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Homework Help: Height of a Heron's fountain

  1. Dec 26, 2017 #1
    1. The problem statement, all variables and given/known data
    I have this homework on how a Heron's fountain works. The exact assigment is as follows: "Construct a Heron's fountain and explain what (which variables) influences the height of the fountain."

    I've already constructed it and it works well, but I'm not so sure about the second part of the problem.

    I understand that the water is making a fountain due to the increased pressure in the "bottle b" (see picture) which pushes the water up through the tube d. I also understand that the air is getting to the bottle b via tube e, because the air itself in "bottle c" is being pushed by the water that is coming down from "bottle a" (tube f).
    bottles-md.jpg

    But in the end I totally confused myself: so what does the pressure in bottle b depend on? Is it the hydrostatic force? Am I even asking the right question? I got into a loop and I need to get out of it, help me please!
    2. Relevant equations
    p=F/A
    ph=mgh=Vρgh
    3. The attempt at a solution
    I suggest from these equations that the height of the heron's fountain directly depends on pressure in "bottle b" and that the relevant parameters are V (volume of the liquid) ρ (viscosity of the liquid) and Δh (difference in heights - the lenght of a tube connecting "bottle a" and "bottle c".

    But I'm not very sure of anything right now, I would be very glad if someone gets me onto the right track.
     
  2. jcsd
  3. Dec 26, 2017 #2

    haruspex

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    Assign unknowns to the various heights and write down expressions for the pressures. For this purpose, suppose you have your finger on the top of tube d so that nothing flows.
     
  4. Dec 26, 2017 #3
    I wrote this down and I'm not sure where are we heading.
    At the top of the "bottle a" ph= h1*ρ*g = 0 (because the height is 0)
    down the bottom of the "bottle a" ph = h2*ρ*g
    at the bottom of "bottle c" [p][/h] = h3*ρ*g
    ...
    I could go on (I have it on paper right next to me)

    I feel stupid. I don't get how this could help me. (or where did I go wrong?)
     
  5. Dec 26, 2017 #4

    haruspex

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    I cannot check that because you have not defined those heights. Can you post a diagram with heights or reference points marked?
    Remember that the tubes containing air make some pressures equal. Your aim is to find the pressure near the top of tube d.
     
  6. Dec 28, 2017 #5
    I'm now not at home, sorry for not replying so long. I have it written on a paper, because it's almost impossible to write on my very old IPhone. I attached the photos here.

    I think I must've made a mistake somewhere but I'm not sure where.
     

    Attached Files:

  7. Dec 28, 2017 #6

    haruspex

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    I can read the diagram, but I do not follow the working.
    The first hurdle is to realise which heights are relevant. There are five types of level in the set up:
    • Top of container
    • Top surface of water
    • Bottom of container
    • Top of tube
    • Bottom of tube
    Clearly not all are relevant. Start with the type that is most clearly relevant and see how far you can get with that type only. Hint: very few of the types are relevant.
     
  8. Dec 28, 2017 #7
    Oh well, I tought you advised me to define various random heights, my bad!

    I think that 3 heights are relevant in this problem.

    The height at the top of the tube (tube d) - let's call it h(t)
    The height at the top surface of the water - let it be h(0)
    The height at the bottom of the container (bottle c) - h(1)
     
  9. Dec 28, 2017 #8

    haruspex

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    Not random.
    If the top of the tube were made a little bit higher or lower, would that be likely to change the height the fountain reaches? How?
    Imagine adding another water-filled section below the base bottle, c, then removing the existing base so that it becomes part of bottle c. Would that change any pressures or flows?
     
  10. Dec 30, 2017 #9
    Ohh...The height of the fountain reach wouldn't really change if we made the tube lower or higher.
    The water would just stay inside the tube if we made the tube high enough, and if we made it lower it would seemingly make a higher fountain.

    hmmmm... If we did this I suppose the pressures won't change because the actual gradient would remain the same.

    Am I correct?
     
  11. Dec 30, 2017 #10

    haruspex

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    Correct.
    What about my other question, making the bottom of the base bottle lower compared with everything else? Or making the top or bottom of any tube higher or lower, as long as it stays in the same region and medium (air versus water)?
     
  12. Dec 30, 2017 #11
    I think that is a similar problem, If we made the bottom lower it wouldn't really change anything since the gradient would stay the same.
    Same applies to your second question.

    Uh, this must imply that the only height that matters is the height of the top surface of the water relative to the tube d?
     
  13. Dec 30, 2017 #12

    haruspex

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    Surfaces, plural.
    Haven't we established that the precise start and end of a tube is irrelevant?
     
  14. Dec 31, 2017 #13
    Yes of course, I was tired and my brain was working slowly yesterday.
    So the parameters that matter are the differences between the heights of water surfaces. Is that right?
     
  15. Dec 31, 2017 #14

    haruspex

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    Yes.
     
  16. Dec 31, 2017 #15
    And anything else that matters?
    Perhaps the viscosity or the width of the tube?
     
  17. Dec 31, 2017 #16

    haruspex

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    Yes, those will matter of course.
    Can you find the relationship between the surface heights and the height of the fountain?
     
  18. Jan 1, 2018 #17
    I was going over this and couldn't work it out.
    It seems to me that if we made the surface height at the bottom bottle lower, the pressure would actually increase, and if we made the surface height lower in the middle bottle, the pressure would drop.
    What to do now?
     
  19. Jan 2, 2018 #18

    haruspex

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    Call the surface heights in the three bottles A, B, C. What is the pressure at C?
     
  20. Jan 2, 2018 #19
    I have literally no idea.
    It's equal to the pressure force of air over the area of the water surface, but how big is that force... I really don't have clue how to calculate that
     
  21. Jan 2, 2018 #20

    haruspex

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    You know the pressure at A, and there is a continuous body of water from there, down through tube f, into bottle c, and back to surface C. In terms of those heights, what is the pressure at C?
     
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