Question

A square gate 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 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

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

Torque of this force about the hinge is

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

Net torque experienced by the gate is

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

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

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