Angular frequency

File:AngularFrequency.gif
Angular frequency ω (in radians per second), is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π. This figure uses the symbol ν, rather than f to denote frequency.

In physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit time (e.g., in rotation) or the rate of change of the phase of a sinusoidal waveform (e.g., in oscillations and waves), or as the rate of change of the argument of the sine function.

Angular frequency (or angular speed) is the magnitude of the vector quantity angular velocity. The term angular frequency vector $\vec{\omega}$ is sometimes used as a synonym for the vector quantity angular velocity.[1]

One revolution is equal to 2π radians, hence[1][2]

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

where:

ω is the angular frequency or angular speed (measured in radians per second),
T is the period (measured in seconds),
f is the ordinary frequency (measured in hertz) (sometimes symbolised with ν).

Units

In SI units, angular frequency is normally presented in radians per second, even when it does not express a rotational value. From the perspective of dimensional analysis, the unit hertz (Hz) is also correct, but in practice it is only used for ordinary frequency f, and almost never for ω. This convention helps avoid confusion.[3]

In digital signal processing, the angular frequency may be normalized by the sampling rate, yielding the normalized frequency.

Examples

File:Rotating Sphere.gif
A sphere rotating around an axis. Points farther from the axis move faster, satisfying ω=v/r.

Circular motion

Main article: Circular motion

In a rotating or orbiting object, there is a relation between distance from the axis, tangential speed, and the angular frequency of the rotation:

$\omega = v/r$