems in special relativity it is often the invariant distance s between two

events that is most important where s is defined by

s 2 c t 2 x 2 y 2 z 2

where c 3 108 m s is the speed of light 1

a Consider the motion of a projectile launched with initial speed v0 at angle of 0 above the

horizontal Assume that g the acceleration of free fall is constant for the motion of the

projectile

i Derive an expression for the invariant distance of the projectile as a function of time t as

measured from the launch assuming that it is launched at t 0 Express your answer

as a function of any or all of 0 v0 c g and t

ii The radius of curvature of an object s trajectory can be estimated by assuming that

the trajectory is part of a circle determining the distance between the end points and

measuring the maximum height above the straight line that connects the endpoints As

suming that we mean invariant distance as defined above find the radius of curvature

of the projectile s trajectory as a function of any or all of 0 v0 c and g Assume that

the projectile lands at the same level from which it was launched and assume that the

motion is not relativistic so v0 c and you can neglect terms with v c compared to

terms without

The particle begins at ct x y 0 and takes a path satisfying

x v0 t cos 0

z v0 t sin 0 gt2

s2 ct 2 v0 t cos 0 2 v0 t sin 0 gt2 2

which can be simplified to

s2 c2 v02 t2 gv0 sin 0 t3 g 2 t4

b The particle reaches the ground again at

xf v0 cos tf

We are using the convention used by Einstein

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2014 USA Physics Olympiad Exam Part A 8

and so the invariant distance between the endpoints is

s2 ctf 2 v0 cos tf 2

The maximum height above the ground is

Because zmax s we have from similar triangles

c A rocket ship far from any gravitational mass is accelerating in the positive x direction at a

constant rate g as measured by someone inside the ship Spaceman Fred at the right end

of the rocket aims a laser pointer toward an alien at the left end of the rocket The two are

separated by a distance d such that dg c2 you can safely ignore terms of the form dg c2 2

i Sketch a graph of the motion of both Fred and the alien on the space time diagram

provided in the answer sheet The graph is not meant to be drawn to scale Note that t

and x are reversed from a traditional graph Assume that the rocket has velocity v 0

at time t 0 and is located at position x 0 Clearly indicate any asymptotes and the

slopes of these asymptotes

Since the rocket ship can never exceed the speed of light yet it is always accelerating

in the local frame it must approach an asymptote that has a slope of one on the

space time diagram shown There is a slight challenge to consider however Since the

rocket ship is an extended object do the two ends represented by Fred and the Alien

approach the same asymptote or two different asymptotes

At this point we must remember a consequence of special relativity for a ship moving

at relativistic speeds the ship will contract in length as measured in the original frame

As the speed of the ship approaches that of light the length of the ship will approach

zero The only way for that to happen is for the two ends of the ship to have slightly

different accelerations

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c American Association of Physics Teachers

2014 USA Physics Olympiad Exam Part A 9

If you had assumed somewhat incorrectly that the two ends of the ship have the same

acceleration then the two trajectories would be approaching two different asymptotes

separated by a constant horizontal distance But this would mean the apparent length

of the ship was constant regardless of speed In the instantaneous rest frame of the ship

we then require that Fred and the Alien be moving apart This means that the ship

must be stretching and eventually breaking

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c American Association of Physics Teachers

2014 USA Physics Olympiad Exam Part A 10

ii If the frequency of the laser pointer as measured by Fred is f1 determine the frequency

of the laser pointer as observed by the alien It is reasonable to assume that f1 c d

If the spaceship is uniformly accelerating then we can choose a reference frame which is

instantaneously at rest with respect to the spaceship at t 0

Consider two instantaneous flashes from the astronaut Flash 1 is emitted by Fred at

t 0 flash 2 is emitted by the Fred at t 1

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2014 USA Physics Olympiad Exam Part A 11

Flash 1 travels down towards the alien who is accelerating upward Let t1 be the time

at which the alien sees flash 1 Equate the distances

ct1 d gt1 2

Flash 2 is emitted at 1 Flash 2 travels down towards the alien who is still accelerating

upward Let t2 be the time at which the alien sees flash 2 Equate the distances

c t2 1 g 1 2 d gt2 2

Notice that the pulse travel for a time t2 1 Defining t2 1 2 and then expanding

to keep terms linear in t1

c t1 2 1 h gt1 2 gt1 2

c 2 1 gt1 2

but t1 d c so

In terms of frequency this is

Alternatively we can follow the motion of two wave crests from Fred to the alien We

can work in the reference frame where Fred is stationary when the first crest is emitted

Because f1 c d we can assume that Fred remains stationary during the period be

tween the first and second crests then because the first crest moves towards the alien

at a speed c they are separated by a distance c f1

Because dg c2 the time taken for the crests to reach the alien is due almost entirely

to the motion of the crests and is d c In this time the spaceship accelerates to a speed

gd c and in Fred s frame of reference the relative speed of the crests and the alien is

c The time between crests reaching the alien is thus

While this time interval is measured in Fred s reference frame time dilation effects are

of the order gd

and can thus be ignored

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c American Association of Physics Teachers

2014 USA Physics Olympiad Exam Part A 12

Question A4

A positive point charge q is located inside a neutral hollow spherical conducting shell The shell

has inner radius a and outer radius b b a is not negligible The shell is centered on the origin

a Assume that the point charge q is located at the origin in the very center of the shell

i Determine the magnitude of the electric field outside the conducting shell at x b

Conducting shell is neutral so there is equal but opposite charge on surface r a and

r b The electric field inside of a static conductor is zero so the charge on inner surface

is equal but opposite to q by Gauss s Law Spherical symmetry requires a spherically

symmetric electric field so by Gauss s law outside the shell we have

and then at x r b we have

ii Sketch a graph for the magnitude of the electric field along the x axis on the answer

sheet provided

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2014 USA Physics Olympiad Exam Part A 13

iii Determine the electric potential at x a

The shell is a conductor so it is an equipotential surface This means potential at r a

is same as r b For points outside the shell spherical symmetry and Gauss s Law

makes the problem reducible to a point charge at the origin so

But V a V b so V x a is

iv Sketch a graph for the electric potential along the x axis on the answer sheet provided

b Assume that the point charge q is now located on the x axis at a point x 2a 3

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2014 USA Physics Olympiad Exam Part A 14

i Determine the magnitude of the electric field outside the conducting shell at x b

The problem for r a maintains the spherical symmetry of above so the answer is

ii Sketch a graph for the magnitude of the electric field along the x axis on the answer

sheet provided

iii Determine the electric potential at x a

The problem for r a maintains the spherical symmetry of above so the answer is

iv Sketch a graph for the electric potential along the x axis on the answer sheet provided

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2014 USA Physics Olympiad Exam Part A 15

v Sketch a figure showing the electric field lines if any inside within and outside the

conducting shell on the answer sheet provided You should show at least eight field lines

in any distinct region that has a non zero field

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c American Association of Physics Teachers