1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Period of pendulum motion

  1. Jan 28, 2017 #1
    1. The problem statement, all variables and given/known data

    A pendulum obeys the equation [tex] \ddot{\theta} = -\sin(\theta) [/tex] and has amplitude [tex] \theta_0 [/tex]. I have to show that the period is
    [tex] T = 4 \int_{0}^{\frac{\pi}{2}} \frac{d\phi}{\sqrt{1-\alpha \sin^2(\phi)}} [/tex] where [tex] \alpha = \sin^2(\frac{\theta_0}{2})[/tex]

    2. The attempt at a solution

    I derived an expression for time:

    [tex] \dot{\theta}\frac{d\dot{\theta}}{d\theta} = -\sin(\theta) [/tex]

    I said that the pendulum starts out at the height of its amplitude [tex] \theta = \theta_0 [/tex] where it also has zero velocity

    [tex] \int_{0}^{\dot{\theta}} \dot{\theta}d\dot{\theta} = \int_{\theta_0}^{\theta} -\sin(\theta)d\theta [/tex]

    [tex] \dot{\theta} = \frac{d\theta}{dt}= \pm \sqrt{2(\cos(\theta)-\cos(\theta_0))} [/tex]

    So for the period we can integrate from [tex] \theta_0 \text{ to } 0 [/tex], which is a quarter of the period, then multiply by 4 to get the whole period.

    [tex] T = 4 \int_{\theta_0}^{0} \frac{d\theta}{\sqrt{2(\cos(\theta)-\cos(\theta_0))}} [/tex]

    By the half-angle identity,

    [tex] T = 2 \int_{\theta_0}^{0} \frac{d\theta}{\sqrt{\sin^2(\theta_0 /2)-\sin^2(\theta /2)}} = 2 \int_{\theta_0}^{0} \frac{d\theta}{\sqrt{\alpha-\sin^2(\theta /2)}} [/tex]

    And this is where I'm stuck. It looks similar to the desired answer, but I can't think of any identity or substitution that would give the right integrand and limits.
     
    Last edited: Jan 28, 2017
  2. jcsd
  3. Jan 29, 2017 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Period of pendulum motion
  1. Period of a pendulum (Replies: 2)

  2. Period of a Pendulum (Replies: 3)

Loading...