Arbitrary-amplitude Period
For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method (Eq. 2),
and then integrating over one complete cycle,
or twice the half-cycle
or 4 times the quarter-cycle
which leads to
This integral can be re-written in terms of elliptic integrals as
where is the incomplete elliptic integral of the first kind defined by
Or more concisely by the substitution expressing in terms of ,
-
(Eq. 3)
where is the complete elliptic integral of the first kind defined by
For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (g = 9.80665 m/s2) at initial angle 10 degrees is . The linear approximation gives . The difference between the two values, less than 0.2%, is much less than that caused by the variation of g with geographical location.
From here there are many ways to proceed to calculate the elliptic integral:
Read more about this topic: Pendulum (mathematics)
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