Question

A square of size $1m\times 1m$ is hinged at its midpoint. A fluid of density $\rho$ fills the space to the left of the gate. The force $F$ required to hold the fluid stationary is $\frac{\rho g}{n}$. The value of $n$ is

Answer 1

At a depth $y$ from the surface of the fluid, the net force acting on the gate element of width $dy$ is
$dF=\left(p_0+\rho gy-p_0\right)\times 1dy=\rho gydy$

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

${\tau }_{net}=\int d\tau +F\times \frac{1}{2}$

$={\int }_0^1\rho gydy\left(\frac{1}{2}-y\right)+F\times \frac{1}{2}=0$

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

So the value of $n$ is 6.