A hemispherical container is placed upside down on a smooth horizontal surface. Through a small hole at the bottom of the container, water is then poured in. Exactly when the container gets full, water starts  leaking from between the table and the edge of the container. Find the mass of the container if water has  density  $\rho$ and radius of the hemisphere is R.

Initially the force on the container is given by the total force of gravity that is acting on the  hemispherical container, or in other words.
  $F_{bef}=Mg+\rho gV=Mg+\frac{2}{3}\pi \rho g{R}^2$
 When the water leaks out, the containers weight no longer acts on the surface. This means the new  pressure is equal to pressure of the water times the area of the base of the hemispherical container, the water pressure is the same on all places of the container so  $P=\rho gR.\pi R^2$. From force balancing, we have 
 $\left(\rho gR\right)\pi R^2=Mg+\frac{2}{3}\pi \rho g{R}^3\Rightarrow M=\frac{1}{3}\pi \rho R^3$