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 (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 is sometimes used as a synonym for the vector quantity angular velocity.
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 ?. This convention helps avoid confusion.
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:
? is referred to as the natural frequency (which can sometimes be denoted as ?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
The resonant angular frequency in a series LC circuit equals the square root of the reciprocal of the product of the capacitance (C measured in farads) and the inductance of the circuit (L, with SI unit henry):
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 .