Angular frequency


In physics, angular frequency ω is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.
Angular frequency is the magnitude of the vector quantity angular velocity. The term angular frequency vector is sometimes used as a synonym for the vector quantity angular velocity.
One revolution is equal to 2π radians, hence
where:

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 is also correct, but in practice it is only used for ordinary frequency f, and almost never for ω. This convention is used to help avoid the confusion that arises when dealing with frequency or the Planck constant because the units of angular measure are omitted in SI.
In digital signal processing, the angular frequency may be normalized by the sampling rate, yielding the normalized frequency.

Examples of Angular 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. 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:

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 frequency.
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 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

Angular frequency is often loosely referred to as frequency, although in a strict sense these two quantities differ by a factor of.