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Homework Help: Finding Angular Frequency of Small Oscillations about an Equilibrium

  1. Feb 19, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider a system of one generalized coordinate theta, having the following Lagrangian equation of motion:

    r and b are constants
    m is mass

    (1/3)mb[tex]^{2}[/tex][tex]\ddot{\theta}[/tex] = r(r+b)[tex]\theta[/tex] + r[tex]^{2}[/tex][tex]\theta[/tex][tex]^{3}[/tex] + gr[tex]\theta[/tex]

    And this potential energy (if it matters):

    U = mg(r+b) - mgr[tex]\theta[/tex][tex]^{2}[/tex]


    There is an equilibrium point where theta is equal to zero.

    Find the angular frequency of small oscillations about [tex]\theta[/tex] = 0.


    2. Relevant equations



    3. The attempt at a solution

    Using the potential energy, can't I just say

    U = (1/2)k[tex]\theta[/tex][tex]^{2}[/tex]
    where
    k = 2mgr
    so that I can write
    [tex]\omega[/tex] = (k/m)^(1/2)
    [tex]\omega[/tex] = (2gr)^(1/2)
    and call that the angular frequency?

    But the problem asks me to do it the Lagrangian way.

    So
    [tex]\omega[/tex] = (2[tex]\pi[/tex])/[tex]\tau[/tex]

    How can I find tau?
     
  2. jcsd
  3. Feb 19, 2010 #2

    ideasrule

    User Avatar
    Homework Helper

    Clever, but it's not right. You can see this by comparing the units of U=(1/2)kx^2 with those of U = (1/2)k[tex]\theta[/tex][tex]^{2}[/tex]: both U's have the same unit, both x doesn't have the same unit as theta, so the two k's must have different units. That means the equation omega=(k/m)^1/2 is not correct.

    To start, do you know the characteristic differential equation for simple harmonic motion? Try to get the Lagrangian equation of motion into that form, remembering that theta^3 is much smaller than theta for small values of theta.
     
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