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, as it repeats itself at standard intervals in a specific manner - described as being sinusoidal, with constant amplitude. It is characterized by its amplitude which is always positive and depends on how motion starts initially, its period which is the time for a single oscillation, and its phase which depends on displacement as well as velocity of the moving object.

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", or $a\propto -x$.

A general equation describing simple harmonic motion is $y(t)=A\sin \left(2\,\pi \,ft+\phi \right)$, where y is the displacement, A is the amplitude of oscillation, f is the frequency, t is the elapsed time, and $\phi$ is the phase of oscillation. If there is no displacement at time t = 0, the phase $\phi =0$. A motion with frequency f has period $T={\frac {1}{f}}$.

Simple harmonic motion can serve as a mathematical model of a variety of motions and provides the basis of the characterisation of more complicated motions through the techniques of Fourier analysis.

Realizations

Simple harmonic motion is exhibited in a variety of simple physical systems and below are some examples:

Mass on a spring: A mass M attached to a spring of spring constant k exhibits simple harmonic motion in space with

$\omega =2\pi f={\sqrt {\frac {k}{M}}}.\,$

Alternately, if the other factors are known and the period is to be found, this equation can be used:

$T=2\pi {\sqrt {\frac {M}{k}}}.$

Uniform circular motion: Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular speed $\omega$ around a circle of radius $R$ centered at the origin of the x-y plane, then its motion along the x and the y coordinates is simple harmonic with amplitude $R$ and angular speed $\omega$.

Mass on a pendulum: In the small-angle approximation, the motion of a pendulum is shown to approximate simple harmonic motion. The period of a mass attached to a string of length $\ell$ with gravitation acceleration $g$ is given by

$T=2\pi {\sqrt {\frac {\ell }{g}}}.$

For an exact solution not relying on a small-angle approximation, see pendulum (mathematics).

See also

• Isochronous
• Uniform Circular Motion
• complex harmonic motion