6. Simple Harmonic Motion

A pendulum is a body that swings freely back and forth under the influence of gravity. When a pendulum is displaced from its resting position, it is subject to a restoring force due to gravity that brings it back towards equilibrium. This motion is called simple harmonic motion (SHM). The mathematics of pendulums can be quite complicated, but for small-angle oscillations, the equations of motion can be solved analytically.

A simple gravity pendulum is an idealized mathematical model of a real pendulum. It consists of a weight on the end of a massless cord suspended from a pivot, without friction. When given an initial displacement, it will swing back and forth at a constant amplitude. The period of a pendulum, which is the time for one complete oscillation, depends only on the length of the string and the strength of the gravitational acceleration. The mass of the bob does not affect the period.

The motion of a pendulum can be described as simple harmonic motion when the amplitude of the oscillations is small. In this case, the displacement of the pendulum from equilibrium follows a sine wave. The period of a simple pendulum is given by the equation T = 2π√(L/g), where T is the period, L is the length of the string, and g is the acceleration due to gravity. This equation holds true as long as the amplitude of the oscillations is less than 15 degrees. For larger amplitudes, the period deviates from this equation and the motion is no longer sinusoidal.

In conclusion, a pendulum exhibits simple harmonic motion when it swings back and forth under the influence of gravity. The period of a pendulum depends only on the length of the string and the strength of the gravitational acceleration. For small-angle oscillations, the motion of a pendulum follows a sine wave. However, for larger amplitudes, the motion deviates from simple harmonic motion.