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Simple Harmonic Motion, object floating in a liquid

  1. Sep 15, 2010 #1
    (apologies for the formatting, mixing LaTeX and regular text looks ugly, but I don't know how to do it otherwise)

    1. The problem statement, all variables and given/known data

    This is problem 3-7 from Thornton and Marion's Classical Dynamics.

    A body of uniform cross-sectional area A = 1 cm2 and of mass density [tex]\rho[/tex]=0.8 g/cm3 floats in a liquid of density [tex]\rho[/tex]0=1 g/cm3 and at equilibrium displaces a volume V = 0.8 cm3. Show that the peroid of small oscillations about the equilibrium position is given by:

    [tex]\tau[/tex] = 2[tex]\pi[/tex] [tex]\sqrt{V/gA}[/tex]

    where g is the gravitational field strength. Determine the value of [tex]\tau[/tex]

    2. Relevant equations

    [tex]\tau[/tex] = 2[tex]\pi[/tex][tex]\sqrt{k/m}[/tex] (equation 3.13 in the book)

    k[tex]\equiv[/tex] -(dF/dx)

    3. The attempt at a solution

    It appears to me that if I can show V/gA = m/k or k = mgA/V, then the equation in the problem would be equal to equation 3.13 quoted above. I tried coming up with an equation for the restoring force F in terms of mg A and V, and then take the x-derivative of k, but it doesn't equal mgA/V.

    I'll be using p = rho from here, because I don't want to mix LaTeX and regular text, and p has no other meaning.

    The restoring force I came up with was F = Vp0g - mg where m is the mass of the block.

    In terms of x, I get F = Axpg - mg where x is the depth of the submerged block.

    Taking the x-derivative of F, from the definition of k, I get Apg, which isn't equal to mgA/V

    Any suggestions?
     
  2. jcsd
  3. Sep 15, 2010 #2
    I'm an idiot, p= m/V by definition, and everything falls together.

    I can't believe I got stuck on such a trivial step. I'm embarrassed that I even posted this thread.
     
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