1.
Obtain an expression for the fraction of a complete period
that a simple harmonic oscillator spends within a small interval ?*x* at a position *x*. Sketch the curves of
this function versus *x* for
several different amplitudes. Discuss
the physical significance of the results. Comment on the areas under the various curves.

2.
A particle of mass *m* is
at rest at the end of a spring (force constant *k*) hanging from a fixed support. At time *t* = 0, a constant downward force *F* is applied to the mass and acts for a time *t*_{0}. Show that, after the force is removed, the displacement of the mass from
its equilibrium position (*x* = *x*_{0}, where *x* is down) is

where

3. For a damped driven oscillator, show that the average kinetic energy is the same at a frequency of a given number of octaves (an octave is a frequency interval in which the highest frequency is just twice the lowest frequency) above the kinetic energy resonance as at a frequency of the same number of octaves below the resonance.

4.
Use the general solutions *x*(*t*) to the differential
equation

for under damped,
critically damped, and over damped motion and choose the constants of
integration to satisfy the initial conditions *x* = *x*_{0} and *v* = *v*_{0} = 0 at *t* = 0. Use a computer to plot the results for *x*(*t*)/*x*_{0} as a function of *w*_{0}*t* for the following three cases:

*b*= (1/2)*w*_{0}*b*=*w*_{0}*b*= 2*w*_{0}

Show all three curves on a single plot.

5. A damped linear oscillator, originally at rest in its equilibrium position, is subjected to a forcing function given by

Find the response
function. Allow *t* -> 0 and show that the solution becomes that for a step function.

Practice the material and concepts discussed in Chapter 3.