Question

Water stands at a depth h behind the vertical face of a dam. It exerts a resultant horizontal force on the dam tending to slide it along its foundation and a torque tending to overturn the dam about the point $a$. Find torque about $O$,

Answer 1

Perpendicular distance of about force from point O is
$r_1=h-y$
$\therefore$ $d\tau =\left(dF\right)r_{\perp }=\left(\rho gly\right)(h-y)dy$
=total torque=${\int }_0^{h}d\tau$$Aftersubstitutingthevaluesweget,$\displaystyle \tau =\frac{\rho glh^{3}}{6}$