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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

ω →

displaystyle vec omega

is sometimes used as a synonym for the vector quantity angular velocity.[1] One revolution is equal to 2π radians, hence[1][2]

ω =

2 π

T

=

2 π f

,

displaystyle omega = 2pi over T = 2pi f ,

where:

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).

Contents

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[edit] 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.[3] In digital signal processing, the angular frequency may be normalized by the sampling rate, yielding the normalized frequency. Circular motion[edit] 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

/

r .

displaystyle omega =v/r.

Oscillations of a spring[edit]

Part of a series of articles about

Classical mechanics

F →

= m

a →

displaystyle vec F =m vec a

Second
Second
law of motion

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dynamics Euler's equations

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Rotation

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reactive

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Angular acceleration / displacement / frequency / velocity

Scientists

Galileo Huygens Newton Kepler Horrocks Halley Euler d'Alembert Clairaut Lagrange Laplace Hamilton Poisson Daniel Bernoulli Johann Bernoulli Cauchy

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[4]

ω =

k m

,

displaystyle omega = sqrt frac k m ,

where

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 = −

ω

2

x ,

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

π

2

f

2

x .

displaystyle a=-4pi ^ 2 f^ 2 x.

LC circuits[edit] The resonant angular frequency in a series LC circuit
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):[5]

ω =

1

L C

.

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[edit]

Mean motion Orders of magnitude (angular velocity) Simple harmonic motion

References and notes[edit]

^ 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). Physics
Physics
for Dummies. Hoboken, New Jersey: Wiley Publishing Inc. p. 201. ISBN 978-0-7645-5433-9.  ^ Lerner, Lawrence S. (1996-01-01). Physics
Physics
for scientists and engineers. p. 145. ISBN 978-0-86720-479-7.  ^ Serway, Raymond A.; Jewett, John W. (2006). Principles of physics (4th ed.). Belmont, CA: Brooks / Cole - Thomson Learning. pp. 375, 376, 385, 397. ISBN 978-0-534-46479-0.  ^ Nahvi, Mahmood; Edminister, Joseph (2003). Schaum's outline of theory and problems of electric circuits. McGraw-Hill Companies (McGraw-Hill Professional). pp. 214, 216. ISBN 0-07-139307-2. (LC1)

Related Reading:

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. 

External

.