Simple harmonic motion

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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.