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

Answer: (D)

$\because f\propto \sqrt{T}$
$\frac{f_{\text{air }}}{f_{\text{water }}}=\sqrt{\frac{W_{\text{air }}}{W_{\text{water }}}}=\sqrt{\frac{V\rho g}{V\rho g-V{\rho }_{w}g}}$
$=\frac{f}{\frac{f}{2}}=\sqrt{\frac{\rho }{\rho -{\rho }_w}}$
$\Rightarrow 2=\sqrt{\frac{\rho }{\rho -{\rho }_w}}$
$\Rightarrow 4\rho -4{\rho }_w=\rho$
$\Rightarrow \rho =\frac{4}{3}{\rho }_w$
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 $S$ )
$\Rightarrow 9=\frac{4}{4-3S}$
$\Rightarrow 36-27S=4$
$\Rightarrow S=\frac{32}{27}$