Chapter 12.7, Problem 30PPS

a.

To determine

To find: Amplitude, Period and Frequency.

Answer to Problem 30PPS

Amplitude $=20$ cm

Period $=2$ seconds

Frequency $=\frac{1}{2}$ vibrations per second

Explanation of Solution

Given information:

  $y=20\cos \pi t$

  $y$ Is the distance in centimeters from its equilibrium position at time $t$ .

Given

  $y=20\cos \pi t$

On comparing with $y=a\cos bx$

We get

  $\begin{array}{l}a=20\\ b=\pi \end{array}$

Therefore,

Amplitude $=\left|a\right|=\left|20\right|=20$

Period $=\frac{2\pi }{\left|b\right|}$

  $\Rightarrow$ Period $=\frac{2\pi }{\pi }$

  $\Rightarrow$ Period $=2$

We know frequency is reciprocal of Period.

  $\Rightarrow frequency=\frac{1}{2}$

Therefore,

We got

Amplitude $=20$ cm

Period $=2$ seconds

Frequency $=\frac{1}{2}$ vibrations per second

b.

To determine

To find: The Distance of the object from its equilibrium position at $t=\frac{1}{4}$ second.

Answer to Problem 30PPS

The distance of the object from the equilibrium position is about $14.14$ cm at $t=\frac{1}{4}$ second.

Explanation of Solution

Given information:

  $y=20\cos \pi t$

  $y$ Is the distance in centimeters from its equilibrium position at time $t$ .

  $t=\frac{1}{4}$

  $y=20\cos \pi t$

On substituting $t=\frac{1}{4}$

We get

  $\begin{array}{l}y=20\cos \pi t\\ \Rightarrow y=20\cos \left(\pi \times \frac{1}{4}\right)\\ \Rightarrow y=20\cos \left(\frac{\pi }{4}\right)\\ \Rightarrow y=20\left(0.7071\right)\\ \Rightarrow y=14.14\end{array}$

Therefore,

The distance of the object from the equilibrium position is about $14.14$ cm at $t=\frac{1}{4}$ second.

c.

To determine

To find: The velocity at $t=\frac{1}{4}$ second.

Answer to Problem 30PPS

The Velocity is about $-44.4$ cm/s.

Explanation of Solution

Given information:

  $v=-20\pi \sin \pi t$

  $v$ Is the velocity of the object at time $t$ .

  $t=\frac{1}{4}$

  $v=-20\pi \sin \pi t$

On substituting $t=\frac{1}{4}$

We get

  $\begin{array}{l}v=-20\pi \sin \pi t\\ \Rightarrow v=-20\pi \sin \left(\pi \times \frac{1}{4}\right)\\ \Rightarrow v=-20\pi \sin \left(\frac{\pi }{4}\right)\\ \Rightarrow v=-20\pi \left(0.7071\right)\\ \Rightarrow v=-44.4\end{array}$

Therefore,

The Velocity is about $-44.4$ cm/s.