angular frequency

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angular frequency
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Angular frequency ω (in radians per second), is larger than frequency ν (in cycles per second, also called Hz), by a factor of 2π. This figure uses the symbol ν, rather than f to denote frequency.
(File:Rotating Sphere.gif|right|thumb|A sphere rotating around an axis. Points farther from the axis move faster, satisfying {{nowrap|ω{{=}}v/r}}.)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 vec{omega} is sometimes used as a synonym for the vector quantity angular velocity.BOOK
, Cummings
, Karen
, Halliday, David
, Understanding physics
, John Wiley & Sons Inc., authorized reprint to Wiley – India
, 2007
, New Delhi
, 449, 484, 485, 487
, 978-81-265-0882-2, (UP1)
One revolution is equal to 2Ï€ radians, henceBOOK
, Holzner
, Steven
, Physics for Dummies
, Wiley Publishing Inc
, 2006
, Hoboken, New Jersey
, 201
, 978-0-7645-5433-9,

omega = {{2 pi} over T} = {2 pi 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).


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.BOOK,weblink Physics for scientists and engineers, Lawrence S., Lerner, 145, 978-0-86720-479-7, 1996-01-01, In digital signal processing, the angular 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:
omega = v/r.

Oscillations of a spring

{{Classical mechanics|rotational}}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 byBOOK
, Serway
, Raymond A.
, Jewett, John W.
, Principles of physics
, 4th
, Brooks / Cole – Thomson Learning
, 2006
, Belmont, CA
, 375, 376, 385, 397
, 978-0-534-46479-0,

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 = -omega^2 x,
where x is displacement from an equilibrium position.Using "ordinary" revolutions-per-second frequency, this equation would be
a = -4 pi^2 f^2 x.

LC circuits

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):BOOK
, Nahvi
, Mahmood
, Edminister, Joseph
, Schaum's outline of theory and problems of electric circuits
, McGraw-Hill Companies (McGraw-Hill Professional)
, 2003
, 214, 216
, 0-07-139307-2, (LC1)

omega = sqrt{frac{1}{LC}}.
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.

See also

References and notes

{{reflist}}Related Reading:
  • BOOK

, Olenick ,
, Richard P.
, Apostol, Tom M., Goodstein, David L.
, The Mechanical Universe
, Cambridge University Press
, 2007
, New York City
, 383–385, 391–395
, 978-0-521-71592-8,

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