Chapter 5.3, Problem 78E

Sound Vibrations A tuning fork is struck, producing a pure tone as its tines vibrate. The vibrations are modeled by the function

$v(t)=0.7\sin (880\pi t)$

where v(t) is the displacement of the tines in millimeters at time t seconds.

1. (a) Find the period of the vibration.
2. (b) Find the frequency of the vibration, that is, the number of times the fork vibrates per second.
3. (c) Graph the function v.

To determine

To find: The period of the vibration for the given function.

Answer to Problem 78E

One period of the function $v\left(t\right)=0.7\sin \left(880\pi t\right)$ is $\frac{1}{440}$ seconds.

Explanation of Solution

Given:

The vibration are modeled by the function $v\left(t\right)=0.7\sin \left(880\pi t\right)$.

Formulas used:

The sine curve $y=a\sin kx\left(k>0\right)$ have amplitude $\left|a\right|$ and period $\frac{2\pi }{k}$.

Calculation:

Rewrite the function.

That is, $v\left(t\right)=0.7\sin \left(880\pi t\right)$.

Here, $v\left(t\right)$ is the displacement of the tines in millimeters at time t seconds.

The period of the wave is computed as follows,

$\begin{array}{c}\frac{2\pi }{k}=\frac{2\pi }{880\pi }\\ =\frac{1}{440}\end{array}$

Therefore, one period of the function $v\left(t\right)=0.7\sin \left(880\pi t\right)$ is $\frac{1}{440}$ seconds.

To determine

To find: The frequency of the vibration. That is, the number of times for k vibrates per second.

Answer to Problem 78E

The frequency of the vibration is 440.

Explanation of Solution

Calculation:

From part (a), One period of the function $v\left(t\right)=0.7\sin \left(880\pi t\right)$ is $\frac{1}{440}$ seconds.

That is, one vibration takes time $\frac{1}{440}$ seconds.

Thus, there are 440 vibration per second.

To determine

To sketch: The graph of the function v.

Explanation of Solution

Use the online graphing calculator to draw the function as shown below in Figure 1.

From Figure 1, it is observed that the function has maximum value occurs at $0.7$ and minimum value occurs at $-0.7$.