Complete Elliptic Integral and Pendulum Period
Use the arithmetic–geometric mean to measure the nonlinear pendulum period.
Free theory articleUse the arithmetic–geometric mean to measure the nonlinear pendulum period.
Concept & interpretation
The period of a simple pendulum is strictly constant only in the small-angle approximation. When the amplitude is large, the exact period T depends on the initial amplitude \theta_0 through a complete elliptic integral of the first kind, K(k). Here the modulus k is related to the amplitude by k = sin(\theta_0 / 2). The period is given by T = 4√(L/g)K(k).
Experiment
Explore the exact period by varying the modulus k and the length L. Observe how the period diverges as k → 1, which represents a pendulum released from the straight up position (a separatrix orbit). You can interact with the experiment below.
- Set Modulus k to 0.5 and Length L to 1.
- Observe that the calculated Period is approximately 2.1528 s, and the Period ratio is 1.073.