Question

A block is being hung in air from a wire which is streached over a sonometer. The bridges of the sonometer are L cm apart when the wire is in unison with a tuning fork of frequency N. When the block is completely immersed in water, the length between the bridges is I cm for re-establishing unison. The specific gravity of the material of the block is

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

Answer 1

Answer: (C)

$\frac{L^2}{L^2-I^2}$

When the block is being hung in air

$N=\frac{1}{2L}\sqrt{\frac{Mg}{\mu }}$

Now, consider the case when the block is in water

From the above diagram we have,

$T_1+F_B=Mg$

$\Rightarrow T_1+\rho \frac{M}{\sigma }=Mg$

$\therefore T_1=Mg(1-\frac{\rho }{\sigma })$

where $\sigma$ : Density of block

$\rho$ : Density of water

$N=\frac{1}{2I}\sqrt{\frac{T_1}{\mu }}=\frac{1}{2I}\sqrt{\frac{Mg(1-\frac{\rho }{\sigma })}{\mu }}$ .....(ii)

If ${\sigma }^{\prime }$: Specific gravity of block then from Eqs. (i) and (ii)

$\frac{I}{L}=\sqrt{1-\frac{1}{{\sigma }^{\prime }}}$

$\Rightarrow \frac{I^2}{L^2}=1-\frac{1}{{\sigma }^{\prime }}$

$\Rightarrow \frac{1}{{\sigma }^{\prime }}=1-\frac{I^2}{L^2}$

$\Rightarrow {\sigma }^{\prime }=\frac{L^2}{L^2-I^2}$