Angular Frequency

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## Units

## Examples

### Circular motion

### Oscillations of a spring

*x* is displacement from an equilibrium position.
### LC circuits

## Terminology

## See also

## References and notes

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Angular Frequency

In physics, **angular frequency** * ?* (also referred to by the terms

One revolution is equal to 2*?* radians, hence^{[1]}^{[2]}

*?*is the angular frequency (measured in radians per second),*T*is the period (measured in seconds),*f*is the ordinary frequency (measured in hertz) (sometimes symbolised with*?*).

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 is used to help avoid the confusion^{[3]} that arises when dealing with frequency or the Planck constant because the units of angular measure (cycle or radian) are omitted in SI.^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}

In digital signal processing, the angular frequency may be normalized by the sampling rate, yielding the normalized frequency.

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^{[9]}

where

*k*is the spring constant,*m*is the mass of the object.

*?* is referred to as the natural frequency (which can sometimes be denoted as *?*_{0}).

As the object oscillates, its acceleration can be calculated by

whereUsing "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):^{[10]}

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 2?.

- Cycle per second
- Radian per second
- Degree (angle)
- Mean motion
- Orders of magnitude (angular velocity)
- Simple harmonic motion

- ^
^{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 for Dummies*. Hoboken, New Jersey: Wiley Publishing Inc. pp. 201. ISBN 978-0-7645-5433-9.angular frequency.

**^**Lerner, Lawrence S. (1996-01-01).*Physics for scientists and engineers*. p. 145. ISBN 978-0-86720-479-7.**^**Mohr, J. C.; Phillips, W. D. (2015). "Dimensionless Units in the SI".*Metrologia*.**52**(1): 40-47. arXiv:1409.2794. Bibcode:2015Metro..52...40M. doi:10.1088/0026-1394/52/1/40. S2CID 3328342.**^**Mills, I. M. (2016). "On the units radian and cycle for the quantity plane angle".*Metrologia*.**53**(3): 991-997. Bibcode:2016Metro..53..991M. doi:10.1088/0026-1394/53/3/991.**^**"SI units need reform to avoid confusion". Editorial.*Nature*.**548**(7666): 135. 7 August 2011. doi:10.1038/548135b. PMID 28796224.**^**P. R. Bunker; I. M. Mills; Per Jensen (2019). "The Planck constant and its units".*J Quant Spectrosc Radiat Transfer*.**237**: 106594. Bibcode:2019JQSRT.23706594B. doi:10.1016/j.jqsrt.2019.106594. S2CID 201264843.**^**P. R. Bunker; Per Jensen (2020). "The Planck constant of action_{A}".*J Quant Spectrosc Radiat Transfer*.**243**: 106835. Bibcode:2020JQSRT.24306835B. doi:10.1016/j.jqsrt.2020.106835. S2CID 214138755.**^**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.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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