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Homework Help: Height of a Heron's fountain

  1. Jan 4, 2018 #26

    haruspex

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    What will happen if the pressures at B and C differ?
     
  2. Jan 7, 2018 #27
    Oh I'm very sorry! I thought replied, and was wondering why you aren't writing back.
    The pressures at B and C can't differ because that would mean the water would be pushed into the tube (with lower pressure).
    Hence the pressure at B and C must be the same, that is P=h3ρg+Pa.
    That is all, or no?
     
  3. Jan 7, 2018 #28

    haruspex

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    They are the same, but not for that reason. What tube connects B and C? What is in that tube?
     
  4. Jan 7, 2018 #29
    Well the bottles are connected with a tube full of air, so I guess the pascal law comes in, saying that pressure applied on the air must be same in every point in the given connected medium.
     
  5. Jan 7, 2018 #30

    haruspex

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    There will be a very small pressure difference because of the weight of air in the tube, but we can ignore it.
    If the tube were full of water then the pressures would be different (and the fountain would not work).

    Edit: what I previously wrote in this next sentence was not what I meant to ask. See next post.
     
  6. Jan 7, 2018 #31

    haruspex

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    Next, what is the pressure inside tube d at A? Look at what connects this to B.
     
  7. Jan 7, 2018 #32
    The tube is full of water, so I think the pressure in tube d at A would be
    P = h1ρg+h3ρg ?
    Of course the pressure changes there, but the initial pressure from the air in bottle B remains the same in the whole liquid no?
     
  8. Jan 7, 2018 #33

    haruspex

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    Is the pressure inside the tube d at A more or less than the pressure at B?
    Not sure what you mean.
    Remember we started by supposing you have your finger on top of the tube d to inhibit the fountain so that everything is static. In that arrangement, there are two rules you can apply:
    • If two points are connected by a path passing only through air then the pressures are near enough equal;
    • If two points are connected by a path passing only through water then the difference in pressures will be ρwgh, where h is the height difference between the two points.
     
  9. Jan 8, 2018 #34
    Wow, I can't get my head around it now... I will go get some sleep and respond tommorow.

    But let me try at least, since the height A is smaller than height B I guess the pressure in the tube d at A must be smaller than at B.
    Then the pressure at A in the tube would be:

    P = h3ρg-(h2-h1)ρg ?
     
  10. Jan 8, 2018 #35

    haruspex

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    To be clear, the way you are using h1 ... h3 they are depths from the top, not heights from the bottom. So you mean that the depth at A is less than the depth at B.
    Yes.
    Yes.
    So what is the pressure difference between inside the tube and outside the tube at A?
    How does that pressure difference relate to the height of the fountain?
     
  11. Jan 8, 2018 #36
    Thanks for that height-depth note, I was getting a bit tangled in this.

    As for the pressure difference, the bigger the difference between in and outside the tube, the bigger the fountain would be.
    And it can be expressed ΔP = h3ρg-(h2-h1)ρg - h1ρg = (h3-h2-2h1)ρg??
    This looks so wrong... so I'll bet there must be something I missed in this calculation?
     
  12. Jan 8, 2018 #37

    haruspex

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    Try that step again.
     
  13. Jan 8, 2018 #38
    Oh, stupid mistake,
    ΔP=(h3-h2)ρg

    Is that it?
     
  14. Jan 8, 2018 #39

    haruspex

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    Yes.
     
  15. Jan 8, 2018 #40
    Well, it's been a pleasure and honor sir!
    I think I have all I need. It took me ages , but now I understand it very well.
    Thanks a lot :)
     
  16. Feb 10, 2018 #41

    haruspex

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    @Physicist1011 (who asked me about this thread): do you understand it down to finding the pressure difference at post #38?
     
  17. Feb 10, 2018 #42
    No I don't. I don't understand how the differences in heights of the water affect the water height and I also don't understand how you got that equation.

    Edit: or does the heights of the water affect the water fountain not only the differences between the heights of the water in the containers?
     
    Last edited: Feb 10, 2018
  18. Feb 10, 2018 #43

    haruspex

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    First thing is to understand how the pressures in the different airspaces are related.
    If two airspaces are connected by a tube of air, what can you say about the two pressures?
     
  19. Feb 10, 2018 #44
    The pressures are the same? but why?
     
  20. Feb 10, 2018 #45

    haruspex

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    If they were different, what would the air do?
     
  21. Feb 10, 2018 #46
    Move towards lower air pressure.
     
  22. Feb 10, 2018 #47

    haruspex

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    Right.
    Early in this thread I advised Dami to consider the set-up with a finger on top of the top tube so that no fountain occurs. This is so that everything is static, which makes the analysis easier.
    So we can assume no air is flowing, so the pressures are the same. (When the fountain is flowing there will be a small difference in the air pressures.)

    Next, consider a tube that is filled with water, connecting two reservoirs. For simplicity, we can just think about the pressures at the surfaces of the reservoirs.
    What can you say about those two pressures? How are they related?
     
  23. Feb 10, 2018 #48
    I am not sure.
     
  24. Feb 10, 2018 #49

    haruspex

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    If two reservoirs are connected by a filled pipe they are effectively a single reservoir. What relates the pressures at two points in a reservoir?
     
  25. Feb 10, 2018 #50
    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?
     
    Last edited: Feb 10, 2018
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