A stone is hung in air from a wire which is stretched over a sonometer. The bridges of the sonometer are…
A stone is hung in air from a wire which is stretched over a sonometer. The bridges of the sonometer are $L\text{ cm}$ apart when the wire is in unison with a tuning fork of frequency $N$. When the stone is completely immersed in water, the length between the bridges is $l\text{ cm}$. For re-establishing unison, the specific gravity of the material of the stone is
$\frac{L^2}{L^2 + l^2}$
$\frac{L^2 - l^2}{L^2}$
$\frac{L^2}{L^2 - l^2}$
$\frac{L^2 + l^2}{L^2}$
Solution
Frequency of vibration stretched string, $n = \frac{1}{2l} \sqrt{\frac{T}{m}}$
When the stone is completely immersed in water, length changes but frequency does not change.
$l \propto \sqrt{T} \Rightarrow \frac{L}{l} = \sqrt{\frac{V\rho g}{V(\rho - 1)g}}$
$\Rightarrow \frac{L}{l} = \sqrt{\frac{\rho}{\rho - 1}} \Rightarrow \rho = \frac{L^2}{L^2 - l^2}$