In physics, angular frequency ω (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
displaystyle vec omega
is sometimes used as a synonym for the vector quantity angular velocity. One revolution is equal to 2π radians, hence
2 π f
displaystyle omega = 2pi over T = 2pi f ,
ω is the angular frequency or angular speed (measured in radians per second), T is the period (measured in seconds), f is the ordinary frequency (measured in hertz) (sometimes symbolised with ν).
1 Units 2 Circular motion
2.1 Oscillations of a spring 2.2 LC circuits
3 See also 4 References and notes 5 External links
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 (Hz) is also correct, but in practice it is only used for ordinary frequency f, and almost never for ω. This convention helps avoid confusion. In digital signal processing, the angular frequency may be normalized by the sampling rate, yielding the normalized frequency. Circular motion Main article: 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:
ω = v
displaystyle omega =v/r.
Oscillations of a spring
Part of a series of articles about
displaystyle vec F =m vec a
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v t e
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
displaystyle omega = sqrt frac k m ,
k is the spring constant, m is the mass of the object.
ω is referred to as the natural frequency (which can sometimes be denoted as ω0). As the object oscillates, its acceleration can be calculated by
a = −
displaystyle a=-omega ^ 2 x,
where x is displacement from an equilibrium position. Using "ordinary" revolutions-per-second frequency, this equation would be
a = − 4
displaystyle a=-4pi ^ 2 f^ 2 x.
The resonant angular frequency in a series
displaystyle omega = sqrt frac 1 LC .
Adding series resistance (for example, due to the resistance of the wire in a coil) does not change the resonate frequency of the series LC circuit. For a parallel tuned circuit, the above equation is often a useful approximation, but the resonate frequency does depend on the losses of parallel elements. See also
Mean motion Orders of magnitude (angular velocity) Simple harmonic motion
References and notes
^ a b Cummings, Karen; Halliday, David (2007). Understanding physics.
New Delhi: John Wiley & Sons Inc., authorized reprint to Wiley -
India. pp. 449, 484, 485, 487.
ISBN 978-81-265-0882-2. (UP1)
^ Holzner, Steven (2006).
Olenick ,, Richard P.; Apostol, Tom M.; Goodstein, David L. (2007). The Mechanical Universe. New York City: Cambridge University Press. pp. 383–385, 391–395. ISBN 978-0-521-71592-8.