PENDULUM PERIOD CALCULATIONSPH 421 Fall 2009restart:with(plots):This worksheet is intended for your benefit, but it is not a beautifully crafted piece! Its purpose is to save you time. Play with it, so it suits your needs.The period of a pendulum oscillating to a maxmum angle NiMmJSZ0aGV0YUc2IyUkbWF4Rw== can be written as:NiMvJSJURyoqLSUlc3FydEc2IyIiIyIiIiUjUGlHISIiJkYkNiMiIiFGKi0lJEludEc2JComRipGKi1GJzYjLCYtJSRjb3NHNiMlJnRoZXRhR0YqLUY4NiMmRjo2IyUkbWF4R0YsRiwvRjo7Ri9GPUYqwhere NiMmJSJURzYjIiIh is the period for small angle. Maple can calculate this integral exactly as shown below, but it can be quite slow if used in a plot function for example. If we recognize it as an elliptical integral (which you are unlikely to have encountered, and that doesn't matter), then we can use the more efficient call to the complete ellpitic integral of the first kind K(k) defined by NiMvLUkqRWxsaXB0aWNLRzYkSSpwcm90ZWN0ZWRHRidJKF9zeXNsaWJHNiI2I0kia0dGKS1JJEludEdGJjYkKigiIiJGMC1JJXNxcnRHRiY2IywmRjBGMCokSSJ0R0YpIiIjISIiRjgtRjI2IywmRjBGMComRitGN0Y2RjdGOEY4L0Y2OyIiIUYw.where NiMvJSJrRy0lJHNpbkc2IyomJiUmdGhldGFHNiMlJG1heEciIiIiIiMhIiI= in our case. The period can then be written asNiMvJSJURyoqIiIjIiIiJkYkNiMiIiFGJyUjUGlHISIiLSUqRWxsaXB0aWNLRzYjLSUkc2luRzYjKiYmJSZ0aGV0YUc2IyUkbWF4R0YnRiZGLEYn
The approximate period is NiMvSSJURzYiKiYmRiQ2IyIiISIiIiwmRipGKiomJkkmdGhldGFHRiU2I0kkbWF4R0kqcHJvdGVjdGVkR0YxIiIjIiM7ISIiRipGKg==.This is the evaluation of the integral that detrmines the period (relative to the small angle period). A is imeasured in radians. A:=120*Pi/180:evalf(sqrt(2)/Pi*Int(1/sqrt(cos(x)-cos(A)),x=0..A));This is the elliptic integral form; you can see it gives the same result.
evalf(2/Pi*EllipticK(sin(A/2)));plot([evalf(2/Pi*EllipticK(sin(B/2))),1+B^2/16,1], B=0*Pi/180..140*Pi/180,thickness=2,color=[red,blue,green],font=[helvetica,plain,11]);