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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.
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