Chapter 9.2, Problem 67E

Simple Harmonic Motion and Damped Harmonic Motion In mechanics an object whose position relative to a rest position is given by a generalized cosine (or sine) function

$p\left(t\right)=A\cos \left(\omega t+d\right)$ is called a simple harmonic oscillator. Examples of simple harmonic oscillators are a mass suspended from a spring and a pendulum swinging through a small angle, in the absence of frictional damping forces. When we take damping forces into account, we obtain a damped harmonic oscillator:

$p\left(t\right)=A{e}^{-bt}\cos \left(\omega t+d\right)$

(assuming that the damping forces are not so large as to pre- vent the system from oscillating entirely). Exercises 67–70 are based on these concepts.

A mass on a spring is undergoing simple harmonic motion so that its vertical position at time t seconds is given by

$p\left(t\right)=1.2\cos \left(5\pi t+\pi \right)$ centimetersbelow the rest position.

a. What is its vertical position at time $t=0$?

b. How fast is the mass moving, and in what direction, at times $t=0$ and $t=0.1$?

c. The frequency of oscillation is defined as the reciprocal of the period. What is the frequency of oscillation of the spring?