Chapter 1.11, Problem 47E

Frequency of Vibration The frequency f of vibration of a violin string is inversely proportional to its length L. The constant of proportionality k is positive and depends on the tension and density of the string.

1. (a) Write an equation that represents this variation.
2. (b) What effect does doubling the length of the string have on the frequency of its vibration?

To determine

To express: The given variation as equation.

Answer to Problem 47E

The equation of the given variation $f=k/L$.

Explanation of Solution

Inverse variation:

If the quantities $a$ and $b$ are related by the equation $a=\frac{C}{b}$ then for $C\ne 0$, $a$ is inversely proportional to $b$ and $C$ is called constant of proportionality.

The frequency of vibration of a violin string is inversely proportional to its length.

Mathematically,

$f=k/L$ (1)

Thus, the equation of given variation is $f=k/L$.

To determine

To find: The change in frequency when length of the string is doubled.

Answer to Problem 47E

The frequency of the vibration becomes half.

Explanation of Solution

Given:

The length of the string is doubled.

Calculation:

To calculate frequency for double the length of string, substitute $2L$ for $L$ in equation (1).

$f_1=k/2L$ (2)

Here, $f_1$ is the frequency of vibration when length is doubled.

Compare equation (1) and equation (2).

$f_1=\frac{f}{2}$

Thus, the frequency for double the length becomes half.