Fluid Statics Problem: Calculating Resultant Force on a Dam - Solution Attempt

[NATURE.M]

Homework Statement

Water stands at a depth D behind the vertical upstream face of a dam. Let W be the width of the dam. Find the resultant horizontal force exerted on the dam by the water.

The Attempt at a Solution

So I know how to solve the problem, I was just unsure of whether we should be considering the force caused by the absolute pressure (pressure of the liquid & the pressure of the air above it) or just simply the gauge pressure (pressure of the liquid). My intuition tells me the latter (since a force from the pressure of the air would act vertical), but I would just like to verify.

 

[Chestermiller]

Homework Statement

Water stands at a depth D behind the vertical upstream face of a dam. Let W be the width of the dam. Find the resultant horizontal force exerted on the dam by the water.

The Attempt at a Solution

So I know how to solve the problem, I was just unsure of whether we should be considering the force caused by the absolute pressure (pressure of the liquid & the pressure of the air above it) or just simply the gauge pressure (pressure of the liquid). My intuition tells me the latter (since a force from the pressure of the air would act vertical), but I would just like to verify.

Your intuition gave you the right answer, but for the wrong reason. The pressure doesn't just act in the vertical direction. It acts in all direction. Otherwise, how does the water pressure exert a horizontal force on the dam? The reason you use the gage pressure is that the air is exerting a horizontal force on the other side of the dam, so the air pressures cancel out.

Chet

 

[NATURE.M]

So there's air pressure on both sides of the dam that cancel?
I thought their was only air pressure on the side of the dam with no water.
And just for emphasis the net force of the air pressure in the vertical direction cancels as well right?

 

[Chestermiller]

So there's air pressure on both sides of the dam that cancel?

Yes.

I thought their was only air pressure on the side of the dam with no water.

No. There is a contribution of air pressure on both sides of the dam. On the water side, the absolute pressure at any depth d is p_a+ρgd (where p_a is atmospheric pressure). On the air side of the dam, the absolute pressure at any depth d is just p_a. The difference between these two pressures is just ρgd, which is the gage pressure.

And just for emphasis the net force of the air pressure in the vertical direction cancels as well right?

The air pressure in the vertical direction is supported separately on each side of the dam by the vertical columns at those locations.

 

[NATURE.M]

Thanks Chestermiller, it makes a lot of sense now.