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

(a) Horizontal force,(b) torque about O, (c ) the height at which the resultant force would have to act to produce the same torque,l=cross-sectional length and `rho=`density of water.

Answer

A, B, C

Explanation

(a) `dF=(p_y)(dA)=(rho g y)(ldy)`

`:. F=int_(0)^(h)dF=(rho glh^(2))/2`
(b) Perpendicular distance of about force point O is
`r_(_|_)=h-y`
`:. dtau=(dF)r_(_|_)=(rho g y)(h-y)dy`
=total torque `=int_(0)^(h) d tau`
After substituting the value we get.
`tau = (rho glh^(3))/(6)`
(c) `Fxxr_(_|_)=(tau)`
`:.r_(_|_) = (tau)/(F)`
Substituting the values we get,
`r_(_|_) = (h)/(3)`