A sonometer wire is in unison with a tuning fork of frequency ' \(n\) ' when it is stretched by a weight of specific gravity ' \(d\) '. When the weight is completely immersed in water, ' \(x\) ' beats are produced per second, then

For Sonometer,
We know, relative density \(\sigma=\frac{\rho_{\text {load }}}{\rho_{\text {water }}}\)
Also, \(\mathrm{n} \propto \sqrt{\mathrm{T}}\)
Tension in air \(\mathrm{T}_{\text {air }}=\rho_{\text {load }} \cdot \mathrm{V} \cdot \mathrm{g}\)
Tension in water \(\mathrm{T}_{\text {water }}=\left(\rho_{\text {load }}-\rho_{\text {water }}\right) \cdot \mathrm{V} \cdot \mathrm{g}\)
\(=\rho_{\text {water }} \cdot(\sigma-1) \cdot \mathrm{V} \cdot \dot{\mathrm{~g}}\)
\(\therefore \quad \frac{\mathrm{n}_{\text {load in air }}}{\mathrm{n}_{\text {load inmersed in water }}}=\frac{\sqrt{\mathrm{T}_{\text {air }}}}{\sqrt{T_{\text {water }}}}\)
On substituting the respective values,
\(\frac{\mathrm{n}_{\text {load in air }}}{\mathrm{n}_{\text {load immersed in water }}}=\frac{\sqrt{\rho_{\text {load }} \cdot \mathrm{V} \cdot \mathrm{g}}}{\sqrt{\rho_{\text {water }} \cdot(\sigma-1) \cdot \mathrm{V} \cdot \mathrm{g}}}\)
\(=\sqrt{\frac{\sigma_{\text {load }}}{\sigma_{\text {load }}-1}}...(i)\)
where, \(\sigma=\) relative density or specific gravity of load
Given, \(\mathrm{n}_{\text {load }}=\) frequency
\(\therefore \quad\) ' \(x\) ' beats are produced when weight is completely immersed in water
\(\mathrm{n}_{\text {load }}\) in water \(=\mathrm{n}-\mathrm{x}\)
Given, Specific gravity of load \(\left(\sigma_{\text {load }}\right)=d\)
\(\Rightarrow \frac{\mathrm{n}}{\mathrm{n}-\mathrm{x}}=\sqrt{\frac{\mathrm{d}}{\mathrm{~d}-1}}\)
...[From(i)]