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Small Oscillations around equilibrium

  1. Mar 8, 2009 #1
    1. The problem statement, all variables and given/known data

    The problem is: A point pendulum is being accelerated at a constant acceleration of a. Basically what's required is to find the equations of motion, the equilibrium point, and to show that the frequency of small oscillations about the e.p. is: [tex]\omega=L^{-1/2} (a^2+g^2)^{-1/4}[/tex]

    2. Relevant equations

    The equation of motion I've arrived at is:

    [tex] \ddot{\theta} + \frac{a\cos\theta + g\sin\theta}{L} = 0[/tex]

    So the e.p is:

    [tex]\theta_0 = -\arctan(a/g)[/tex]

    3. The attempt at a solution

    If we do a Taylor expansion for [tex]\eta << 1[/tex] around [tex]\theta_0[/tex]:

    [tex]\cos(\theta_0+\eta) = \cos\theta_0 - \eta \sin\theta_0+\ldots[/tex]
    [tex]\sin(\theta_0+\eta) = \sin\theta_0 + \eta \cos\theta_0+\ldots[/tex]

    We end up with:

    [tex]\ddot{\eta} + \frac{\cos\theta_0}{L}\frac{g^2+a^2}{g} \eta = 0[/tex]

    Then what am I missing here?
     

    Attached Files:

  2. jcsd
  3. Mar 8, 2009 #2
    Correction:

    [tex]
    \omega=L^{-1/2} (a^2+g^2)^{1/4}
    [/tex]
     
  4. Mar 9, 2009 #3
    Anyone?
     
  5. Mar 9, 2009 #4
    Seriously, can no one say anything?
     
  6. Mar 10, 2009 #5

    Redbelly98

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    Just saw your question.

    You're almost there.

    Express θ0 in terms of a and g, using the arctan relation you came up with earlier. Things will simplify.
     
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