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Buoyancy Force on a Bubble

  1. Nov 19, 2008 #1
    First off, thanks in advance to anyone who can help at all. I've attempted to do a few things, and none of them seem to work...let me know what you think!

    THANK YOU!!!

    1. The problem statement, all variables and given/known data
    A sperical bubble of radius r has a volume v and a pressure p when just below the surface of a liquid of density d (top of bubble is tangent to the water level).

    Let F1 be the magnitude of the buoyant force on the bubble when it is at this location.. Assume P is the atmospheric pressure and that the temperatureof the liquid and bubble do not change with depth.

    Let the y-axis be vertical. Let the x-axis be the water level. The bubble (though it doesn't matter) is on the positive side of the x-axis).

    a) Show that the magnitude of the buoyant force (F2) on the bubble as a function of its vertical position y in the liquid is: F2 = F1 [ p / (p - d*g*y) ]

    b) If the liquid is water, show that the buoyant force is half the surface value at a depth of 10.3m.

    2. Relevant equations
    Buoyant Force = (density of liquid)(volume displaced)(gravity) = d*v*g

    3. The attempt at a solution

    a) We know that as the bubble descends in depth, it is moving in the negative-y direction. We also know that:

    Buoyant Force = d*v*g = d*(cross-sectional area)(depth)*g = d*A*(-y)*g

    What I don't understand, is why you would need to go any further, given the information that we have. However, the question has been asked so I must solve it, right : P

    If it helps at all, this question is in a chapter about the Ideal Gas Law (PV = nRT)

    My next best attempt was the following:

    we know that F2 = d*A*(-y)*g and that (by Ideal Gas Law) v = nRT/p
    F2 = d*A*(-y)*g = d*v*g
    F2 = d*A*(-y)*g = d*(nRT/p)*g (this gets P in the denominator)
    F2 = A*(-y) = nRT/p
    F2 = (nRT/p)/(A*(-y)
    F2 = (nRT)/[A*(-y)*(p)]
    F2 = (nRT)/[(pi*r^2)(-y)(p)]

    And thats as far as I get before I realize I'm really not getting anywhere!

    There must be an equation or formula or form that I am missing somewhere, because this is really not even close to the form we are asked to show...and I am positive if I had this "missing equation" I could finish this...


  2. jcsd
  3. Nov 20, 2008 #2

    Doc Al

    User Avatar

    Staff: Mentor

    How does pressure change with depth? How does the bubble's volume change with depth? Hint: Use the ideal gas law to get a simple ratio.
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