Chapter 3.5, Problem 13E

a.

To determine

To find: The body’s velocity, speed, and acceleration at the time $t$ .

Answer to Problem 13E

Velocity is $3\cos t$ ,speed is $\left|3\cos t\right|$ and the acceleration is $-3\sin t$ .

Explanation of Solution

Given information: A body is moving in simple harmonic motion with position $s=2+3\sin t$ .

Calculation: Velocity is derivative of distance.

  $s\text{'}=3\cos t$ .

Speed is absolute value of velocity.

So speed is $\left|\mathrm{velocity}\right|=\left|3\cos t\right|$ .

The acceleration is derivative of velocity.

Acceleration is $s\text{'}\text{'}=3(-\sin t)=-3\sin t$ .

Thus, velocity is $3\cos t$ ,speed is $\left|3\cos t\right|$ and the acceleration is $-3\sin t$ .

b.

To determine

The body’s velocity, speed, and acceleration at the time $t=\frac{\pi }{4}$ .

Answer to Problem 13E

Velocity is $\frac{3\sqrt{2}}{2}$ ,speed is $\frac{3\sqrt{2}}{2}$ and the acceleration is $\frac{-3\sqrt{2}}{2}$ .

Explanation of Solution

Given information: A body is moving in simple harmonic motion with position $s=2+3\sin t$ .

Calculation: Velocity is derivative of distance.

  $s\text{'}=3\cos t$ .

Put $t=\frac{\pi }{4}\mathrm{in}s\text{'}$ we get:

  $\begin{array}{l}s\text{'}=3\cos \frac{\pi }{4}\\ s\text{'}=3(\frac{1}{\sqrt{2}})=\frac{3}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}\\ s\text{'}=\frac{3\sqrt{2}}{2}\end{array}$

Speed is absolute value of velocity.

Put $t=\frac{\pi }{4}$ So speed is

  $\begin{array}{l}=\left|3\cos t\right|\\ =\left|3\cos \frac{\pi }{4}\right|\\ =\left|3\times \frac{1}{\sqrt{2}}\right|=\left|\frac{3}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}\right|\\ =\left|\frac{3\sqrt{2}}{2}\right|=\frac{3\sqrt{2}}{2}\end{array}$ .

The acceleration is derivative of velocity.

Acceleration is $s\text{'}\text{'}=3(-\sin t)=-3\sin t$ .

Put $t=\frac{\pi }{4}\mathrm{in}\mathrm{s}\text{'}\text{'}$ we get:

  $\begin{array}{l}\mathrm{s}\text{'}\text{'}=-3\sin \frac{\pi }{4}=\frac{-3}{\sqrt{2}}\\ s\text{'}\text{'}=\frac{-3}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{-3\sqrt{2}}{2}\end{array}$

Thus, velocity is $\frac{3\sqrt{2}}{2}$ ,speed is $\frac{3\sqrt{2}}{2}$ and the acceleration is $\frac{-3\sqrt{2}}{2}$ .

c.

To determine

Describe the motion of the body.

Answer to Problem 13E

The period of $s=2+3\sin t$ is $2\pi$ .

Explanation of Solution

Given information: A body is moving in simple harmonic motion with position $s=2+3\sin t$ .

Calculation: A body is moving in simple harmonic motion with position $s=2+3\sin t$ .

Draw the graph:

  

We observe from the graph, body swing between $-1\mathrm{and}5$ . So period is $2\pi$ .