Question

A stone is hung in air from a wire which is stretched over a sonometer. The bridges of the sonometer are $Lcm$ 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 $lcm$ for re-estabilishing the unison. The specific gravity of the material of the stone is

• A. $\frac{L^2}{L^2+l^2}$
• B. $\frac{L^2-l^2}{L^2}$
• C. $\frac{L^2}{L^2-l^2}$
• D. $\frac{L^2+l^2}{l^2}$

Answer 1

The correct option is C $\frac{L^2}{L^2-l^2}$

When immersed in liquid,


for equilibrium
$T^1+F_B=mg\Rightarrow T^1+\frac{{\rho }_w}{{\rho }_s}mg=mg\Rightarrow T^1=mg[1-\frac{{\rho }_w}{{\rho }_s}]$
Here,
${\rho }_w$ = density of water and ${\rho }_s$= density of stone

As in both the cases sonometer is in unison with tuning fork
$\Rightarrow N=\frac{1}{2L}\sqrt{\frac{mg}{\mu }}=\frac{1}{2l}\sqrt{\frac{mg[1-\frac{{\rho }_w}{{\rho }_s}]}{\mu }}\Rightarrow \frac{l}{L}=\sqrt{1-\frac{1}{{\sigma }^1}}({\sigma }^1=\text{specific gravity of stone})\Rightarrow \frac{l^2}{L^2}=1-\frac{1}{{\sigma }^1}\Rightarrow \frac{1}{{\sigma }^1}=\frac{L^2-l^2}{L^2}\Rightarrow {\sigma }^1=\frac{L^2}{L^2-l^2}$