The relationship between the amplitude and the time period of a pendulum is not direct. The amplitude of a pendulum swing refers to the maximum displacement from its equilibrium position, while the time period refers to the time it takes for the pendulum to complete one full oscillation.
For a simple pendulum, the time period is primarily determined by the length of the pendulum string and the acceleration due to gravity. The longer the length of the pendulum string, the longer it takes for the pendulum to swing back and forth, resulting in a longer time period.
On the other hand, the amplitude of a pendulum swing depends on the initial conditions and the amount of energy imparted to the pendulum. The amplitude does not directly affect the time period of the pendulum. Regardless of whether the pendulum has a small or large amplitude, the time it takes for one complete oscillation remains constant as long as the other factors, such as length and gravitational acceleration, are unchanged.
It's worth noting that for large amplitudes, the period of a pendulum can be slightly different from the period of a small amplitude pendulum, due to the nonlinearity of the pendulum's motion. This is known as the "small angle approximation" where the period is independent of the amplitude. However, for small amplitudes (generally less than around 20 degrees), the time period of a pendulum is effectively constant regardless of the amplitude.