SHM for the Simple Pendulum

A simple example of a pendulum is a mass on a string. The forces on the mass are gravity and the string tension. Applying Newton's second law for rotation about the pivot point

Στ = Iα   →  - mg L sin θ = Iα

Note negative sign because torque is opposite to angular displacement.

For small angles we use the approximation:   sin θ ≈ θ

This gives the small angles equation of motion:   - mg Lθ ≈ Iα.

This equation of motion has the SHM form:              - ω2θ = α.
So the angular frequency is   ω = (
m g L
) ½

For a simple pendulum the rotational inertia is by:   I = mL2
This gives   ω = (
) ½   independent of the pendulum mass.