Chapter 12.7, Problem 31PPS

a.

To determine

To find: Equation and graph of given model.

Answer to Problem 31PPS

Equation of given graph is

  $y=\frac{3}{2}\cos \theta$

Explanation of Solution

Given information:

Frequency of piano string $=130$ hertz

Amplitude $=a=1$

We need to write equation using Cosine model.

Given

  $a=1$

Frequency of piano string $=130$ hertz

We know

Period $=\frac{1}{frequency}$

  $\Rightarrow$ Period $=\frac{1}{130}$

  $\begin{array}{l}\Rightarrow \frac{2\pi }{b}=\frac{1}{130}\\ \Rightarrow b=130\times 2\pi \\ \Rightarrow b=260\pi \end{array}$

Given it is Cosine function.

Therefore,

On substituting values

We get

  $\begin{array}{l}y=a\cos bt\\ \Rightarrow y=1\cos 260\pi t\\ \Rightarrow y=\cos 260\pi t\end{array}$

Therefore,

Equation of given situation is

  $y=\cos 260\pi t$

Graph of above equation is

  

b.

To determine

To find: Does the amplitude and period change when frequency doubles.

Answer to Problem 31PPS

Amplitude remains the same.

Period decreases.

Explanation of Solution

Given information:

Frequency is doubled

We know

Period $=\frac{1}{frequency}$

From above formula we can say that Frequency and Period are reciprocal to each other.

So, if Value of Frequency is doubled it means increased then the value of period will decrease as they are reciprocal to each Other.

As there is no relation between Frequency and Amplitude.

So Amplitude will remain same when Frequency is doubled.

Therefore

Amplitude remains the same.

Period decreases.