al mech

There is a rigorous mathematical analysis on the period of a simple pendulum with an arbitrary initial angle of displacement [theta]. The period T is a function of the angle [theta], which is given by

T = 2.pi.square-root[L/g].F(theta)
where L is the length of the pendulum, g is the acceleration due to gravity, pi = 3.14159...., and F(theta) is a function in [theta]. You may refer to the following web-page for the exact equation (see the equation just before the paragraph on "Compound Pendulum").

http://en.wikipedia.org/wiki/Pendulum_(mathematics)

The period T is independent of the angle [theta] only applies when the angle is small, typically below 10 degrees. In such situation, the oscillation of the pendulum is "simple harmonic". For large angle of displacement, the oscillation is periodic, but not simple harmonic.

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