Theory

A simple pendulum is constructed by placing a mass m at the end of a rod of length L with negligible mass. The system oscillates about the lower vertical position due to a torque τ about the pivot produced by gravity acting on the mass. Although a pendulum oscillates, the angle cannot be described by simple trigonometric functions except for small angles. Newton's Law for planar rotation states that the angular acceleration α of an object is proportional to the torque τ applied to that object

τ = I α .

The constant of proportionality I is known as the moment of inertia and can be shown to be I = mL² for a mass that is a distance L from the point of rotation. Applying Newton's Second Law for rotation to the pendulum leads to the following second-order differential equation

d²θ / dt² = -(g/L) sin( θ ) .

Comparing this dynamical equation to the simple harmonic oscillator differential equation, we see that the pendulum equation undergoes simple harmonic motion for small angles when the approximation θ ~ sin( θ ) is valid. The angular frequency ω= 2πf for this small angle motion is ω= (g/L)^1/2.