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# Simple harmonic motion

Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped.

The motion is periodic and can be described as that of a sine function (or equivalently a cosine function), with constant amplitude. It is characterised by its amplitude, its period and its phase.

One definition of simple harmonic motion is "motion in which the acceleration of the oscillator is proportional to, and opposite in direction to the displacement from its equilibrium position".

A motion with period T has frequency $f=\frac{1}{T}$.

Its importance is that it can serve as a mathematical model of a variety of systems and provides the basis of the characterisation of more complicated motions through the techniques of Fourier analysis.

Simple harmonic motion can in some cases be considered to be the one-dimensional projection of two-dimensional circular motion.

Consider a long pendulum swinging over the turntable of a record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line. It is possible to change the frequency of rotation of the turntable in order to have a perfect synchronization with the motion of the pendulum.

The angular speed of the turntable is the pulsation of the pendulum.

In general, the pulsation of a straight-line simple harmonic motion is the angular speed of the corresponding circular motion.

Therefore, a motion with period T and frequency f=1/T has pulsation

$\omega=2\pi\cdot f = \frac{2\pi}{T}$

In general, pulsation and angular speed are not synonymous. For instance the pulsation of a pendulum is not the angular speed of the pendulum itself, but it is the angular speed of the corresponding circular motion.

03-10-2013 05:06:04