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.

## Contents

Angular frequency (or angular speed) is the magnitude of the vector quantity *angular velocity*. The term **angular frequency vector**

One revolution is equal to 2π radians, hence

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.

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

## 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:

## Oscillations of a spring

An object attached to a spring will oscillate. Assuming that the spring is ideal and massless with no damping then the motion will be simple and harmonic with an angular frequency given by:

where

*k*is the spring constant

*m*is the mass of the object.

ω is referred to as the natural frequency (which can sometimes be denoted as ω_{0}).

As the object oscillates, its acceleration can be calculated by

where x is displacement from an equilibrium position.

Using 'ordinary' revolutions-per-second frequency, this equation would be

## LC circuits

The resonant angular frequency in a series LC circuit equals the square root of the inverse of the product of the capacitance (*C* measured in farads) and the inductance of the circuit (*L* in henrys).

Adding series resistance, for example due to the resistance of the wire in a coil, does not change the resonate frequency of the series LC circuit. For a parallel tuned circuit, the above equation is often a useful approximation, but the resonate frequency does depend on the losses of parallel elements.