Chapter 1, Problem 136RE

To determine

To calculate:The equation that depicts the frequency of a given vibrating string which is under constant tension $f$ and varies inversely with respect to its length $l$ and use it to find out the length the string must be shortened to vibrate with $f=660$ if the violin string is $12$ inches long when $f=440$

Answer to Problem 136RE

The string must be shortened to $l=8$ inches to vibrate with $f=660$ times per second.

Explanation of Solution

Given information:

Here, the frequency of a given vibrating string which is under constant tension $f$ and varies inversely with respect to its length $l$

Also, violin string is $12$ inches long when $f=440$

New frequency is $f=660$

Formula used:

For 2 variables say, $x$ and $y$ , the statement $x$ is directly proportional to $y$ can be written as:

  $x\alpha y$

Which can be written as:

  $x=ky$

Where $k$ denotes the proportionality constant.

Similarly the statement $x$ is inversely proportional to $y$ can be interpreted as:

  $x\alpha \frac{1}{y}$

Which can be written as:

  $x=k\left(\frac{1}{y}\right)$

Where $k$ denotes the proportionality constant.

Calculation:

As the frequency of a given vibrating string which is under constant tension $f$ and varies inversely with respect to its length $l$

Recall, For 2 variables say, $x$ and $y$ , the statement $x$ is inversely proportional to $y$ can be interpreted as:

  $x\alpha \frac{1}{y}$

Which can be written as:

  $x=k\left(\frac{1}{y}\right)$

Where $k$ denotes the proportionality constant.

Hence, this variation can be expressed as follows:

  $f\alpha \frac{1}{l}$

  $f=\frac{k}{l}$ $\left(1\right)$

Where $k$ denotes the proportionality constant.

It is also given $f=440$ times per secwhen $d=12$ inches

Put these values in $\left(1\right)$ to get:

  $\begin{array}{l}440=\frac{k}{12}\\ k=440*12\\ k=5280\end{array}$

Therefore, the proportionality constant $k=5280$ inches per second.

Now for new frequency of vibrating string $f=660$ times per second

Replace the value of $k=5280$ and $f=660$ in $\left(1\right)$ to get:

  $\begin{array}{l}660=\frac{5280}{l}\\ l=\frac{5280}{660}\\ l=8\end{array}$

Thus, the string must be shortened to $l=8$ inches to vibrate with $f=660$ times per second.