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93/11(a) Semester 1, 2009 Page 1 of 1
THE UNIVERSITY OF SYDNEY
PHYS 1901
PHYSICS 1A (ADVANCED)
Solutions
JUNE 2009
Time allowed: THREE Hours
MARKS FOR QUESTIONS ARE AS INDICATED
TOTAL: 90 MARKS
INSTRUCTIONS
**- All questions are to be answered.
- Use a separate answer book for section A and section B.
- All answers should include explanations in terms of physical principles.**
DATA
Density of fresh water =
3 3 1.000 10 kg.m
Free fall acceleration at earth's surface g =
2 9.80 m.s
Gravitational constant G =
11 2 2 6.67 10 N.m .kg
Speed of light in a vacuum c =
8 1 3.00 10 m.s
Speed of sound in air v =
1 344 m.s
Avogadro constant N A
=
23 1 6.023 10 mol
Universal gas constant R =
1 1 8.314 J.mol .K
Boltzmann constant k =
23 1 1.380 10 J.K
Stefan-Boltzmann constant σ =
8 2 4 5.67 10 W.m .K
ADV_Q
It has been proposed that future space stations should be rotating cylinders, providing the
astronauts with artificial gravity. Consider such a cylinder of radius R rotating with an angular
velocity of in deep space, far away from any gravitational fields.
(a) Identify the forces acting on an astronaut who is rotating with the space station. By
comparing to an astronaut standing on the surface of the Earth, briefly describe how the
astronaut in the rotating space station experiences “weight”.
(b) If an astronaut in a rotating space station of radius R is to experience the same weight as
she would on Earth, show that the space station must be rotated at an angular velocity of
g
R
Ensure that you justify your answer.
(c) Suppose an astronaut rotating with the space station releases a ball from her hand.
Qualitatively describe the motion of the ball as observed by an external observer (not
rotating with the space station).
(5 marks)
ADV_Q02=REG_Q
An astronaut constructs an accelerometer for his rocket using two identical springs of
unextended length L and spring constant k , and two identical masses of mass m. He hangs the
springs and masses in the rocket as shown below.
Assume the rocket remains within the gravitational field of the Earth and that the acceleration
due to gravity is g at all times.
(a) After launch, the rocket accelerates vertically upwards with an acceleration a. Show that
the length of the accelerometer (the total length of the springs, assuming the masses have
negligible length) is:
m g a L k
(b) Eventually the rocket runs low on fuel and the motors are shut down. What is the total
length of the accelerometer after shutdown of the motors? Explain your result.
(5 marks)
Solution
(a)
Use Newton’s second law and take upwards to be positive:
Hookes law states that
F k x
where x is the extension of the spring beyond its natural length and k is the spring constant.
For Mass A:
F 1 (^) F 2 m g ma
For Mass B:
F 2 m g ma
(1 mark)
F 2 m g a
Using the expression for F 2 we get
F 1 (^) 2 m g a .
Hookes law gives the extensions for each spring:
1 1
1
F k x m g a
m g a x k
and
2 2
2
F k x m g a
m g a x k
(1 mark)
Ignoring the signs which just indicate direction then the total length is given by
REG_Q03=ADV_Q
Two solid spherical balls (A and B) are released at the same time from the top of the inclined
ramp. Both have the same mass M , but Ball A has twice the radius of Ball B. They both roll
down the ramp without slipping. The dimensions of the ramp are much larger than the radius of
either ball.
(a) Which ball reaches the bottom of the ramp first? Justify your answer.
Hint: Consider the potential, rotational, and translational kinetic energies.
In a second experiment, a solid spherical ball of radius R and a solid cylinder with a circular
cross section of radius R are released at the same time from the top of an inclined ramp. Both
have mass M. The dimensions of the ramp are much larger than the radius R of the ball or
cylinder.
Which object reaches the bottom of the ramp first if:
(b) The ramp is frictionless. Justify your answer.
(c) There is a frictional force between each object and the ramp so that they roll without
slipping. Justify your answer.
Moment of Inertia Data
Moment of Inertia for Solid Sphere
I M R
Moment of Inertia for Solid Cylinder
I M R
(5 marks)
Solution
(a)
The potential energy lost as an object comes down the ramp is equal to the sum of the
kinetic energy of translation and the kinetic energy of rotation at the bottom. If the vertical
height of the ramp is h , the final velocity of the object (of mass M and moment of inertia I ) at the
bottom of the ramp is v , and the rotational angular velocity of the object is then we have:
M g h M v I .
If the object rolls without slipping then
v r
where r is the radius of the object.
Apply this to:
Object 1: solid sphere of radius R rolling without slipping
2 2 2
2 2 2
v M g h M v MR R
g h v v v
Object 2: solid sphere of radius 2R rolling without slipping
Same answer as for Object 1 because the answer does not depend on radius.
Hence the two objects will have the same velocity (at all times going down the ramp as h is
arbitrary). They will reach the bottom of the ramp at the same time.
(2 marks)
(b)
Apply energy conservation equation to:
Object 1: solid sphere of radius R on ramp with no friction
Object slides so
2
2
M g h M v
g h v
Object 2: solid cylinder of radius R on ramp without friction
Object slides and answer is the same as for Object1 because the answer does not depend on the
moment of inertia of the object.
Hence the two objects will have the same velocity (at all times going down the ramp as h is
arbitrary). They will reach the bottom of the ramp at the same time.
(1 mark)
(c)
Apply energy conservation equation to:
Object 1: solid sphere of radius R on ramp rolling without slipping
ADV_Q
(a) Why does the water temperature at your favourite beach only vary by a few degrees
between peak summertime and mid-winter, while the daytime air temperature can vary
by as much as 30 degrees between seasons?
(b) Whilst lying on the beach on a hot day, it is common to feel a breeze blowing towards
you from the sea. Explain briefly why this occurs.
(c) In what fundamental way does a Stirling engine differ from an internal combustion
engine, such as that used in cars?
(d) Suppose you place an ideal Stirling engine on a block of ice. If the surrounding air
temperature is22.0 C
, calculate the maximum efficiency of the engine.
(5 marks)
Solution
(a)
The large heat capacity of water means that a relatively large amount of heat needs to be
absorbed or lost to produce even a small temperature change. So the temperature of the water
will vary to a lesser extent than the air temperature.
(1 mark)
(b)
During daytime, land is hotter than the sea so hot air rises up from land creating a region of
lower pressure over the land. Air then moves from the region of higher pressure over the sea to
that of lower pressure over the land. This is the sea breeze, often experienced later in the day.
(1½ marks)
(c)
The heat sink in a Stirling engine is external to the working fluid, so it is an external combustion
engine.
(1 mark)
(d)
Use Carnot efficiency:
c
h
T
e T
(1½ marks)
ADV_Q
Harry stands near a bird’s nest. The bird tries to scare him away by flying in circles around his
head while emitting a sound wave with a constant frequency. The bird keeps a constant distance
from Harry.
Sally, standing at a large distance from Harry, also hears the bird (see the above diagram).
(a) At which position(s) of the bird (A,B,C,D) does Sally hear the highest frequency?
Briefly explain your answer.
(b) At which position(s) of the bird (A,B,C,D) does Harry hear the highest frequency?
Briefly explain your answer.
(c) At which position(s) of the bird (A,B,C,D) do Harry and Sally hear the same frequency?
Briefly explain your answer.
(5 marks)
Solution
(a)
At position B as the bird is approaching Sally with the greatest speed. The doppler shift( towards
higher frequency) of the sound emitted by the bird is largest at this position.
(2 marks)
(b)
All positions (A,B,C,D) are the same. The bird has no component of velocity towards Harry and
this is no doppler shift of the sound heard by him.
(1½ marks)
(c)
Positions A and C. At these positions there is no velocity towards Sally and likewise no velocity
towards Harry. There is no doppler shift and both Harry and Sally hear the same frequency.
(1½ marks)
ADV_Q
A proton of mass m moves in one dimension in a potential energy function given by
2
U x ( ) x x
where and are positive constants. The proton is released from rest at x 0
In the following, ensure that you justify your answers.
(a) Show that the potential can be written as
2
0 0 2 0
x x U x x x x
^
(b) Sketch a graph of the potential as a function of x and identify the point x 0.
(c) Give a qualitative description of the motion, identifying the minimum and maximum
values of x reached during the motion.
(d) What is the force on the proton as a function of x?
(e) Let the proton now be released from rest at 1
x
. Give a qualitative description of
the motion; how does this differ from the motion described in (c)?
(10 marks)
Solution
(a)
In the potential energy function
2
U x ( ) x x
put 0
x
2
2 2
2 2
2 2
2 2 0 0 2 2
2 0 0 2 2 0
U x x x
x x
x x
x x
x x
x x x
^
^
(2 marks)
(b)
0
U x ( x ) 0
(2 marks)
(c)
Starts from rest and moves outwards as force accelerates proton to higher speed (force is
outwards). Past the minimum, the force is negative and so the proton decelerates but gets to
infinity with ever decreasing speed.
(2 marks)
(d)
Force is the negative gradient of the potential.
2 0 0 2 2 0
x x U x x x x
so
2 0 0 2 3 2 0
dU x x F x dx x x x
or, in terms of the potential function originally given in the question
2
U x ( ) x x
3 2 3 2
dU F x dx x x x x
^ ^ ^
(2 marks)
(e)
ADV_Q
On a warm summer’s day at the South Pole, a worker is pushing boxes up a rough plank inclined
at an angle above the horizontal. The plank is partially covered in ice, with more ice near the
bottom of the plank than near the top. He pushes the boxes at the bottom and each slides up the
plank to a height that depends on how hard he pushed.
The coefficient of friction increases with distance x up the plank such that:
x
where is a positive constant. The bottom of the plank is at x = 0. Assume that the coefficients
of static and kinetic friction are equal, so that s (^) k .
(a) Sketch a diagram showing all the forces acting on a box while on the plank.
(b) Show that as a box of mass m slides from x = 0 to x = A , the work done by friction on the
box is
2
cos 2
A
W m g
(c) If the box has an initial velocity of v 0 as it starts up the plank at x = 0, show that the
maximum distance along the plank that the mass travels, B , is the solution to the equation
2 2
g B cos 2 g B sin v 0 0
Derive this equation, but do not solve it.
(d) Again consider the mass sliding up the slope with an initial velocity v 0. Show that when
the box first comes to rest, it remains at rest if
2 2 0
3 sin
cos
g v
(10 marks)
Solution
(a)
(1 mark)
(b)
Take the x direction upwards along the plank and the y direction upwards and perpendicular to
the plank.
Decomposing forces in the y direction:
cos 0
cos
N m g
N m g
(1 mark)
Kinetic friction force is
Ff N mg cos.
The question tells us that varies along the plank as follows:
x
Hence kinetic friction force is given by
Ff xmg cos
Work done is given by
0
cos
A W (^) F dxf (^) m g x dx
(1 mark)
Therefore
2
cos 2
A
W mg
(1 mark)
(c)
Conservation of energy at point B gives
2 2 0
cos sin 2 2
B
mv mg mg h mg B
(1 mark)
This can be rearranged as
ADV_Q
It is a hot day and the air temperature in your room is 30.0 C so you turn on the air conditioner,
which brings the temperature down to 22.0 C. The room is sealed and contains 2500 moles of
air.
(a) Calculate the change in entropy in the room. Use
1 1 cair 29.11 J.mol .K
.
(b) Using the refrigerator statement, explain how the room can continue to cool below the
surrounding outside temperature (30C) without violating the 2
nd law of thermodynamics.
(c) Suppose once the temperature is constant in your room that you open a door leading to an
adjacent room of identical volume that you’ve kept in a vacuum for the purpose of
conducting “secret physics experiments”. Calculate the entropy change as the N
molecules of air expand freely to occupy the volume of both rooms. Assume there is no
change in the air temperature.
(10 marks)
Solution
(a)
Use
2
1
dQ S
T
(1 mark)
2 2 2 3 1
1 1 1
ln (2500)(29.11)ln 1.947 10 J.K 303
dQ dT T S nc nc T T T
(2 marks)
(b)
The refrigerator statement says that energy cannot flow from a cooler body to a hotter body
without work being done. As the room cools to a temperature below the surrounding
temperature, it can continue to cool only because work is being done by the air conditioning unit
to continually remove heat from the room to outside.
(3 marks)
(c)
Use
S k ln
w 2
w 1
where w 1 is the number of microscopic states of the system when it occupies volume 1 and w 2 is
the number of microscopic states of system when it occupies volume 2.
(1 mark)
So
2 2 1 ln2^ ln
N N w w S k N k
(1 mark)
and
23 27 N n NA (2500)(6.02 10 ) 1.51 10 molecules.
So
4 1 S 1.44 10 J.K
.
(2 marks)
Can also use
2
1
ln ln(2) ln(2)
V
S n R n R N k V
which corresponds to an equivalent reversible process – isothermal expansion, with Q W , so
2
1
V
V
dQ W dV S n R T T V
.
