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Rate of change of the phase angle
Angular frequency ? (in radians per second), is larger than frequency ? (in cycles per second, also called Hz), by a factor of 2?. This figure uses the symbol ?, rather than f to denote frequency.
A sphere rotating around an axis. Points farther from the axis move faster, satisfying .
In physics, angular frequency? (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 (for example, in rotation) or the rate of change of the phase of a sinusoidal waveform (for example, 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.
In SIunits, 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 ?. 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 (cycle or radian) are omitted in SI.
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
Adding series resistance (for example, due to the resistance of the wire in a coil) 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.
Angular frequency is often loosely referred to as frequency, although in a strict sense these two quantities differ by a factor of 2?.