# Periodic motion

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Periodic motion is the regular, repetitive motion of a body which continually retraces its path at regular intervals.

Period T of a periodic motion is the time to make one complete cycle.

Frequency f of a periodic motion is the number of cycles per unit time.

$T = \frac{1}{f}$

Angular frequency $\omega$ of a periodic motion is the rate of change of angular displacement with respect to time.

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

Displacement $x$ of an object is the distance of the oscillating particle from its equilibrium position at any instant.

Amplitude $x_{o}$ of a periodic motion is the magnitude of the maximum displacement of the oscillating particle from the equilibrium position.

$x = x_{o} \, sin \left( t \frac{2 \pi}{T} \right)$, used when motion starts from equilibrium position.

$x = x_{o} \, cos \left( t \frac{2 \pi}{T} \right)$, used when motion starts from extreme displacement.

If motion starts at somewhere between the amplitude and equilibrium, use:

$x = x_{o} \, sin \left( t \frac{2 \pi}{T} \right) \, + \, \phi$              OR           $x = x_{o} \, cos \left( t \frac{2 \pi}{T} \right) \, + \, \phi$, where $\phi$ is the distance from equilibrium