To what height should a cyclindrical vessel be filled with a homogeneous liquid to make the force with which the liquid pressure on the sides of the vessel equal to the force exerted by the liquid on the bottom of the vessel ?

Explanation

let h be the height of liquid column of density `rho` in the cylindrical vessel of radius r. The froce acting on the bottom of vessel = weight of liquid filled in cylinder ` = mg = (pi r^2 h rho)g` …. (i) The area of walls incontact with liquid ` = 2pi rho` Average pressure of liquid on walls is `P = ("pressure at bottom "+ "pressrue at top of liquid column")/(2)`
` = (rho gh _ 0)/(2) = (rho gh)/(2)`
Force on the walls of vessel `= (rho g h)/(2) xx 2pi r h = pi rho grh^2` ..... (ii)
As per question the forces (i) and (ii) are equal so, `pi r^2 h rho g = pi rho g rh^2` or `h =r` it means height of liquid coluum in vessel should be equal to its radius.