Editing Angular frequency (section)

== Examples ==

=== Circular motion ===
{{main|Circular motion}}
In a rotating or orbiting object, there is a relation between distance from the axis, <math>r</math>, [[tangential speed]], <math>v</math>, and the angular frequency of the rotation. During one period, <math>T</math>, a body in circular motion travels a distance <math>vT</math>. This distance is also equal to the circumference of the path traced out by the body, <math>2\pi r</math>. Setting these two quantities equal, and recalling the link between period and angular frequency we obtain: <math>\omega = v/r.</math> Circular motion on the unit circle is given by
<math display="block">\omega = \frac{2 \pi}{T} = {2 \pi f} , </math>
where:
* ''ω'' is the angular frequency (SI unit: [[radians per second]]),
* ''T'' is the [[Frequency|period]] (SI unit: [[second]]s),
* ''f'' is the [[ordinary frequency]] (SI unit: [[hertz]]).

=== Oscillations of a spring ===
{{Classical mechanics|rotational}}
An object attached to a spring can [[Oscillation|oscillate]]. If the spring is assumed to be ideal and massless with no damping, then the motion is [[Harmonic oscillator|simple and harmonic]] with an angular frequency given by<ref name=PoP1>
{{cite book
  | last = Serway
  | first = Raymond A.
  | author2 = Jewett, John W.
  | title = Principles of physics
  | edition = 4th
  | publisher = Brooks / Cole – Thomson Learning
  | year = 2006
  | location = Belmont, CA
  | pages = 375, 376, 385, 397
  | url = https://books.google.com/books?id=1DZz341Pp50C&q=angular+frequency&pg=PA376
  | isbn =978-0-534-46479-0 }}</ref>
<math display="block"> \omega = \sqrt{\frac{k}{m}}, </math>
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 ''ω''<sub>0</sub>).

As the object oscillates, its acceleration can be calculated by
<math display="block" qid=Q11376>a = -\omega^2 x, </math>
where ''x'' is displacement from an equilibrium position.

Using standard frequency ''f'', this equation would be
<math display="block"> a = -(2 \pi f)^2 x. </math>

=== LC circuits ===
The resonant angular frequency in a series [[LC circuit]] equals the square root of the [[multiplicative inverse|reciprocal]] of the product of the [[capacitance]] (''C'', with SI unit [[farad]]) and the [[inductance]] of the circuit (''L'', with SI unit [[Henry (unit)|henry]]):<ref name=LC1>
{{cite book
  | last = Nahvi
  | first = Mahmood
  | author2 = Edminister, Joseph
  | title = Schaum's outline of theory and problems of electric circuits
  | publisher = McGraw-Hill Companies (McGraw-Hill Professional)
  | year = 2003
  | pages = 214, 216
  | url = https://books.google.com/books?id=nrxT9Qjguk8C&q=angular+frequency&pg=PA103
  | isbn = 0-07-139307-2
}} (LC1)</ref>
<math display="block">\omega = \sqrt{\frac{1}{LC}}.</math>

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.