A smooth gate is kept in equilibrium by applying a horizontal force. What is the value of y so that no horizontal reaction force acts at the pivot?

The thrust force due to pressure act at center of gravity'∴Fth=PA=ρgh2hw       here w=width of gateFor translational equilibrium :  Fth+Rx=F If horizontal reaction force is absent, Rx=0 ⇒Fth=FFor rotational equilibrium about hinge point : Torque due to Fth=Torque due to F ⇒∫0hρgh-y'wdy'y'=Fy ⇒ρgw∫0hhy'-y'2dy'=Fy ⇒ρgwh6=Fy ⇒ρgh2hw  h3=Fy ⇒Fthh3=Fy ⇒y=h3The center of liquid thrust force passes through y=h3