Question

A sonometer wire is vibrating with a frequency of $30\text{Hz}$ in the fundamental mode. If the length of the wire is increased by $20\%$, the change in the frequency of the fundamental mode is

• A. $5\text{Hz}$
• B. $10\text{Hz}$
• C. $15\text{Hz}$
• D. $20\text{Hz}$

Answer 1

The correct option is A $5\text{Hz}$

If the length is increased by $20\%$, the new length is
$L^{{}^{\prime }}=L+\frac{20L}{100}=1.2L$
The original frequency is
$v=\frac{1}{2L}\sqrt{\frac{T}{\mu }}$
Since $T$ and $\mu$ are the same, the new frequency will be
$v^{{}^{\prime }}=\frac{1}{2L^{{}^{\prime }}}\sqrt{\frac{T}{m}}=\frac{1}{1.2\times 2L}\sqrt{\frac{T}{\mu }}=\frac{v}{1.2}$

Given that $v=30\text{Hz}$,
Therefore, $v^{{}^{\prime }}=\frac{30}{1.2}=25\text{Hz}$.

Thus, the frequency decreases by $5\text{Hz}$.
Hence the correct choice is (a).