So then, according to Hooke’s Law,
And so the period of a pendulum is
equal to
As you can see, both the mass (m) and amplitude of angular displacement (Ɵ) are canceled out in the above derivation and the only factor upon which the period of a pendulum is dependent is the length of the wire (l).
Any pendulum undergoes simple harmonic motion when the amplitude of angular displacement Ɵ is small.
Simple harmonic motion is an idealized expression that assumes that a mass displaced from equilibrium responds with a restoring force that is proportional to the displacement.
But what happens for large amplitude displacements? The pendulum still oscillates, but the motion is no longer simple harmonic because the restoring force is not proportional to the displacement force. For large angular displacements, the graph is no longer sinusoidal.
Click to see the difference!