Does Hydrostatic Pressure Follow the Same Rules for All Shapes of Containers?

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Hydrostatic pressure is defined as pgh, where p is water density, g is gravitational acceleration, and h is the depth below the surface. This principle applies universally, regardless of container shape, but the method of calculating weight can differ based on geometry. In a cylindrical beaker, pressure multiplied by area equals the weight of the water, but this relationship changes in non-vertical containers like an hourglass. The pressure at the bottom of the hourglass does not account for the vertical components of pressure acting on its sides, leading to discrepancies in weight calculations. Understanding these forces clarifies the logic behind hydrostatic pressure in various container shapes.
brett351
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My understanding is that the pressure below the surface of water is pgh.

p - density of water, g - accel of gravity, h - dist below surface

And that this relationship holds regardless of the shape of the container of water. (also, I'm neglecting the atmospheric pressure at the surface)

If I have a cylindrical beaker of water of height H and area A and put it on a scale, I can calc the reading of the scale two ways...

1) pressure at bottom times area is pgHA.

2) density times volume times g is also pgHA.

Both ways give the same reading. Now if the shape of the beaker is an hourglass which has an area at top and bottom of A (same as top and bottom of cylindrical beaker), the two ways don't yield the same result.

Method 1) yields the same result for both shapes but method 2) yields a smaller result for the hourglass. What wrong with my logic? Thanks, Brett.
 
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I don't understand how you are measuring things. The two pressures are equal, but your method for measuring them is wrong.
 
brett351 said:
My understanding is that the pressure below the surface of water is pgh.

p - density of water, g - accel of gravity, h - dist below surface

And that this relationship holds regardless of the shape of the container of water. (also, I'm neglecting the atmospheric pressure at the surface)
OK.

If I have a cylindrical beaker of water of height H and area A and put it on a scale, I can calc the reading of the scale two ways...

1) pressure at bottom times area is pgHA.

2) density times volume times g is also pgHA.
The scale just measures the weight of the beaker of water. Method 1 gives the water pressure*area on the inside bottom of the beaker--this only equals the weight in the special case where the walls are vertical.

Both ways give the same reading. Now if the shape of the beaker is an hourglass which has an area at top and bottom of A (same as top and bottom of cylindrical beaker), the two ways don't yield the same result.
Because the pressure*area on the bottom of the hourglass beaker does not equal the net force on the beaker due to the water. The water also pushes up on other areas of the beaker surface. If you added up the net force that the water exerts on the entire beaker, that would equal the weight of the water.
 
Thanks Doc Al. Now I see that I forgot to take into account the vertical components of the pressures on the side of the hourglass, Brett.
 

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