Physics 120 Christmas Exam

Physics 120 Christmas Exam - 1992

1.

``QUICKIES'' [20 marks -- 2 each]

(a)

At t=0 you have $2 \times 10^4$ atoms of radioactive isotope A (whose half-life is 4 hours) and $8 \times 10^4$ atoms of radioactive isotope B (whose half-life is 2 hours). How many hours does it take until you are most likely to have equal numbers of A and B atoms?

(b)

What paradoxical question did the astronomer Olbers ask 150 years ago?

(c)

A hailstone of mass m falling under gravity (at the Earth's surface) experiences a drag force F_d = k v² where k is a constant and v is the hailstone's speed. What is the terminal speed of the hailstone in terms of m, g and k?

(d)

If all the mass of the Earth were compressed into a uniform sphere with one tenth of the present Earth's radius, what would be the value of ``g'' at its surface?

(e)

In the ``Newton's rings'' experiment, a circular lens sits on a flat glass plate as shown (side view). Light shines down normal to the flat glass surface. At the centre, where the lens is in contact with the plate, is the reflected light brighter or darker than average?

$\begin{figure}\epsfysize 0.25in \begin{center} \mbox{\epsfbox{/home/jess/P120/PS/newtons_rings.ps} } \end{center}\end{figure}$

(f)

If two sound emitters produce 9 beats per second when heard together, what is the difference in their frequencies (in Hz)?

(g)

A mass on a spring oscillates with a 2 cm amplitude. If we suddenly increase the velocity of the mass by a factor of 4 at the instant when it passes its equilibrium point, what is the new oscillation amplitude?

(h)

A set of 8 equally-spaced slits is illuminated by a parallel beam of light; the resulting interference pattern is then observed on a distant screen. How many times does the light intensity on the screen fall to zero between each pair of principal maxima?

(i)

An organ pipe open at both ends is 3.80 m long and has the same fundemental frequency as a second pipe which is closed at one end. How long is the second pipe?

(j)

If you were to move along a straight line from infinity to the centre of the Earth, at what point would your gravitational potential energy be the lowest?

2.

P115 Question #4 [20 marks] An object oscillates with an amplitude of 0.6 m on a horizontal spring of force constant 2.0 N/m. Its maximum speed is 2.40 m/s.

(a): [4 marks] What is the total energy of the object?
(b): [6 marks] What is the frequency of oscillations of the object?
(c): [4 marks] What is the mass of the object?
(d): [2 marks] Write an equation describing the displacement x(t) of the object from the equilibrium position at any time. [Give numerical values for any known constants in your equation.]
(e): [4 marks] If at t=1.0 second the object is found to be at x=.02m, what is the initial phase $\phi$ ?

3.

Wave Motion [20 marks] The displacement y of a string is given as a function of position x and time t as

$\begin{displaymath}y(x,t) \; = \; 0.4 \sin(5x - 7t) \; + \; 5 \cos(5x - 7t) \end{displaymath}$

where all the constants are given in SI units.

(a): [3 marks] What is the direction of propagation of the wave? (Explain.)
(b): [4 marks] What is the amplitude of the travelling wave?
(c): [3 marks] What is the wavelength of the wave?
(d): [3 marks] What is the frequency of the ${\cal SHM}$ of each point of the string [in Hz]?
(e): [3 marks] What is the velocity of wave propagation down the string?
(f): [4 marks] What is the maximum velocity of any point in the string?

4.

P115 Question #6 [20 marks]

$\begin{figure}\epsfysize 0.5in \begin{center} \mbox{\epsfbox{/home/jess/P120/PS/x92_wire.ps} } \end{center}\end{figure}$

A wire of mass 0.05 kg and total length 4.0 m is held fixed at points 2.0 m apart, as shown above. It is noted that the wire is vibrating in its fundamental mode and that the time for 100 oscillations is 0.5 s.

(a): [4 marks] What is the time period of the oscillations?
(b): [6 marks] What is the tension in the wire?
(c): [6 marks] By what factor must the tension be changed to halve the time for 100 oscillations?
(d): [4 marks] Sketch the first three harmonics of this wire.

5.

P115 Question #7 [20 marks]

$\begin{figure}\epsfysize 2.0in \begin{center} \mbox{\epsfbox{/home/jess/P120/PS/x92_speakers.ps} } \end{center}\end{figure}$

A student sits 10 m from a loudspeaker (L₁) which emits 50 watts of power in all directions equally.

(a): [4 marks] What is the intensity at the student's position from L₁ alone?
(b): [4 marks] What is the sound level [in decibels] heard by the student?
(c): [6 marks] Now a second speaker (L₂) is placed 13 m from the student as shown. The two speakers operate in phase and can be tuned over the audible range (20 Hz to 20,000 Hz). What are the two lowest frequencies for which the student hears a minimum. [Take the sound velocity to be 340 m/s.]
(d): [4 marks] What are the two lowest frequencies for which the student hears a maximum?
(e): [2 marks] If the first speaker had its wires interchanged so that it operated 180 $^\circ$ out of phase with the other, what would change in parts (a), (b), (c) and (d)?

6.

Diffraction Grating [20 marks]

$\begin{figure}\epsfysize 2.5in \begin{center} \mbox{\epsfbox{/home/jess/P120/PS/x92_grating.ps} } \end{center}\end{figure}$

The drawing above shows the central region of the light intensity pattern on a screen 10 m away from an array of N identical slits illuminated with light of wavelength 500 nm. [Note: 1 mrad $\equiv 10^{-3}$ radian.]

(a): How many slits are illuminated?
(b): What is the distance between the centres of adjacent slits?
(c): Estimate the width of one slit.

-- FINIS --

Some extra questions that were NOT on the exam:

1.

P115 Question #1 [20 marks] A diverging lens with a focal length of -20 cm is located 10 cm to the left of a converging lens having a focal length of +15 cm . A real object is located 40 cm to the left of the diverging lens ( see the figure).

(a): CALCULATE the properties of the image formed by the diverging lens. ( i.e. calculate the image distance, lateral magnification and state whether the image is real or virtual, upright or inverted) ( 2 points )
(b): Taking the image formed by the diverging lens as the object for the converging lens CALCULATE the properties of the image formed by the converging lens. ( 3 points )
(c): CALCULATE the properties of the final image.( 2 points )
(d): DRAW an appropriate ray diagram for the two-lens system (use the given figure). ( 3 points )

2.

P115 Question #2 [20 marks] A 180 g copper pot contains 200 litre of water at an initial temperature of 20 $^\circ$ C. A copper rod heated to 900 $^\circ$ C is dropped into the water. This causes the water to boil with 5.0 g of water converted to steam. What is the mass of the copper rod?

Specific heat of copper = 386 J/kg.K

Specific heat of water = 4190 J/kg.K

Heat of vaporisation of water = 2256000 J/kg

3.

P115 Question #3 [20 marks] An igloo, a hemispherical enclosure built of ice, has an inner radius of 2.5m . The thickness of the ice is 50cm. On a cold winter day the outside temperature is -40 $^\circ$ C. At what rate must thermal energy be generated to maintain the temperature inside the igloo at -5 $^\circ$ C? ( 5 points )

Thermal conductivity of ice = 0.592 W/m.K

A quantity of gas ( $\gamma$ = 1.4 ) expands adiabatically and quasi-statically from an initial pressure of 2 atm and volume of 2L at temperature of 20 $^\circ$ C to twice its original volume.

(a): What is the final pressure of the gas? ( 3 points )
(b): What is the final temperature of the gas? ( 2 points )

4.

Work and Power [20 marks] An automobile manufacturer advertises a 1000 kg car with a newly designed engine and drivetrain that can deliver 100 hp to the wheels. [1 hp $\equiv$ 550 ft-lb/s $\equiv$ 0.7457 kW] Assume that the wheels never slip.

(a): What would be the acceleration of such a car at 60 kph? [kph $\equiv$ kilometers per hour]
(b): What would be its acceleration starting from rest?
(c): What is wrong with the advertisement?