Question

A square gate of size $1\text{m}\times 1\text{m}$ 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 gate stationary is

• A. $\frac{\rho g}{3}$
• B. $\frac{\rho g}{2}$
• C. $\frac{\rho g}{6}$
• D. $\frac{\rho g}{8}$

Answer 1

The correct option is C $\frac{\rho g}{6}$

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

$dF=(p_0+\rho gy-p_0)\times 1\times dy=\rho gydy$

Torque of this force about the hinge is

$d\tau$ = $\rho gydy\times (\frac{1}{2}-y)$

Net torque on the gate should be zero for the gate to remain stationary.

${\tau }_{net}=\int d\tau +(F\times \frac{1}{2})=0$

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

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

Hence, option $\left(\text{C}\right)$ is correct.