A square of size 1 m × 1 m is hinged at its midpoint. A fluid of density r fills the space to the left of the gate. The force F required to hold the gas stationary is

 

At a depth y from the surface of the fluid, the net force acting on the gate element of width dy is $dF=(p_0+\rho gy–p_0)\times 1dy=\rho gydy$. Torque of this force about the hinge is $d\tau =\rho gydy\times \left(\frac{1}{2}-y\right)$. 

Net torque experienced by the gate is ${\tau }_{net}=\int d\tau +F\times \frac{1}{2}$$\Rightarrow \underset{0}{\overset{1}{\int }}\rho gydy\left(\frac{1}{2}-y\right)+F\times \frac{1}{2}=0$$\Rightarrow F=\frac{\rho g}{6}$