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?