Question

The frequency of sonometer wire is $f$, but when the weights producing the tensions are completely immersed in water, the frequency becomes $f/2$ and on immersing the weights in a certain liquid, the frequency becomes $f/3$. The specific gravity of the liquid is :

• A. $4/3$
• B. $16/9$
• C. $15/12$
• D. $32/27$

Answer 1

The correct option is D $32/27$

$\because f\propto \sqrt{T}$

$\frac{f_{air}}{f_{water}}=\sqrt{\frac{W_{air}}{W_{water}}}=\sqrt{\frac{V\rho g}{V\rho g-V{\rho }_{w}g}}$

$=\frac{f}{\frac{f}{2}}=\sqrt{\frac{\rho }{\rho -{\rho }_w}}$

$\Rightarrow 2=\sqrt{\frac{\rho }{\rho -{\rho }_w}}$

$\Rightarrow 4\rho -4{\rho }_w=\rho$

$\Rightarrow \rho =\frac{4}{3}{\rho }_w$

Similarly in second case,

$\frac{f}{\frac{f}{3}}=\sqrt{\frac{\rho }{\rho -{\rho }_L}}$

$\Rightarrow 3=\sqrt{\frac{\frac{4}{3}{\rho }_w}{4-\frac{3{\rho }_L}{{\rho }_w}}}=\sqrt{\frac{4}{4-\frac{3{\rho }_L}{{\rho }_w}}}$

Here, $\frac{{\rho }_L}{{\rho }_w}=$ specific gravity (say $S$)

$\Rightarrow 9=\frac{4}{4-3S}$

$\Rightarrow 36-27S=4\Rightarrow S=\frac{32}{27}$