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Calculate time for water to leak completely

  1. Jan 24, 2016 #1
    1. The problem statement, all variables and given/known data
    Given the dimensions of a cylinder container with a hole, the dimensions of the hole, and the volume of water, calculate the time for water inside the container to completely leak out.

    2. Relevant equations
    Bernoulli equation: P + 1/2pv^2 + pgh = constant https://en.wikipedia.org/wiki/Bernoulli's_principle
    Continuity equation: A1V1 = A2V2 http://theory.uwinnipeg.ca/mod_tech/node65.html

    3. The attempt at a solution
    - Assume the water is ideal liquid
    - Assume container has an open top, so pressure cancels

    So far I am able to find the relationship between height of the water in the container and the velocity at the top of the water. Since both height, volume, and velocity changes at each instant, I think that integration needs to be used. Can someone enlighten me on how I should incorporate calculus in the equation?

    See attachment for attempt

    Thank you!
     

    Attached Files:

  2. jcsd
  3. Jan 24, 2016 #2

    SteamKing

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    You need to figure out which variables change with time.
     
  4. Jan 24, 2016 #3
    Both height and velocity change with time. Can I replace velocity with dh/dt?
     
  5. Jan 24, 2016 #4

    SteamKing

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    You can't replace velocity directly and completely with dh/dt, but you should be able to write how velocity changes with time, given dh/dt, since you know how velocity is affected by h.
     
  6. Jan 25, 2016 #5
    Can you elaborate a little bit more? Thank you for your time :)
     
  7. Jan 25, 2016 #6

    SteamKing

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    If you analyze this problem using Toricelli's Law (which can be derived from Bernoulli's equation using certain simplifications), the velocity of the stream coming out the hole is
    $$v = \sqrt{2g ⋅ h}$$
    https://en.wikipedia.org/wiki/Torricelli's_law

    Since the volume of fluid coming out is also proportional to the velocity, you should be able to use the formula for velocity to determine how velocity changes with time. What happens if you differentiate the velocity equation w.r.t. time?

    I've given you some pretty substantial hints. It's time for you to start showing what you are doing with them to solve this problem.
     
  8. Jan 31, 2016 #7
    Hey SteamKing,
    After following your suggestions, I was able to find the relationship between velocity and time:

    $$h = \frac{T-v \pi r^2t}{\pi R^2}$$
    where T is the volume of the water, r is the radius of the hole, R is the radius of the container, and t is time

    $$v = \sqrt{2g\frac{T-v \pi r^2t}{\pi R^2}}$$

    How do I integrate if velocity is on both sides?

    Really appreciate your help :c
     
  9. Jan 31, 2016 #8

    haruspex

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    I assume T is the initial volume. To get that equation you have to assume v is constant, but it is not.
     
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