The equation for a swinging pendulum is , where is the angle of the pendulum at time , is the acceleration due to gravity, and is the length of the pendulum arm. The linearized approximation replaces by , which is valid for small .

Plots are shown for both the linear (blue) and nonlinear (pink) solutions. These solutions are close for small initial angles but diverge as increases. The mass of the ball does not affect the equations governing the system so the response is the same, which might seem counterintuitive.

This is the example An Oscillating Pendulum from [1], Section 1.4, Modeling with First Order Equations. The equation can be solved numerically using NDSolve in Mathematica.

Reference

[1] J. R. Brannan and W. E. Boyce, Differential Equations with Boundary Value Problems: An Introduction to Modern Methods and Applications, New York: John Wiley and Sons, 2010.