Question

The upper edge of a gate in a dam runs along the water surface. The gate is $2\text{m}$ high and $4\text{m}$ wide and is hinged along the horizontal line through its top. Find the torque( in $\text{kN~m}$) due to the force from water about the top edge of the gate.

Answer 1

Consider a strip of thickness $dy$ at a height $y$ from bottom of the gate.
Area of this strip is $4dy$
Force on the strip
$dF=\rho g(2-y)4dy;$
Torque due to this force about top edge of the plate is
$d\tau =4\rho g(2-y{)}^{2}dy$
$\tau =\rho g4{[4y+\frac{y^3}{3}-2y^2]}_0^2$
$={10}^3\times 10\times 4[8+\frac{8}{3}-8]$
$=10.67\times {10}^4$ N-m.