The period of a simple pendulum refers to the time it takes for the pendulum to complete one full oscillation, typically measured as the time for it to swing from one extreme to the other and back. When the amplitude of a simple pendulum increases, meaning the pendulum is swinging with a larger angle, the period of the pendulum increases as well. This phenomenon is known as the increase in period with increasing amplitude.
There are a few reasons why this occurs:
Conservation of energy: A simple pendulum oscillates due to the interplay between potential energy (at the highest point of its swing) and kinetic energy (at the lowest point of its swing). As the amplitude increases, the pendulum reaches higher heights and thus gains potential energy. According to the law of conservation of energy, this increase in potential energy must come at the expense of kinetic energy. Since the pendulum's velocity decreases with increasing amplitude, it takes more time for it to complete an oscillation, resulting in a longer period.
Non-linearity of the restoring force: In a simple pendulum, the restoring force that brings the pendulum back to its equilibrium position is provided by gravity. However, the relationship between the restoring force and the displacement is not linear for large amplitudes. At larger angles, the sine of the angle, which represents the restoring force, deviates more significantly from the linear approximation (small-angle approximation). This non-linearity causes the period to increase as the amplitude becomes larger.
Elliptical motion: When the amplitude of a pendulum becomes larger, the motion of the pendulum deviates from a pure back-and-forth motion, and it starts to trace an elliptical path rather than a simple arc. The elliptical motion introduces additional complexities, such as variations in the effective length of the pendulum, which affect the period and cause it to increase with increasing amplitude.
These factors collectively contribute to the increase in the period of a simple pendulum as the amplitude of its swing becomes larger.