# Document 10703062

```Problem Set #3, Due March 11th
1. Problem 1 For the two systems below, find frequencies of normal modes for small
oscillations and time period for beats. Also show the ratio of the amplitude of the left mass
M to that of the right mass M for the two normal modes is 1 and -1.
2. A transverse wave travelling along a string given by:

 z, t   0.5m cos 50 s 1  t  4 m 1  z

where the units within the parentheses are meters and seconds
a) Verify that this expression is a solution for the wave equation
b) Determine the amplitude, wavelength and period of the wave.
c) Determine the wave phase velocity.
d) Find the transverse displacement and transverse velocity of the string at z=5m and
t=0.1sec.
e) Given that tension on the string is supplied by a pulley with a weight attached to a pulley.
The weight has a mass 10 times the mass of the string, calculate the length of the string
3. Please provide proof for the following Fourier Transform theorems.
{ f (at )}  F ( /a) / a
 f (t  a)  exp(i a) F ( )
{ f '(t )}  i F ( )
```