The net force acting on the gate element of width dy at a depth $y$ from the surface of the fluid, is

$dy  =\left( p _0+\rho_{ g } y - p _0\right) \times 1 dy$

$=\rho \text { gydy }$

Torque about the hinge is

$d \tau=\operatorname{pgydy} \times\left(\frac{1}{2}- y \right)$

Net torque experienced by the gate is

$\tau_{\text {net }}=\int d \tau+ F \times \frac{1}{2}=0$

$\Rightarrow F =\frac{\rho g }{6}$