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Mathematical pendulum - Arbitrary-amplitude period

  1. Oct 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi! I have a problem with mathematical pendulum. I had to prove that the right expression for the period of the mathematical pendulum is:
    [itex]T=\sqrt{\frac{8l}{g}}\int_0^{\theta_0}\frac{d\theta}{\sqrt{\cos \theta- \cos \theta_0}}[/itex]
    I did that via energy, but now I have to transform that integral with:
    [itex]\cos\theta=1-2\sin^2(\theta/2)[/itex] and with this substitution
    [itex]\sin x =\sin(\theta/2)/\sin(\theta_0/2)[/itex] so that I could expand that integral into Taylor series.

    The attempt at a solution

    I've tried with substitution and I get:
    [itex]2\sqrt{\frac{l}{g}}\int_0^{\theta}\frac{d\theta}{\sqrt{\sin^2(\theta_0/2)-\sin^2(\theta/2)}}[/itex]
    Now I have seen what the expansion is on wikipedia, but there is no solution to how to get to there. If I try to use the sine substitution, I get stuck at changing the integral variables, and the integral becomes a mess. I saw that the complete elliptic integral of 1st order appears, but I don't know what to do with that.

    Can anyone help with this clue? How to get that Taylors expansion?

    Thank you!
     
  2. jcsd
  3. Oct 17, 2009 #2
    Anyone?
     
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