A sonometer wire stretched by weight ' $w$ ' is in unison with a tuning fork. The corresponding resonating…
A sonometer wire stretched by weight ' $w$ ' is in unison with a tuning fork. The corresponding resonating length is ' $\mathrm{L}_1$ ' if the weight is increased by ' $3 w$ ' the corresponding resonating length of the sonometer in unison tuning fork becomes ' $\mathrm{L}_2$.' . The ratio $\left(\frac{\mathrm{L}_1}{\mathrm{~L}_2}\right)$ is
$4: 1$
$2: 2$
$1: 2$
$1: 4$
Solution
Velocity of the sound on the string under tension $\mathrm{T}$ and with mass per unit length $\mu$ is given by
$\begin{aligned} & \mathrm{v}=\sqrt{\frac{\mathrm{T}}{\mu}} \\ & \therefore \mathrm{v}_1=\sqrt{\frac{\mathrm{w}}{\mu}} \mathrm{v}_2=\sqrt{\frac{(3 \mathrm{w}+\mathrm{w})}{\mu}}\end{aligned}$
We know $\lambda \propto L \& f=\frac{v}{\lambda}$
Therefore, $\mathrm{f}_1 \propto \frac{\sqrt{\mathrm{w}}}{\mathrm{L}_1}$ and $\mathrm{f}_2 \propto \frac{\sqrt{4 \mathrm{w}}}{\mathrm{L}_2}$
$\because \mathrm{f}_1=\mathrm{f}_2 \Rightarrow \frac{\mathrm{L}_1}{\mathrm{~L}_2}=\frac{1}{\sqrt{4}}=\frac{1}{2}$