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Height of a Heron's fountain

  • #51
haruspex
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Sorry I am really not sure as this is what I am confused about.
How does the pressure at d relate to the difference in heights since the pressure is what relates to the fountain's height right?
Consider a tank of water. How does the pressure at height y1 from the bottom relate to the pressure at height y2 from the bottom?
 
  • #52
These 2 heights of water are connected in a closed space (except for the water surface at the top). Such that the pressure of the lower height will be larger than that of the higher water height.
 
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  • #53
haruspex
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These 2 heights of water are connected in a closed space (except for the water surface at the top). Such that the pressure of the lower height will be larger than that of the higher water height.
Yes, but by how much?
 
  • #54
Pressure difference = pgh2(higher water height) - pgh1(lower water height)

Edit: oh so change in pressure from upper height to lower height = change in height *pg
 
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  • #55
haruspex
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Pressure difference = pgh2(higher water height) - pgh1(lower water height)

Edit: oh so change in pressure from upper height to lower height = change in height *pg
Right.
There are four locations to consider: the surface of the water in each container, and the base of the fountain. (By "base of fountain" I mean the point inside the top tube that is the same height as the surface of the top reservoir.)
If you can move from one such location to another only moving through air then you know the pressures at those locations are the same; if you can move from one to another only moving through water then you can relate the pressures by the height differences.
And don't forget, you have a finger on top of the top tube.

What equations can you write relating these four pressures?
 
  • #56
Pressure difference between B and A = pg*height difference between water surface B and A
Pressure difference between B and C - same
Pressure difference between C and A = pg*height difference between C and A
I am getting a little confused - is this correct?
 
  • #57
haruspex
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Pressure difference between B and A = pg*height difference between water surface B and A
Is there a path between those two surfaces that is entirely in water?
 
  • #58
Yes there is.. that's why there is a pressure difference.
 
  • #59
haruspex
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Yes there is.. that's why there is a pressure difference.
I am not seeing it.
Tube f (water) connects A and C.
Tube e (air) connects B and C.
Tube d (water) connects fountain base with B.
 
  • #60
Oh the pressure at A is atmospheric pressure so the difference in pressure between B and A is PB=Patm
 
  • #61
haruspex
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Oh the pressure at A is atmospheric pressure so the difference in pressure between B and A is PB=Patm
Atmospheric pressure is a background common to all points. We only need concern ourselves with gauge pressure, i.e. the excess above atmospheric.
But you still have not answered my question about A and B. What path is there between those two surfaces which either only passes through water or only passes through air?
 
  • #62
Well the water in B passes through a tube of water and comes out into air. So neither of those things you mentioned above happen because the water passes through water (in B) and then air (coming out of the tube d).
 
  • #63
haruspex
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Well the water in B passes through a tube of water and comes out into air. So neither of those things you mentioned above happen because the water passes through water (in B) and then air (coming out of the tube d).
As I reminded you, we are considering a static arrangement. Your finger is stopping the fountain, so nothing is actually passing anywhere.
My questions are in relation to paths that exist within this static arrangement. Could you thread a path from the one surface to the other without changing medium?
 
  • #64
Yes, the water is flowing in the tube to the 2nd surface so the medium does not change since the path is the same medium on both surfaces.
 
  • #65
haruspex
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Yes, the water is flowing in the tube to the 2nd surface so the medium does not change since the path is the same medium on both surfaces.
Which tube? They are labelled.
 
  • #66
tube d
 
  • #67
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tube d
As I wrote in post #59, tube d connects reservoir B to the base of the fountain. It does not connect with reservoir A. There is no water path from the surface of reservoir B to the surface of reservoir A.
 
  • #68
Oh, tube e - tube of air connecting to 2 air mediums.
 
  • #69
haruspex
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Oh, tube e - tube of air connecting to 2 air mediums.
That connects the airspaces at B and C, not A and B..
 
  • #70
ok. tube f is a tube full of water which connects a water medium to another water medium.
 
  • #71
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ok. tube f is a tube full of water which connects a water medium to another water medium.
Yes, A and C, not A and B.
Have you lost track of where we are at?
You need to make a list of the pairs of surfaces which are connected by a water only path, and a list of pairs connected by an air only path.
In post #56 you claimed there was such a path connecting A and B. Are you ready to agree there is not one?
 
  • #72
oh sorry tube d definitely isn't one.
 
  • #73
haruspex
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oh sorry tube d definitely isn't one.
Ok.
In post #56, you correctly had A and C connected by water.
For B and C you wrote "same", but it is not clear whether you meant they are the same pressure or that the connection is the same type as AB. Please clarify.
Finally, you need to say what the base of the fountain (the point inside tube d that's at the same height as surface A) connects to, and through what medium.
 
  • #74
B and C are of the same pressure. The base of the fountain connects from b into the atmosphere - through both water and air.
 
  • #75
haruspex
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B and C are of the same pressure.
Right.
The base of the fountain connects from b into the atmosphere - through both water and air.
No, remember that we are supposing you have a finger on top of that tube, so the fountain base is isolated from the atmosphere.
Now, what equations can you write relating the four pressures?
 

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