The amplitude of a simple pendulum is the maximum angle that the pendulum swings away from its equilibrium position. In the case of a simple pendulum, which consists of a mass (bob) attached to a string or rod, there are a few reasons why it is generally recommended to keep the amplitude under 10 degrees:
Small Angle Approximation: The motion of a simple pendulum is most accurately described using the small angle approximation, which assumes that the amplitude is small (usually less than about 10 degrees). This approximation simplifies the mathematical equations governing the pendulum's motion and provides accurate results. When the amplitude exceeds a certain threshold, the small angle approximation becomes less accurate, leading to deviations from the expected behavior.
Linear Restoring Force: The motion of a simple pendulum relies on a linear restoring force that is proportional to the displacement of the pendulum from its equilibrium position. This linear relationship holds true for small angles, but as the amplitude increases, the restoring force becomes nonlinear, deviating from the linear relationship. This can introduce errors in the motion and affect the accuracy of any calculations or predictions based on the pendulum's behavior.
Conservation of Energy: A simple pendulum exhibits the conservation of mechanical energy, where the sum of the pendulum's kinetic energy (motion) and potential energy (height) remains constant. However, as the amplitude increases, the pendulum swings higher, resulting in a larger potential energy component. This energy transfer can lead to inaccuracies in energy calculations and affect the overall behavior of the pendulum system.
Nonlinearity of Period: The period of a simple pendulum, which is the time it takes to complete one full oscillation, is directly related to the pendulum's length. However, as the amplitude increases, the period becomes slightly longer due to the nonlinear relationship between amplitude and period. For small angles, this deviation is negligible, but as the amplitude grows larger, the period becomes more affected by nonlinear effects, introducing errors in timing calculations.
While the specific threshold for an acceptable amplitude may vary depending on the context and precision required, keeping the amplitude of a simple pendulum under 10 degrees ensures that the small angle approximation remains valid, linear relationships are maintained, energy conservation holds, and period calculations remain accurate within a reasonable margin of error.