The motion of an undamped Pendulum approximates to simple harmonic motion if the amplitude is very small relative to the length of the rod.
In the small-angle approximation, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a string of length ℓ with gravitational acceleration g is given by
This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not the acceleration due to gravity (g), therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational acceleration.
This approximation is accurate only in small angles because of the expression for angular acceleration being proportional to the sine of position:
where I is the moment of inertia. When θ is small, sin θ ≈ θ and therefore the expression becomes
which makes angular acceleration directly proportional to θ, satisfying the definition of simple harmonic motion
