Question

A square gate of size $1m\times 1m$ is hinged at its mid-point. 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

Answer: (C)

$\frac{\rho g}{6}$

The net force acting on the gate element of width $dy$ at a depth $y$ from the surface of the fluid, is
$dy=(p_0+{\rho }_{g}y-p_0)\times 1dy$
Torque about the hinge is
$d\tau =pgydy\times (\frac{1}{2}-y)$
Net torque experience by the gate is
${\tau }_{net}=\int d\tau +F\times \frac{1}{2}$
$={\int }_0^1\rho gydy(\frac{1}{2}-y)+F\times \frac{1}{2}=0$
$\Rightarrow F=\frac{\rho g}{6}$
i.e., The force $F$ required to hold the gate stationary is
$\frac{pg}{6}$.