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Water Flowing through U-tubes (Velocity Given Cross-Sectional Areas)

  1. Jan 28, 2013 #1
    1. The problem statement, all variables and given/known data
    http://www.aapt.org/physicsteam/2010/upload/2010_FmaSolutions.pdf
    See question 23

    A' = 1/2A

    2. Relevant equations
    Av = A'v'

    3. The attempt at a solution
    I thought this was just a simple Av = A'v' problem
    which would lead to v' = 2v. But there is apparently more to it, as the answer is √2v.
    It could have to do with the momentum because F_net = 0 so it is conserved.
    In that case,
    mv = mv but I'm confused about how to use that to solve the problem.
     
  2. jcsd
  3. Jan 28, 2013 #2
    Any thoughts?
     
  4. Jan 28, 2013 #3
    Use conservation of momentum. In terms of the fluid density, the area, and the velocity v, what is the rate of change of momentum of the fluid passing through the U tube on the left? Same question for the U tube on the right.
     
  5. Jan 28, 2013 #4
    momentum = mv
    dp/dt = m/t(v)
    dp/dt = (ρAv)v
    dp/dt = ρAv^2 - This is the left

    Right:
    dp/dt = pA'v'^2
    A' = .5A
    dp/dt = ρAv'^2/2

    The two rate changes are equal ( net forces are zero).
    v^2 = v'^2/2
    2v^2 = v'^2
    v' = √2v

    This is the right answer. Thanks.

    Just a question; from where did you get the motivation to use rate change of momentum?
     
  6. Jan 28, 2013 #5
    The problem said that the assembly of u tubes is in equilibrium. That means that the forces exerted by the fluids on the u tubes must be equal. The force exerted by each fluid on each u tube is equal to the force exerted by the u tube on the fluid. The latter is equal to the rate of change of momentum of the fluid.
     
  7. Jan 28, 2013 #6
    Ok I see where you were going.
    Essentially, the conceptual link is:
    Equlibrium → ƩF = 0 → FL = FR
    F = dp/dt
    dp/dt = mv/dt
    Given, that v is constant, m is changing
    dp/dt = dm/dt*v
    dm/dt = ρAv
    dp/dt = ρAv^2
    And that's it right.

    I'm just making sure so I don't end up memorizing scenarios.
     
  8. Jan 28, 2013 #7
    Yes. Well done. I hold in very high regard a student like you who focuses on fundamentals. You can look forward to a bright future.

    Chet
     
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