Physics - Wave on String Question with Solution | TestHub

The fundamental frequency of a sonometer wire is given by $n=\frac{1}{2L}\sqrt{\frac{T}{m}}$

The linear mass density $m=\text{volume per unit length}\times \text{density}=\pi {\left(\frac{D}{2}\right)}^2\rho$

Therefore, $n=\frac{1}{2L}\sqrt{\frac{4T}{\pi D^2\rho }}=\frac{1}{LD}\sqrt{\frac{T}{\pi \rho }}$

This implies $n\propto \frac{\sqrt{T}}{LD}$

Given $T^{\prime }=3T$, $L^{\prime }=3L$, and $D^{\prime }=2D$

The new frequency is $n^{\prime }\propto \frac{\sqrt{3T}}{3L\times 2D}$

$n^{\prime }=\frac{\sqrt{3}}{6}\left(\frac{\sqrt{T}}{LD}\right)=\frac{\sqrt{3}}{6}n=\frac{n}{2\sqrt{3}}$

Answer: $\frac{\mathrm{n}}{2\sqrt{3}}$