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Cylinder oscillating in water.

  1. Jul 14, 2012 #1
    ω1. The problem statement, all variables and given/known data

    A cylinder of diameter d floats with l of its length submerged. The total height is L. Assume no damping. At time t = 0 the cylinder is pushed down a distance B and released.

    What is the frequency of oscillation?
    2. Relevant equations

    [itex] f = ω/2\pi [/itex]
    [itex] Ma = F_{(bouyancy)} [/itex]
    Writing this in our differential form, making proper substitutions, and noting that bouyancy is affected by the distance that our cylinder is submerged we come to.

    [itex] dx^2 (M_{(cylinder)}) + x (\rho_{(water)} g Area_{(cylinder face)})= 0 [/itex]

    we know that [itex]M_{(cylinder)} = V_{(cylinder)}\rho_{(cylinder)} [/itex]

    hence we should have
    [itex] ω^2 = (\rho_{(water)} g Area_{(cylinder face)}) / V_{cylinder}\rho_{(cylinder)} = g\rho_{(water)} / l \rho_{(cylinder)}[/itex]

    however the solution in my problem set has ω^2 = g/l. Can anyone shed some light on why the densities may cancel??
  2. jcsd
  3. Jul 14, 2012 #2


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    hi ozone! :smile:
    (that's the same as g/L)

    i haven't followed what you've done, but i'd guess you've used the wrong expression for the mass of the cylinder :confused:

    just use a (vertical) force equation for the cylinder (at depth l + x) :wink:
  4. Jul 15, 2012 #3
    the mass of the cylinder is the density of the cylinder times the area.. but the force from the water depends only on the density of water.. that is why i don't understand how the densities are cancelled out.
  5. Jul 15, 2012 #4


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    In the denominator of your final expression for ω2, is that a small l or a capital L?

    You can find an expression for the ratio of the two densities in terms of the ratio of l and L by considering the condition for equilibrium when length l of the cylinder is submerged.
  6. Jul 16, 2012 #5
  7. Jul 16, 2012 #6


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    But in the expression that you derived: ω2 = gρw/lρc, you should have a capital L rather than a lower case l in the denominator. Then you should be able to show that this expression reduces to the correct answer.
  8. Jul 18, 2012 #7
    We never learned about fluid dynamics in my mechanics class, but I am guessing that the water displaced in equilibrium is equal to the mass of the cylinder?
  9. Jul 18, 2012 #8


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    Yes. Archimedes Principle: Buoyant force = weight of fluid displaced

    In equilibrium, Buoyant force equals the weight of the floating object.
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