Angular frequency

In physics, angular frequency (symbol ω), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine function (for example, in oscillations and waves). Angular frequency (or angular speed) is the magnitude of the pseudovector quantity angular velocity. Angular frequency can be obtained multiplying rotational frequency, ν (or ordinary frequency, f) by a full turn (2π radians): ω=2π radν. It can also be formulated as ω=dθ/dt, the instantaneous rate of change of the angular displacement, θ, with respect to time, t. In SI units, angular frequency is normally presented in radians per second, even when it does not express a rotational value. The unit hertz (Hz) 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 (such as cycle or radian) are considered to be one and hence may be omitted when expressing quantities in SI units. In digital signal processing, the 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. 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: 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 k is the spring constant, m is the mass of the object. ω is referred to as the natural angular frequency (sometimes be denoted as ω0).