SCIENCE 1 PHYSICS

Christmas Exam - 1998

- 2.5 hours -

Instructors: Jess H. Brewer & Domingo Louis-Martinez

1.

``QUICKIES''   [24 marks -- 4 each]

(a)

A particle has a position (x) as a function of time (t) given by   $x = b \, t^3$  where  b = 2  m/s³. What is the acceleration of the particle at  t = 1 s?

(b)

If you have a thousand radioactive atoms with a half-life of 3 hours, how many would you expect to decay in the first six hours?

(c)

A hailstone of mass m falling from the sky over Vancouver experiences a drag force given by F_d = k v² where k is a constant and v is the hailstone's airspeed. What is the terminal airspeed of the hailstone in terms of m, g and k?

(d)

Two sound emitters have frequencies $\nu_1$ and $\nu_2$ whose average is 400 Hz. Their combined intensity exhibits 2 beats per second. What are $\nu_1$ and $\nu_2$?

(e)

One organ pipe (A) is open at both ends; another (B) is open at one end and closed at the other. If both have the same fundamental frequency, what is the ratio of their lengths,  L_A / L_B?   Hint: A sketch should help.

(f)

Suppose that you want to make a glass camera lens with a nonreflective coating of a transparent material whose index of refraction is higher than that of the glass. What is the optimum thickness of the coating?

2.

Right Hemisphere - Left Hemisphere   [15 marks]

Match up each of the following situations with one or more of the graphs shown: for each match, draw a connecting line and label the axes on the graph.

•   F_x = 0
  F_y = -mg
    $\vec{v}$₀ $\cdot \hat{\jmath}$= 0
• $\tau = - \kappa \theta - \gamma \omega$
  ($\tau$ = torque)
• Mass on a spring
    (no damping)
• Boltzmann
  distribution

3.

Falling Rod   [15 marks] A thin, uniform rod 1 m long is pivoted about a frictionless pin at one end. The rod is released from a vertical position with an infinitesimal angular velocity. Find the angular velocity and angular acceleration of the rod at the instant when it is horizontal.

4.

Interference and/or Diffraction   [15 marks]

Blue-green light of wavelength $\lambda = 500$ nm passes normally through a planar array of parallel slits and makes the above intensity pattern on a screen D = 1.0 m away. Describe the array of slits in as much quantitative detail as you can.

5.

Thermal Physics   [16 marks]

Use words to define any mathematical symbols you use in answering the questions on this page.

(a)

[5 marks] What is the definition of the ENTROPY of a system?

(b)

[5 marks] What is the definition of the TEMPERATURE of a system?

(c)

[6 marks] A very simple organism has only two possible states: asleep (zero energy) and awake (energy $\varepsilon = 4.14 \times 10^{-21}$ J). If this organism is in thermal equilibrium with a heat reservoir at room temperature (300 K), what is the probability that it will be asleep?
Hint: The probabilities of the two possible states must add up to 1.
Boltzmann's constant: $k_{\scriptscriptstyle{\rm B}} = 1.38 \times 10^{-23}$ J/K.  

6.

YOUR CHOICE of DERIVATIONS   [15 marks]

Answer EITHER this question OR the one on the next page!

(a)

Power in a Wavy String

If a continuous transverse sinusoidal wave of amplitude A is propagating down a taut string of linear mass density $\mu$ at velocity v with frequency $\omega$, show that the power transmitted in the wave is   $P = {1\over2} \mu \, \omega^2 A^2 \, v$.

(b)

Doppler Effect

-- Answer EITHER this question OR the one on the previous page!

Sound waves emitted by a source moving directly toward a stationary observer at a speed v (less than the speed of sound c) will be detected at a frequency f which is different from the frequency f₀ at which they are emitted in the rest frame of the source.

Show that f / f₀ = c / (c - v).

Hint: Think in terms of wave crests, as illustrated in the sketch above. How much does the source ``catch up'' with one wave crest before it emits the next one?