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THE UNIVERSITY OF SYDNEY
PHYS 1901
PHYSICS 1A (ADVANCED)
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 = 1.000 10 kg.m ^3 ^3
Free fall acceleration at earth's surface g = 9.80 m.s^2
Gravitational constant G = 6.67 10 11 N.m .kg^2 ^2
Speed of light in a vacuum c = 3.00 10 m.s ^8 ^1
Speed of sound in air v = 344 m.s^1
Avogadro constant N A = 6.023 10 ^23 mol^1
Universal gas constant R = 8.314 J.mol .K^1 ^1
Boltzmann constant k = 1.380 10 ^23 J.K^1
Stefan-Boltzmann constant σ = 5.67 10 ^8 W.m .K^2 ^4
SECTION A
Question 1
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).
Question 3
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^2
I M R
Moment of Inertia for Solid Cylinder^2
I M R
Question 4
(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 is 22.0 C , calculate the maximum efficiency of the engine.
(5 marks)
Question 5
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.
SECTION B (Please use another book to answer this section)
Question 7
A proton of mass m moves in one dimension in a potential energy function given by
U x ( ) x 2 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)?
Question 8
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
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
Question 10
In the figure above, cylinder A is separated from cylinder B by an adiabatic piston which is freely movable. In cylinder A there is 0.010 mole of an ideal monatomic gas with an initial
temperature 300 K and a volume of 1.0 10 4 m^3. In cylinder B, there is 0.010 mole of the
same ideal monatomic gas with the same initial temperature and the same volume as the gas in cylinder A.
Suppose that heat is allowed to flow slowly to the gas in cylinder A, and that the gas in cylinder B undergoes the thermodynamic process of adiabatic compression. Heat flows into
cylinder A until finally the gas in cylinder B is compressed to a volume of 5.0 10 ^5 m^3.
Assume that CV 12.47 J.mol ^1 .K^1 and the ratio of heat capacities is 1. 67.
(a) Calculate the final temperature and pressure of the ideal monatomic gas in cylinder B after the adiabatic compression.
(b) How much work does the gas in cylinder B do during the compression? Explain the meaning of the sign of the work.
(c) What is the final temperature of the gas in the cylinder A? ( Hint: the pressure exerted by the gas in cylinder A on the piston is equal to that exerted by the gas in cylinder B on the piston.)
(d) How much heat flows to the gas in the cylinder A during this process?
Question 11
A heavy chain with mass M and length L is hanging vertically from your hand. Nothing is attached to the bottom of the chain, and the chain is not moving.
(a) The tension in the chain is not the same at all points. Is it greater near the top or near the bottom? Explain briefly.
(b) You generate a transverse pulse by jerking your hand sideways and back again. As the pulse travels down the chain, does it get faster or slower? Explain briefly.
(c) Consider a point on the chain at a distance y from the bottom. Derive an expression for
v y ( ) , the speed of the pulse as a function of y , and show that it does not depend on the
mass of the chain.
(d) You now generate a transverse standing wave in the chain by shaking your hand from side to side at a frequency of 3 Hz. The standing wave has several nodes that are not equally spaced. Is the spacing between the nodes greater near the top or near the bottom. Explain briefly.
