A cylinder of cork is floating in water in a container closed at the top as shown in the figure. Now more air is introduced into the container so that pressure of air on the free surface is increased. Then volume of water displaced by the cork

Let V' be the initial volume of water displaced by the cork and V'' be the final volume of water displace by the same cork. If m be the mass of the cork, then initially, 

$\mathrm{mg}+\mathrm{PA}=\left(\mathrm{P}+\mathrm{\rho gh}\right)\mathrm{A}\Rightarrow \mathrm{mg}=\mathrm{\rho g}\left(\mathrm{hA}\right)$

$\Rightarrow \mathrm{mg}=\mathrm{\rho gV}\text{'}.......\left(1\right)\mathrm{where}\mathrm{V}\text{'}=\mathrm{hA}$

Finally $\mathrm{mg}+\mathrm{P}\text{'}\mathrm{A}=\left(\mathrm{P}\text{'}+{\mathrm{\rho gh}}^1\right)\mathrm{A}\Rightarrow \mathrm{mg}=\mathrm{\rho g}\left({\mathrm{h}}^1\mathrm{A}\right)\Rightarrow \mathrm{mg}=\mathrm{\rho gV}\text{'}\text{'}..........\left(2\right)$

Here h = initial submerged length, h' = final submerged length, From equations (1) and (2), V'' = V