Question

A sonometer wire of length l vibrates in fundamental mode when excited by a tuning fork of frequency 416. Hz. If the length is doubled keeping other things same, the string will
(a) vibrate with a frequency of 416 Hz
(b) vibrate with a frequency of 208 Hz
(c) vibrate with a frequency of 832 Hz
(d) stop vibrating.

Answer 1

(b) vibrate with a frequency of 208 Hz

According to the relation of the fundamental frequency of a string
$\nu =\frac{1}{2l}\sqrt{\frac{F}{\mu }}$
where
l is the length of the string
F is the tension
μ is the linear mass density

We know that ν₁ = 416 Hz, l₁ = l and l₂ = 2l.
$$\begin{gathered} v_1\propto \frac{1}{l_1} \\ {v}_1{l}_1=v_2{l}_2 \\ \left(416\right)l=v_2\left(2l\right) \\ {v}_2=208\mathrm{Hz} \end{gathered}$$