PHYS212 : Homework

## Homework Assignment 1 (due 1/23)

Complex exponentials, simple harmonic motion

1. Write the complex number in Cartesian and polar form.

2. Use complex exponential functions to show that

3. (a) Use trig identities to show that

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

(b) Use Maple to plot the function

and explain why the expression of part (a) is useful. In particular, why are the frequencies and 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 hanging from a massless string of length ).

(b) Find the frequency of small-angle oscillations of a thin ring of mass 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