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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.

This set covers the material discussed in Chapter 3.