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.