Relation of shape of bottle to water pressure

  • #1
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Main Question or Discussion Point

So, I know that according to everything online water pressure is independent from everything except gravity, height, and density of fluid. The situation I'm talking about is slight different.

Imagine a standard water bottle that angles inwards near the top. Now imagine a point on the side of the water bottle, near the on the slanted part. I know that what matters is the height of a "column" of water above your point, but you can't make a column of water here -- once the bottle starts slanting inwards, going straight up would mean you hit the slanted side of the bottle.

I mean, think about the way the equation cancels out.
P = F/A
P = MG/A
P = Density x Volume x Gravity / Area

In the column-of-water scenario, the volume of the column (m^3) cancels out with the area (m^2), leaving behind just M, aka height. But in my scenario, the volume and the area wouldn't cancel out to produce height because the volume isn't just a function of length, width, and height: it's irregular.

I can add some diagrams later if this is a confusing scenario to envision.

All I really need to know is, am I correct in assuming that here, the shape of the bottle is not irrelevant?
 

Answers and Replies

  • #2
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Not entirely sure what you mean, but if you have a known volume of water in a container and weigh it, you will always get the same answer of 'X' kg, no matter what the shape of the container is.
 
  • #3
Spinnor
Gold Member
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Imagine a standard water bottle that angles inwards near the top. Now imagine a point on the side of the water bottle, near the on the slanted part. I know that what matters is the height of a "column" of water above your point, but you can't make a column of water here -- once the bottle starts slanting inwards, going straight up would mean you hit the slanted side of the bottle.
You could make a column of water above that point, just imagine drilling a hole and glueing and sealing a vertical plastic straw to your water bottle and fill it with water to the proper height.
 
  • #4
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So, is that what happens, then? The bottle could be absolutely any shape and all that matters is the height to the top? I have a lab on this, and although I didn't take any data on the slanted part of the bottle I still need to say definitively if it matters whether data is taken on the slanted part or not. If you were graphing pressure against water height on an irregularly-slanted bottle, would the relationship still be linear across all parts of the bottle?
 
  • #5
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The slanted side of the bottle has water pressure pressing against it, and it presses back on the water. The pressure force applied by the slanted side to the water is oriented perpendicular to the slanted side, so it has a component in the downward direction. This applies a downward force on the fluid below.

You also need to be aware of Pascal's law. It is based on observation, and says that, at a given spatial location in the water, the pressure acts equally in all directions. So, at a given depth the pressure is acting horizontally also. If you have a continuous vertical column of water away from the sides, you can get the pressure at the bottom of that column. That pressure is also acting horizontally, and the pressure does not change in the horizontal direction (from a horizontal force balance). So, the pressure at the bottom of the column is also transmitted horizontally to the slanted walls of the bottle. So the only thing that matters is the depth below the upper surface.
 
  • #6
Khashishi
Science Advisor
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There's a variation of the hydrostatic paradox, where the pressure in a barrel is increased by adding a small but long tube to the top and filling the tube with water, until the barrel bursts. It might not be intuitive how a narrow tube of water can contribute so much to the pressure in a much larger barrel. (Of course, if the tube is too narrow, then there will be capillary action which complicates things.)
 
  • #7
There's a variation of the hydrostatic paradox, where the pressure in a barrel is increased by adding a small but long tube to the top and filling the tube with water, until the barrel bursts. It might not be intuitive how a narrow tube of water can contribute so much to the pressure in a much larger barrel. (Of course, if the tube is too narrow, then there will be capillary action which complicates things.)
I didn't understand single bit of it. Pls help what you are trying to say; a schematic diagram would do wonder for me.
 
  • #8
Nidum
Science Advisor
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straw paradox.jpg
 
  • #9
Nidum
Science Advisor
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There is no paradox .

Hint : Think about energy as well as pressure .
 
  • #10
That's why I love physics; very non-intuitive yet fascinating. Thanks for the picture
 
  • #11
CWatters
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So, is that what happens, then? The bottle could be absolutely any shape and all that matters is the height to the top? I have a lab on this, and although I didn't take any data on the slanted part of the bottle I still need to say definitively if it matters whether data is taken on the slanted part or not. If you were graphing pressure against water height on an irregularly-slanted bottle, would the relationship still be linear across all parts of the bottle?
See..
Pressure.png
Pressure2.png
 

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