Pendulum

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

d2 θ / 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.