Simple harmonic motion

Definition

In terms of the equation for position

Simple harmonic motion is defined as a form of periodic/oscillatory motion along a straight line, with the following form of equation describing the position coordinate as a function of time:

$\!x=x(t)=x_{0}+A\sin(\omega t+\varphi )$

Here, $x_{0}$ is the resting position or mean position, $A$ is the amplitude of oscillation (in that it describes the maximum possible magnitude of displacement from the mean position), $\omega$ is a parameter controlling the frequency (in fact, $\omega$ is $2\pi$ times the frequency of completing one full oscillation), $t$ is the time parameter, and $\varphi$ is a phase angle.

As a consequence, we have the following descriptions of the velocity and acceleration functions:

$\!v(t)=A\omega \cos(\omega t+\varphi )$

and

$\!a(t)=-A\omega ^{2}\sin(\omega t+\varphi )$

In terms of the differential equation it solves

Simple harmonic motion is a form of motion of an object/particle along a line subject to the constraint:

$\!x''(t)=-\omega ^{2}x(t)$

where $x(t)$ is the position coordinate of the particle at time $t$ and $x''(t)=a(t)$ is the acceleration (signed) of the particle at time $t$.