Chapter 4.8, Problem 68E

a.

To determine

To graph: Graph the function using graphing utility.

Answer to Problem 68E

  

Explanation of Solution

Given: A ball is bobbing up and down on the end of a spring has a maximum displacement of $3$ inches. Its motion is modeled by $y=\frac{1}{4}\cos 16t,t>0$ . $y$ is measured in feet and $t$ is time measure in seconds.

Graph:

By using the graphing utility the below graph is the graphical representation of the function $y=\frac{1}{4}\cos 16t,t>0$ ,

  

Thus we can use a graphical utility to graph a function.

b.

To determine

To find: The period of oscillation.

Answer to Problem 68E

The period of oscillation is $\frac{\pi }{8}\approx 0.3927$ .

Explanation of Solution

Given: A ball is bobbing up and down on the end of a spring has a maximum displacement of $3$ inches. Its motion is modeled by $y=\frac{1}{4}\cos 16t,t>0$ . $y$ is measured in feet and $t$ is time measure in seconds.

Calculation:

From the given information,

  $\begin{array}{l}y=\frac{1}{4}\cos 16t\to y\text{ in feet, }3\text{inches}=\frac{1}{4}\text{feet}\\ \end{array}$

Therefore the period is,

  $\frac{2\pi }{16}=\frac{\pi }{8}\approx 0.3927$

Thus we can find the period of oscillation.

c.

To determine

The first time the ball passes the point of equilibrium $\left(y=0\right)$ .

Answer to Problem 68E

The first time the ball passes the point of equilibrium when $t=\frac{\pi }{32}\approx 0.09817$ .

Explanation of Solution

Given: A ball is bobbing up and down on the end of a spring has a maximum displacement of $3$ inches. Its motion is modeled by $y=\frac{1}{4}\cos 16t,t>0$ . $y$ is measured in feet and $t$ is time measure in seconds.

Calculation:

From the given information,

  $y=0\to \text{is the equilibirium}$

The equilibrium occurs when the angle of the $\cos \text{is}\frac{\pi }{2}$ ,

  $\begin{array}{c}16t=\frac{\pi }{2}\\ t=\frac{\pi }{32}\end{array}$

Thus we can determine the first time the ball passes the point of equilibirium.