Simple Harmonic Motion – Overview

Simple harmonic motion (SHM) is a special type of oscillation that occurs under certain conditions.  The oscillation in SHM is sinusoidal, which is to say that it can be described by the trigonometric function sine.  Two common examples of SHM are:  the motion of a mass attached to a spring and the motion of a pendulum bob.

In order for simple harmonic motion to occur, the following condition must be true:

where:  F = net force acting on an object

k = a constant (a positive value in the above equation)

x = position of the object relative to equilibrium

If these conditions apply to an object it will oscillate about its point of equilibrium when it is set into motion.  The period of oscillation for any SHM will be given by:

Springs and Masses

Any common steel spring will apply a force that is proportional to its elongation or compression (this is known as Hooke’s Law).  Therefore a mass that is free to move that is attached to a spring will undergo SHM because it will satisfy the above conditions.  In that circumstance the constant, k, is called the “spring constant”.  The value of k represents the amount of change in force per change in length of the spring.  A mass attached to a spring is described by the two above equations with no modifications

Pendulums

A pendulum can also exhibit SHM.  It can be shown that the net force is directly proportional to the displacement from equilibrium to a high degree of accuracy as long as the amplitude of the swing is under 5°.  In this situation the value of k can be shown to be equal to mg ¸ l.  Therefore the period of a pendulum is given by: