Would it take an equal amount of lb-force as the total water weight weighing down on the piston to be at equilibrium? Pascals principle

Question

Pascals principle sounds simple, but I'm a bit confused on this.

A cylinder that's 24" in diameter by 5 ft in height filled with 117 gallons(980 lb) that's connected to 1" diameter pipe going up 300 ft high filled with 12 gallons(100 lb).

I know that with deeper depth the water pressure increases, but I'm a little lost on this question about how much lb-force I'd need to lift this water upwards.

I have 129 gallons of water or (1080 lb) in total weight weighing down on a piston that's set at the very bottom of the 24"D cylinder. Considering the height of the smaller pipe is 300 ft high, would I need an equal amount of weight (1080 lb) force up to have this piston at equilibrium?

And to lift this piston up I'd assume the lb-force should be slightly over equilibrium if I used a lever.

Answer 1

I assume the question is to determine the force required by the piston. Notice that if the piston at the very bottom is in equilibrium, the magnitude of the force from the water on the piston must equal the force exerted by the piston.

So we just need to find the force of the water on the piston. At the very top of the tube the pressure is atmospheric pressure. The change in pressure due to the depth of the water is given by $\Delta P=h\rho g$, where $h$ in this case is 305 feet. So to obtain the pressure at the very bottom, you can add simply add this change to the initial atmospheric pressure.

Finally it suffices to use the expression $F = PA$ where $A$ is the area of the piston in order to determine the force.

A perhaps strange result of static fluid pressure is that it does not depend on the mass nor volume nor shape of the fluid, only the height. If we increased the diameter of the top pipe whilst keeping the water level constant, the mass of fluid would clearly increase yet the force of the piston would remain the same (assuming the area of the piston doesn't change).

Answer 2

The annular region of the cylinder lid exerts a downward force on the fluid below also. So the total force on the piston is equal to the pressure at the piston times the total area of the piston.