Angular frequency


In physics, angular frequency, also called angular speed and angular rate, is a scalar measure of the angle rate or the temporal rate of change of the phase argument of a sinusoidal waveform or sine function.
Angular frequency is the magnitude of the pseudovector quantity angular velocity.
Angular frequency can be obtained by multiplying rotational frequency, ν by a full turn :.
It can also be formulated as, the instantaneous rate of change of the angular displacement, θ, with respect to time, t.

Unit

In SI units, angular frequency is normally presented in the unit radian per second. The unit hertz is dimensionally equivalent, but by convention it is only used for frequency f, never for angular frequency ω. This convention is used to help avoid the confusion that arises when dealing with quantities such as frequency and angular quantities because the units of measure are considered to be one and hence may be omitted when expressing quantities in terms of SI units.
In digital signal processing, the frequency may be normalized by the sampling rate, yielding the normalized frequency.

Examples

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. During one period,, a body in circular motion travels a distance. This distance is also equal to the circumference of the path traced out by the body,. Setting these two quantities equal, and recalling the link between period and angular frequency we obtain: Circular motion on the unit circle is given by
where:
  • ω is the angular frequency,
  • T is the period,
  • f is the ordinary frequency.

    Oscillations of a spring

An object attached to a spring can oscillate. If the spring is assumed to be ideal and massless with no damping, then the motion is simple and harmonic with an angular frequency given by
where
ω is referred to as the natural angular frequency.
As the object oscillates, its acceleration can be calculated by
where x is displacement from an equilibrium position.
Using standard frequency f, this equation would be

LC circuits

The resonant angular frequency in a series LC circuit equals the square root of the reciprocal of the product of the capacitance and the inductance of the circuit :
Adding series resistance does not change the resonant frequency of the series LC circuit. For a parallel tuned circuit, the above equation is often a useful approximation, but the resonant frequency does depend on the losses of parallel elements.

Terminology

Although angular frequency is often loosely referred to as frequency, it differs from frequency by a factor of 2, which potentially leads confusion when the distinction is not made clear.