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Plane pendulum in terms of the x axis

  1. Feb 11, 2008 #1
    1. The problem statement, all variables and given/known data

    consider the free motion of a plane pendulum whose amplitude is not small. Show that the horizontal component of the motion may be represented by the approximate expression ( components through the 3rd order are included)

    [tex]\ddot{x}[/tex]+w[tex]^{2}_{0}[/tex] (1+x[tex]^{2}_{0}[/tex]/L[tex]^{2}[/tex])x - Ex[tex]^{3}[/tex]=0

    where w[tex]^{2}_{0}[/tex]=g/L , E=3g/(2L[tex]^{3}[/tex]) and L is the length of the suspension.
    2. Relevant equations

    we have a second order non-linear equation: [tex]\ddot{\Theta}[/tex]+w[tex]^{2}_{0}[/tex]sin[tex]^{2}[/tex][tex]\Theta[/tex]=0

    3. The attempt at a solution
    I'm pretty stuck on this. It seems that I need to get a function for x in terms of [tex]\Theta[/tex], but I'm not quite sure where to start. I see here that I'm not going to be able to use the small angle approximation sin[tex]\Theta[/tex]=[tex]\Theta[/tex].

    Thanks for any help.
  2. jcsd
  3. Feb 12, 2008 #2

    Shooting Star

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    x = L(sin theta) => theta = arcsin x/L.

    Try to find theta'' in terms of x'', and at some point sub in the value of theta'' from the last eqn, so as to get w0^2.
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