How does a pendulum exhibit SHM?

A pendulum exhibits Simple Harmonic Motion (SHM) due to the restoring force acting towards its equilibrium position.

A pendulum, such as a swing or a grandfather clock, is a classic example of a system that exhibits Simple Harmonic Motion (SHM). This is a type of motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. In the case of a pendulum, the restoring force is gravity, which pulls the pendulum back towards its equilibrium position. For more on the basics of SHM, see Basics of SHM.

When a pendulum is displaced from its equilibrium position and released, it begins to swing back and forth. This is due to the force of gravity acting on the pendulum, pulling it back towards the centre. The force exerted by gravity is proportional to the displacement of the pendulum from its equilibrium position, which is the characteristic of SHM.

The motion of the pendulum can be described by the equation of SHM, which is a sine or cosine function. The amplitude of the motion (the maximum displacement from the equilibrium position) remains constant, as does the period (the time taken for one complete cycle of motion). This is because the force of gravity is constant, and the length of the pendulum, which determines the period of the motion, does not change. An understanding of the energy in SHM can provide deeper insights into how energy transformation underpins the pendulum's motion.

However, it's important to note that the assumption of SHM in a pendulum is an approximation. It only holds true for small angles of displacement, typically less than about 15 degrees. This is because the restoring force is only exactly proportional to the displacement for small angles. For larger angles, the force is not linearly proportional to the displacement, and the motion is not exactly simple harmonic. The effects of damping in SHM can also influence the pendulum's motion, especially in non-ideal conditions.

In summary, a pendulum exhibits SHM because the force of gravity provides a restoring force that is proportional to the displacement from the equilibrium position, causing the pendulum to oscillate back and forth. The motion of the pendulum can be described by the equation of SHM, with a constant amplitude and period. However, this is an approximation that only holds true for small angles of displacement.