Question

A vibrating string has a frequency of 200 Hz. What will its frequency be if:A. Its length is decreased to ¼ of its original length?B. Its tension is quadrupled?

Answer 1

Given data:

* The frequency of the string vibration is f = 200 Hz.

Solution:

(A). The frequency of vibration in terms of the length of the string is,

$f=\frac{\sqrt[]{T}}{L}$

If the length of the string is decreased to 1/4 of its actual value, then the frequency of the string is,

$\begin{gathered} f^{\prime}=\frac{\sqrt[]{T}}{\frac{L}{4}} \\ f^{\prime}=\frac{4\sqrt[]{T}}{L} \\ f^{\prime}=4f \end{gathered}$

Substituting the known values,

$\begin{gathered} f^{\prime}=4\times200\text{ } \\ f^{\prime}=800\text{ Hz} \end{gathered}$

Thus, the frequency of the vibration in the given case is 800 Hz.

(B). If the tension of the string is quadrupled, that is its value becomes 4 times the actual value, then the frequency of the vibration is,

$\begin{gathered} f^{\prime}=\frac{\sqrt[]{4T}}{L} \\ f^{\prime}=\frac{2\sqrt[]{T}}{L} \\ f^{\prime}=2f \end{gathered}$

Substituting the known values,

$\begin{gathered} f^{\prime}=2\times200 \\ f^{\prime}=400\text{ Hz} \end{gathered}$

Thus, the frequency of the vibration becomes 400 Hz.