Physics 120 Christmas Exam

Physics 120 Christmas Exam - 1991

1.

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

(a): A particle has a position (x) as a function of time (t) given by $x = [3\hbox{\sl ~m/s\/}^3] \, t^3$ .
What is the acceleration of the particle at t = 1 s?
(b): What is the angle (in radians) between lines AB and CD on the faces of the cube shown below?
$\begin{figure}\epsfysize 2.0in \begin{center} \mbox{\epsfbox{/home/jess/P120/PS/x1.ps} } \end{center}\end{figure}$
(c): You have $100 in the bank with interest compounded continuously at such a rate that you obtain a balance of $200 in 19 months. What is your balance after 57 months?
(d): The escape velocity for a body on the Earth's surface is $v_e = 1.1 \times 10^5$ m/s. What would be the escape velocity if the mass of the Earth were doubled without changing its radius?
(e): Two sound emitters have frequencies $\nu_1$ and $\nu_2$ whose average is 300 Hz. Their combined intensity exhibits 8 beats per second. What are $\nu_1$ and $\nu_2$ ?
(f): One organ pipe (A) is open at both ends; another (B) is open at one end and closed at the other. If the fundamental frequency of A is 3 times higher than that of B, what is the ratio of their lengths, ${L_A \over L_B}$ ?
(g): On a cello there are two adjacent strings, A and B, whose fundamental frequencies differ by a factor of ${3 \over 2}$ : $\nu_B = {3 \over 2} \nu_A$ . If string A has twice the mass per unit length of string B, what is the ratio of their tensions, ${\tau_A \over \tau_B}$ ?
(h): If you want to minimize the reflection of light of wavelength $\lambda$ from a glass lens by coating the lens with a film of plastic whose index of refraction n is between those of air and glass, what is the minimum thickness d of the plastic coating (in terms of $\lambda$ and n)? [Assume normal incidence.]
(i): An array of equally spaced slits produces an interference pattern with 14 secondary intensity maxima between adjacent principal maxima. If you were to cover up exactly half the slits, how many secondary maxima would then appear between adjacent principal maxima?
(j): Consider an array of 36 equally spaced slits. At the first minimum beyond the central maximum of its intensity pattern, what is the phase difference (in degrees) between light from adjacent slits?

2.

Weird Trajectory [20 marks]

$\begin{figure}\epsfysize 2.25in \begin{center} \mbox{\epsfbox{/home/jess/P120/PS/x2.ps} } \end{center}\end{figure}$

A 1 kg mass is attached by a massless spring of force constant 100 N/m to a massless bracket B which slides without friction along a rail perpendicular to the spring, as shown in the figure at right. The mass is initially at rest in equilibrium 1 m from the rail and 5 m from the wall at the end of the rail. It is then hit with a hammer to give it an instantaneous velocity of 10 m/s in the direction shown.

(a): How long does it take the mass to reach the wall?
(b): What is the minimum kinetic energy of the mass?
(c): What is the maximum potential energy stored in the spring?
(d): What is the maximum distance between the mass and the rail?
(e): Where does the mass hit the wall?

3.

Synchronous Satellite Spy in the Sky [20 marks] You are standing on the Earth's equator and you want to build a spy satellite that appears to hover directly overhead. [Note: the radius of the Earth is $R_E = 6.37 \times 10^6$ m and the acceleration of gravity at the Earth's surface is g = 9.81 m/s².]

(a): What is the radius of the circular orbit of the satellite?
(b): The satellite in question is intended to carry a telescope with an optically perfect circular lens capable of reading your copy of the Ubyssey from outer space. Assuming this means you want to resolve dots 3 mm apart on the paper using green light of wavelength 550 nm, what is the minimum diameter of the telescope lens?

4.

P115 Question #6 [20 marks] A continuous sinusoidal wave is propagating down a string in the x direction at a velocity 0.45 m/s. The linear density of the string is 0.004 kg/m. The displacement y of particles of the string at position x=0.012 m is found to vary with time t according to the equation

$\begin{displaymath}y \; = \; (0.06 \hbox{\sl m\/}) \; \sin [1.4 \, - \, (3.5 \, \hbox{\sl s\/}^{-1}) t] . \end{displaymath}$

(a)

What is the frequency of the wave (in Hz)?

(b)

What is the wavelength $\lambda$ , the angular wave number k and the phase $\phi$ at (x=0 and t=0)?

(c)

For this traveling wave, write down the general equation expressing the displacement as a function of arbitrary position and time; then calculate the displacement at (x=50 cm and t = 0.28 s).

(d)

Suppose a section of the same string is clamped between two points that are 0.73 m apart and a tension of 23 N is applied to the string. Describe (with words and a drawing) the motion of the clamped string at both its lowest frequency and its second lowest frequency.

(e)

What are the three lowest frequencies at which the string (clamped and stretched as described above) could vibrate?

5.

P115 Question #8 [20 marks] Bjossa, the female killer whale in the Vancouver aquarium, is trying to get the attention of her new baby calf, which is floating motionless in the pool. Bjossa swims toward the calf at 7 m/s emitting a high-pitched tone of 12 kHz. The speed of sound in water is 1500 m/s.

(a): What frequency does the baby hear?
(b): The baby does not care for what it hears and swims away at 10 m/s. What frequency does it hear now?
(c): What is the wavelength measured in the water by a stationary scuba diver?
(d): As Bjossa swims at 7 m/s some of the sound waves are reflected from the pool wall ahead of her. What is the frequency that she hears for these reflected sounds?

6.

Diffraction Grating [20 marks]

$\begin{figure}\epsfysize 2.5in \begin{center} \mbox{\epsfbox{/home/jess/P120/PS/x6.ps} } \end{center}\end{figure}$

The drawing shows the central region of the light intensity pattern on a screen 12 m away from an array of N identical slits illuminated with light of wavelength $4.5 \times 10^{-7}$ m. [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): On the drawing, sketch in the intensity pattern that would result if you covered up a single slit at one end of the array. [Label this curve ``c''.]
(d): For each of the positions A, B and C shown on the drawing, make a separate sketch of the phasors yielding the observed intensity.
(e): If the array of N slits were defective so that one of the slits (it doesn't matter which) yielded twice the light amplitude of the others, indicate [by a curve labelled ``e'' on the drawing] the resulting light pattern. Show clearly the positions of the maxima and minima and the light intensity compared with those of the original pattern shown in the drawing. (Hint: use phasors and choose the defective slit to be the first one in the array.)

-- FINIS --

The following questions were NOT on the exam:

1.
Orbiting a Strange Attractor [20 marks] A light mass $m = 2.488 \times 10^{-28}$ kg is attracted to a much heavier mass $M \gg m$ by a peculiar force that drops off exponentially with the distance between the two masses: $F(r) = F_0 \, e^{-\lambda r}$ , with $F_0 = 2.488 \times 10^{-10}$ N and $\lambda = 10^{15}$ m^-1.
What is the period of a circular orbit of radius r = 10^-15 m?

2.
P115 Question #4 [20 marks] An elevator and its load have a combined mass of 1600 kg.

(a)
The elevator initially has an upward speed of 12 m/s. The elevator is then brought to rest with constant acceleration in a distance of 20 m. Find the acceleration.
(b)
What is the tension in the cable while the elevator is brought to rest?
(c)
How much work does the elevator motor do during the slowdown time?
(d)
How much potential energy does the elevator gain during the slowdown time?
(e)
Why is the work done by the elevator motor less than the gain in potential energy of the elevator?
(f)
If a pendulum clock were standing in the elevator, would its period increase or decrease while the elevator is slowing down? Explain your answer.

3.
P115 Question #5 [20 marks]

A particle moves counterclockwise in a circle of radius 5 m with an angular velocity of 10 rad/s. At the time t=0 the particle has an x-coordinate of 2 m.

(a)
What is the amplitude of the oscillation?
(b)
Write an equation for the x-coordinate (the displacement in the x-direction) of the particle as a function of time.
(c)
What is the phase $\Phi$ at the time t=0?
(d)
Find the x-component of the particle's velocity at the time t=3.2 s.
(e)
Find the x-component of the particle's acceleration at the time t=3.2 s.
(f)
What is the tangential velocity of the particle?
(g)
What is the tangential acceleration of the particle?
(h)
If the particle has a mass of 0.35 kg and is held by a string, what is the tension force in the string?

4.
P115 Question #7 [20 marks]

The frequency of a loud speaker S can be varied continuously. It is connected to an acoustical interferometer which has two tubes A and B of different lengths, d₁ and d₂. A maximum transmission occurs at 2100 Hz and the next maximum is at 2800 Hz. (The velocity of sound is 343 m/s.)

(a)
What are the two wavelengths?
(b)
What is the difference in length $\Delta \equiv d_1 - d_2$ between the two tubes?
(c)
How many wavelengths is this path difference

i) at 2100 Hz? ii) at 2800 Hz?
(d)
What is the next frequency above 2800 Hz at which the transmission would be a minimum?