Question

The volume of an air bubble is doubled as it rises from the bottom of a lake to its surface. If the atmospheric pressure is $Hm$ of mercury & the density of mercury is $n$ times that of lake water. The depth of the lake is given as $xnH$. Find $x$.

Answer 1

Let's say $h$ is the depth of the lake $\rho$ is the density of water then

${\rho }_m=n\rho \because$ given

$P_o=H{\rho }_{m}g=Hn\rho g$ .........$\left(I\right)$

and $P_1{V}_1=P_2{V}_2$

$\Rightarrow (P_o+h\rho g)V=P_o\left(2V\right)$

$\Rightarrow P_o+h\rho g=2P_o$

$\Rightarrow P_o=h\rho g$ .........$\left(II\right)$

from $\left(I\right)$ and $\left(II\right)$ we have

$Hn\rho g=h\rho g\Rightarrow h=nH$