A sonometer wire is stretched by hanging a metal bob, the fundamental frequency of the wire is ' \(n_1\) '. When the bob is completely immersed in water, the frequency of vibration of wire becomes ' \(\mathrm{n}_2\) '. The relative density of the metal of the bob is

\(\mathrm{n}=\frac{1}{2 \mathrm{~L}} \sqrt{\frac{\mathrm{~T}}{\mu}}\)
The sonometer wire is stretched by a metal bob.
Hence, we can write, \(T=m g=W\)
where W is the weight of the metal bob.
\(\begin{array}{ll}
\therefore \quad & \text { From(i), } \\
& \mathrm{n}_1=\frac{1}{2 L} \sqrt{\frac{W_1}{\mu}}, \quad n_2=\frac{1}{2 L} \sqrt{\frac{W_2}{\mu}} \\
\therefore \quad & \frac{W_1}{W_2}=\frac{n_1^2}{n_2^2}
\end{array}\)
\(\text {Relative density }(\sigma)=\frac{\text { weight in air }}{\text { loss of weight in water }}\)
\(\therefore \quad \frac{\mathrm{W}_1}{\mathrm{~W}_1-\mathrm{W}_2}=\frac{\mathrm{n}_1^2}{\mathrm{n}_1^2-\mathrm{n}_2^2}\)