THE UNIVERSITY OF BRITISH COLUMBIA
 
Science 1 Physics Assignment # 1:
 
WAVES
 
05 Jan. 2000 - finish by 12 Jan. 2000

1.

DHM Review: A mass  m  is attached to a spring with force constant  k  and set in motion. The amplitude of the resulting oscillation is  x_m. At a certain instant (which we will call  t=0) when the displacement from equilibrium is exactly half the amplitude, ( $x[t=0] = {1\over2} x_m$), a damping force   $F_d = -\kappa v$,  (where  v  is the velocity of the mass) begins to act. One second later the velocity is zero, the acceleration is  a[t=1 s] = 8 cm/s²  and the position is  x[t=1 s] = -2 cm. Another second later the position is  x[t=2 s] = 0.5 cm. Find the initial phase  $\phi$,  the angular frequency  $\omega$, the damping coefficient  $\kappa$  and the initial amplitude  x_m.

(Warning: The approximation $\omega^2 = {k \over m} - {\kappa^2 \over 4 m^2} \approx {k \over m}$ is only accurate to $\sim$ 13%!)

2.

Travelling Wave in a String: We are observing a string whose transverse displacement  y  is given [in ${\cal SI}$ units] as a function of time  t  and position  z  (down the length of the string) by

$$y(z,t) \; = \; 4.73 \sin(7z - 5t) \; + \; 7.22 \cos(7z - 5t)$$

(a)

In what direction is the wave traveling?

(b)

What is the amplitude of the traveling wave?

(c)

What is the frequency of ${\cal SHM}$ at any point on the string?

(d)

What is the wave propagation velocity on the string?

Also try the following problems from Tipler Ch. 13:   6, 7