 Acoustics and Vibration Animations Daniel A. Russell, Ph.D. Graduate Program in Acoustics The Pennsylvania State University All text and images on this page are ©2004-2011 by Daniel A. Russell and may not used in other web pages or reports without permission.  # The Simple Pendulum 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 The mpeg movie at left (39.5 kB) shows two pendula, with different lengths. How many complete oscillations does the shorter (blue) pendulum make in the time for one complete oscillation of the longer (black) pendulum? From this information and the definition of the period for a simple pendulum, what is the ratio of lengths for the two pendula?

## The Real (Nonlinear) Pendulum

When the angular displacement amplitude of the pendulum is large enough that the small angle approximation no longer holds, then the equation of motion must remain in its nonlinear form .
This differential equation does not have a closed form solution, and must be solved numerically using a computer. Mathematica numerically solves this differential equation very easily with the built in function NDSolve[ ].

#### Small Initial Amplitude The small angle approximation is valid for initial angular displacements of about 20° or less. The (0.132MB) mpeg movie at left shows two pendula: the black pendulum assumes the linear small angle approximation of simple harmonic motion, the grey pendulum (hidded behind the black one) shows the numerical solution of the actual nonlinear differential equation of motion. For small initial angular displacements the error in the small angle approximation becomes evident only after several oscillations.

#### Large Initial Amplitude When the initial angular displacement is significantly large that the small angle approximation is no longer valid, the error between the simple harmonic solution and the actual solution becomes apparent almost immediately, and grows as time progresses. In the (0.226MB) mpeg move at left, the dark blue pendulum is the simple approximation, and the light blue pendulum (initially hidden behind the dark blue one) shows the numerical solution of the nonlinear differential equation of motion.

Back to the Vibrations and Waves Demo Page