Chapter 10.1, Problem 33PS

Solution

a.

To determine the initial displacement and the period of the spring and then graph the given function.

The initial displacement is $0.2\text{ feet}$ and the period of the spring is $\frac{\pi }{3}.$

Given:

The given function is:

  $y=A\cos kt$

where,

  $\begin{array}{l}y=Spring\text{'}sverticaldisplacement\\ A=\mathrm{Amplitude}\\ k=\mathrm{constant}\\ t=\mathrm{time}\end{array}$

Calculation:

It is given that a spring whose motion can be modelled by the function is:

  $y=0.2\cos 6t$

Now, the initial displacement is:

  $\begin{array}{l}y=0.2\cos \left(6\times 0\right)\\ y=0.2\times 1\\ y=0.2\end{array}$

And the period for the cosine function is:

  $\begin{array}{l}\text{period}=\frac{2\pi }{b}\\ \text{period}=\frac{2\pi }{6}\\ \text{period}=\frac{\pi }{3}\end{array}$

Hence, the initial displacement is $A=0.2\text{ feet}$ and the period is $\frac{\pi }{3}.$

Graph:

The graph of the given function is shown below:

  

b.

To draw the graph of the given function and what effect does damping have on the motion.

A damping force effects an oscillatory system. Due to damping force, amplitude of the wave decreases with time.

Given:

The given function is:

  $y=0.2e^{-4.5t}\cos 4t$

Graph:

The graph is shown below:

  

Interpretation:

A damping force effects an oscillatory system. Due to damping force, amplitude of the wave decreases with time and finally asymptotes to zero.