PHYS212 : Homework

Homework Assignment 1 (due 1/23)

Complex exponentials, simple harmonic motion

  1. Write the complex number $z = 3e^{i\pi/3}$ in Cartesian and polar form.

  2. Use complex exponential functions to show that

\sin 2 \theta = 2 \sin \theta \cos \theta.

  3. (a) Use trig identities to show that

\cos(\omega_1 t) + \cos(\omega_2 t)
= 2 \cos \left( \frac{(...
\cos \left( \frac{(\omega_1 - \omega_2)}{2} t \right).

    (Technically, this is a trig identity. Use other trig identities to prove it.)

    (b) Use Maple to plot the function

f(t) = \cos(2 \pi t) + \cos(2.2 \pi t)

    and explain why the expression of part (a) is useful. In particular, why are the frequencies $\frac{(\omega_1 + \omega_2)}{2}$ and $\frac{(\omega_1 - \omega_2)}{2}$ helpful in describing the function? The Maple syntax for the plot is

    [> plot(cos(2*Pi*t)+cos(2.2*Pi*t),t=0..20);

    If you want to put two functions on the same plot, separate them by a comma and enclose them in square brackets ([]). For example,

    [> plot([cos(2*Pi*t), cos(2.2*Pi*t)],t=0..10);

  4. (a) Use the expression for the frequency for small angles of a physical pendulum derived in class to find the frequency of a simple pendulum (a mass $m$ hanging from a massless string of length $L$).

    (b) Find the frequency of small-angle oscillations of a thin ring of mass $m$ hanging from a point on its circumference.

    (c) How long must the pendulum in a pendulum clock be in order for it to have a period of 2.0 s (1.0 s between ``tick'' and ``tock'') using the simple pendulum of part (a)?

    (d) What diameter must the ring of part (b) have in order to give the same period?

Copyright © 2003-2009, Lewis A. Riley Updated Tue Apr 14 02:04:45 2009