Thermodynamics Hydrostatic Pressure as Function of Height Question

[f8pc]

How is hydrostatic pressure not a function of the shape of the container? My professor says that the pressure P2 is the same for the figures on the right and left. How does the area of the two sections not matter at all? Clearly, if a plate is inserted of negligible mass, thickness, and buoyancy, the pressure at P2 is not equal (see work under each figure). If the forces without the plate aren't distributed across the area of the bottom pool, then the pressure on the far left or far right of of the bottom pool of the left figure would only be pg*h2 and not pg*(h1+h2).


[PLAIN]http://img29.imageshack.us/img29/6066/hydrostaticpressure.jpg

 

[Doc Al]

Clearly, if a plate is inserted of negligible mass, thickness, and buoyancy, the pressure at P2 is not equal (see work under each figure).

I don't understand your logic with the movable plate (is it surrounded by fluid?). But if you want to understand how the pressure at the bottom of each container can be the same, be sure to consider that the walls of the container also exert a pressure--in particular, the 'ceiling' of the container on the left exerts a downward pressure on the fluid.

 

[f8pc]

Hmm, but the pressure the ceiling puts on the fluid is not that great because it is only a result of the long column of vertical water. The area of the ceiling is quite large so the distribution of the forces from the vertical column would make the force is exerts rather small?Yes, the plates are free to move and are completely surrounded by water. The ceiling in the left picture wouldn't be pushing on the plate as the water above it would push it down.

The ultimate goal of this question is to determine why the pressure at any point on the bottom of the left figure is the same as the pressure at the bottom of the right figure (without the plates. The plates only explain my thinking).

 

[Doc Al]

Hmm, but the pressure the ceiling puts on the fluid is not that great because it is only a result of the long column of vertical water. The area of the ceiling is quite large so the distribution of the forces from the vertical column would make the force is exerts rather small?

No, the pressure turns out to be equal at all points at the same level--the ceiling pushes down with a pressure that is equal to the pressure from the long column of water.

Yes, the plates are free to move and are completely surrounded by water. The ceiling in the left picture wouldn't be pushing on the plate as the water above it would push it down.

But the ceiling would be pushing the water down right above the plate. So you're right back where you started.

 

[f8pc]

What's the theory behind the ceiling pushing down with pressure equal to the long column of water? Where can I read about why that occurs and why it is equal? Thanks for the quick responses by the way. This was driving me nuts.

It's clear the plates change the system then if the ceiling is pushing down so I am happy to throw that out. I was assuming the only pressure would be from the long column above and the water below.

 

[Doc Al]

What's the theory behind the ceiling pushing down with pressure equal to the long column of water? Where can I read about why that occurs and why it is equal? Thanks for the quick responses by the way. This was driving me nuts.

It's essentially Pascal's principle--that the pressure in a fluid (at least in the static case) must be the same in all directions. A fluid cannot support shear stress (at least an ideal fluid cannot).

It's clear the plates change the system then if the ceiling is pushing down so I am happy to throw that out. I was assuming the only pressure would be from the long column above and the water below.

Think about this: If the ceiling didn't press down on the fluid underneath it, then you could remove it with impunity. What do you think would happen then? (The vertical tube just connecting with an open pool of water.)

 

[f8pc]

Okay, that helps a lot! I wish I could thank you for helping me in a way other than just saying thanks so much! I really appreciate the time you took to answer each of my questions and doing so in a clear and polite way. You're truly an asset to this forum.