Question

The length of a stretched string is doubled and tension is increased 100 times. What will happen to the pitch of the sound produced by the string?

• A. Pitch will increase
• B. Pitch will decrease
• C. No change
• D. Depends on the medium of sound propagation

Answer 1

The correct option is A

Pitch will increase

Frequency of a stretched string is given as:
f = $\frac{1}{2L}$ $\sqrt{\frac{T}{m}}$
where, f is frequency, l is the length of the string, T is tension in the string, m is mass per unit length of the string.
Initially, let frequency be $f_1$, length be $L_1$, tension be $T_1$, mass per unit length be $m_1$. Then
$f_1$ = $\frac{1}{2L_1}$ $\sqrt{\frac{T_1}{m_1}}$................(i)
After making changes, let frequency be $f_2$, length be $L_2$, tension be $T_2$, mass per unit length be $m_2$.
Then, $T_2$ = 100 $\times$ $T_1$, $L_2$ = 2 $\times$ $L_1$, $m_2$ = $m_1$
$\Rightarrow$ $f_2$ = $\frac{1}{4L_1}$ $\sqrt{\frac{{100T}_1}{m_1}}$................(ii)
Dividing (ii) by (i), we get, $f_2$ = 5 $\times$ $f_1$
Thus, pitch will increase because frequency is increasing.