Question

The volume of an air bubble is doubled as it rises from the bottom of lake to its surface. The atmospheric pressure is $75cm$ of mercury. The ratio of density of mercury to that of lake water is $\frac{40}{3}$. The depth of the lake in metre is :

• A. $10$
• B. $15$
• C. $20$
• D. $25$

Answer 1

The correct option is A $10$

Let $v^{\prime }$ be the volume of bubble at the surface and $v$ at depth $h$

Ratio of volume of air bubble at top to bottom

$v^{\prime }/v=2\Rightarrow v^{\prime }=2v$

From idle gass equation $p=p^{\prime }\left(\frac{v^{\prime }}{v}\right)=2p^{\prime }$

$\therefore p^{\prime }=P_o$ so $p=2P_o$

Ratio of density of mercury to that of lake water, ${\rho }_{hg}/{\rho }_w=40/3$

Pressure at $h$ depth $=P_o+{\rho }_{w}gh$

$\Rightarrow 2P_o=P_o+{\rho }_{w}gh$

$\Rightarrow P_o={\rho }_{w}gh$

$\Rightarrow {\rho }_{hg}g{h}_{hg}={\rho }_{w}g{h}_w$

$h_w=\frac{{\rho }_{hg}{h}_{hg}}{\rho w}$

$h_w=\frac{40\times .75}{3}$

$h=10m$