Oscillations and Waves

1.

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

2.

Imagine a peculiar sort of ``spring'' that produces a restoring force proportional to the cube of the displacement from equilibrium:   F = - k x³  with  k = 100 N/m³.

(a)

If a mass  m = 1 kg  resting on a frictionless horizontal surface is attached to this ``spring,'' pulled out horizontally a distance of  10 cm  from its equilibrium position and released from rest, plot the position as a function of time over the first full period of the motion. (Use numerical methods to approximate the motion by constant acceleration on small intervals, as in the free fall problem.)

(b)

Would the period be the same for a different amplitude of oscillation?

(c)

Would it be the same if the mass hung vertically from the ``spring'' in the Earth's gravity?

Explain your last two answers qualitatively.

3.

In a test tank of a special fluid, a surface wave is observed to have a maximum height (crest) at point A (located at x = 0) when  t=0  and a minimum (trough) at the same point when  t = 2.2 s. At  t=1 s the surface is moving downward at  12 cm/s at the same position. The wavecrests are moving in the positive x direction at a speed of  24 m/s. What will be the displacement and velocity of the fluid surface at a position  x = 2.5 m  when  t = 1 s?

4.

A uniform rope of mass  m  and length  $\ell$  hangs from a ceiling.

(a)

Show that the speed of a transverse wave in the rope is a function of  y,  the distance from the lower end, and is given by   $v = \sqrt{gy}$.

(b)

Show that the time it takes a transverse wave to travel the length of the rope is given by   ${\displaystyle t = 2 \sqrt{\ell \over g} }$.

5.

A ${\cal SHO}$ is weakly damped so that the amplitude of its oscillations does not decrease appreciably in one period. Show that for a damping force   $F_d = -\kappa v$  the total energy stored in the oscillation decays exponentially at a rate (averaged over one period) of  $\kappa/m$.