Forces in curved surfaces in fluid dynamics

Question

This is a very simple question that I'm having a hard time to conceptualize. I'm doing the following problem from White's Fluids Mechanics:

Gate AB is a quarter-circle 3 meters wide and hinged at B. Find the force F just sufficient to keep the gate from opening. The gate is uniform and weighs 15 kN

I started calculating the forces through the centers of pressure and weight to apply in the torque equation and find F as a result. The force on the curved surface of the gate divides into the horizontal and vertical forces:

My problem is determining $F_V$. I checked the solutions and it says:

The vertical force equals the weight of the missing piece of water above the gate

My question is: Why is that so? Why is it equal to the weight of the missing piece of water above the gate, if there is only air above. If I filled the missing part with water, would $F_V$ remain the same?

Answer 1

If you filled the top with water Fv would be 0.

The pressure on top would balance the pressure below, the gate would remain closed on its own.

Their solution is actually pretty clever in that respect. If you neglect the thickness of the gate (not given so that's a fair assumption) then you know the pressure at those points would have to be the same in a fully submerged system. That means that when that water isn't there, the pressure on it has to be the same or more as if there were water there if you want the force down on the gate to beat the force up.