Question

A sonometer wire is vibrating in resonance with a tuning fork. Keeping the tension applied same, the length of the wire is doubled. The wire will still be in resonance with the tuning fork if the wire vibrates in:

• A. 4 loops
• B. 2 loops
• C. 8 loops
• D. is not possible

Answer 1

Answer: (B)

2 loops

The sonometer frequency is given by

$v=\frac{n}{2L}\sqrt{\frac{T}{\mu }}$

Cae $1$

it vibrates with length L

$n=n_1\therefore v_1=\frac{n_2}{2L}\sqrt{\frac{T}{\mu }}$

Case $2$

It vibrates with a length of $2$L

$\therefore v_2=\frac{n_2}{2\left(2L\right)}\sqrt{\frac{T}{\mu }}\therefore \frac{v_1}{v_2}=\frac{n_1\times 2}{n_2}\{v_1=v_2\}\therefore 1=\frac{n_1\times 2}{n_2}\therefore n_2=2n_1$

Hence, the wire is doubled the number of loops also get doubled to produce the resonance. That is, it resonates in second harmonic.