A simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point P. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. By applying Newton's secont law for rotational systems, the equation of motion for the pendulum may be obtained
,
and rearranged as
.
If the amplitude of angular displacement is small enough that the small angle approximation () holds true, then the equation of motion reduces to the equation of simple harmonic motion
.
The simple harmonic solution is
with being the natural frequency of the motion.
Small Angle Approximation and Simple Harmonic Motion
With the assumption of small angles, the frequency and period of the pendulum are independent of the initial angular displacement amplitude. All simple pendulums should have the same period regardless of their initial angle (and regardless of their masses). This simple approximation is illustrated in the (48 kB) mpeg movie at left. All three pendulums cycle through one complete oscillation in the same amount of time.
The period for a simple pendulum does not depend on the mass or the initial anglular displacement, but depends only on the length L of the string and the value of the gravitational field strength g, according to
,
.
) holds true, then the equation of motion reduces to the equation of simple harmonic motion 
.
being the natural frequency of the motion.
