Editing Angular velocity (section)

{{Short description|Direction and rate of rotation}}
{{Use dmy dates|date=July 2022}}
{{Infobox physical quantity
| name = Angular velocity
| otherunits =
| image = [[File:Vector-omega.svg|class=skin-invert-image|220px]]
| symbols = '''{{math|ω}}'''
| unit = rad{{sdot}}s<sup>−1</sup>
| baseunits = s<sup>−1</sup>
| dimension = wikidata
| extensive = yes
| intensive = yes (for [[rigid body]] only)
| conserved = no
| transformsas = pseudovector
| derivations =  {{math|1='''''ω''''' = d'''θ''' / d''t''}}
}}
{{Classical mechanics|rotational}}

In [[physics]], '''angular velocity''' (symbol '''{{math|ω}}''' or <math>\vec{\omega}</math>, the lowercase Greek letter [[omega]]), also known as the '''angular frequency vector''',<ref name="UP1">{{cite book
  | last = Cummings
  | first = Karen
  |author2=Halliday, David
  | title = Understanding physics
  | publisher = John Wiley & Sons Inc., authorized reprint to Wiley – India
  | date = 2007
  | location = New Delhi
  | pages = 449, 484, 485, 487
  | url = https://books.google.com/books?id=rAfF_X9cE0EC
  | isbn =978-81-265-0882-2 }}(UP1)</ref> is a [[pseudovector]] representation of how the [[angular position]] or [[orientation (geometry)|orientation]] of an object changes with time, i.e. how quickly an object [[rotate]]s (spins or revolves) around an axis of rotation and how fast the axis itself changes [[direction (geometry)|direction]].<ref>{{Cite web |title=Angular velocity {{!}} Rotational Motion, Angular Momentum, Torque {{!}} Britannica |url=https://www.britannica.com/science/angular-velocity |access-date=2024-10-05 |website=www.britannica.com |language=en}}</ref> 

The magnitude of the pseudovector, <math>\omega=\|\boldsymbol{\omega}\|</math>, represents the ''[[angular speed]]'' (or ''angular frequency''), the angular rate at which the object rotates (spins or revolves). The pseudovector direction <math>\hat\boldsymbol{\omega}=\boldsymbol{\omega}/\omega</math> is [[Normal (geometry)|normal]] to the instantaneous [[plane of rotation]] or [[angular displacement]]. 

There are two types of angular velocity:
* '''Orbital angular velocity''' refers to how fast a point object [[Rotation around a fixed axis|revolves about a fixed origin]], i.e. the time rate of change of its angular position relative to the [[Origin (mathematics)|origin]]. {{Citation needed|date=February 2023}}
* '''Spin angular velocity''' refers to how fast a rigid body rotates around a fixed axis of rotation, and is independent of the choice of origin, in contrast to orbital angular velocity.

Angular velocity has [[dimension (physics)|dimension]] of angle per unit time; this is analogous to linear [[velocity]], with angle replacing [[distance]], with time in common. The [[SI unit]] of angular velocity is [[radians per second]],<ref>{{cite book |title=International System of Units (SI) |edition=revised 2008 |first1=Barry N. |last1=Taylor |publisher=DIANE Publishing |year=2009 |isbn=978-1-4379-1558-7 |page=27 |url=https://books.google.com/books?id=I-BlErBBeL8C}} [https://books.google.com/books?id=I-BlErBBeL8C&pg=PA27 Extract of page 27]</ref> although [[degrees per second]] (°/s) is also common. The [[radian]] is a [[dimensionless quantity]], thus the SI units of angular velocity are dimensionally equivalent to [[reciprocal seconds]], s<sup>−1</sup>, although rad/s is preferable to avoid confusion with '''rotation velocity''' in units of [[hertz]] (also equivalent to s<sup>−1</sup>).<ref>{{cite web |url=http://www.bipm.org/en/publications/si-brochure/section2-2-2.html |title=Units with special names and symbols; units that incorporate special names and symbols }}</ref> 

The sense of angular velocity is conventionally specified by the [[right-hand rule]], implying [[clockwise]] rotations (as viewed on the plane of rotation); [[negation (arithmetic)|negation]] (multiplication by −1) leaves the magnitude unchanged but flips the axis in the [[opposite direction (geometry)|opposite direction]].<ref name= EM1>{{cite book
  | last =  Hibbeler
  | first = Russell C.
  | title = Engineering Mechanics
  | publisher = Pearson Prentice Hall
  | year = 2009
  | location = [[Upper Saddle River]], New Jersey
  | pages = 314, 153
  | url =https://books.google.com/books?id=tOFRjXB-XvMC&q=angular+velocity&pg=PA314
  | isbn = 978-0-13-607791-6}}(EM1)</ref>

For example, a [[Geosynchronous orbit|geostationary]] satellite completes one orbit per day above the [[equator]] (360 degrees per 24 hours){{ref|sidereal|a}} has angular velocity magnitude (angular speed) ''ω'' = 360°/24&nbsp;h = 15°/h (or 2π&nbsp;rad/24&nbsp;h ≈ 0.26&nbsp;rad/h) and angular velocity direction (a [[unit vector]]) parallel to [[Earth's rotation axis]] (<math>\hat\omega=\hat{Z}</math>, in the [[geocentric coordinate system]]). If angle is measured in radians, the linear velocity is the radius times the angular velocity, <math>v = r\omega</math>. With orbital radius 42,000&nbsp;km from the Earth's center, the satellite's [[tangential speed]] through space is thus ''v'' = 42,000&nbsp;km &times; 0.26/h ≈ 11,000&nbsp;km/h. The angular velocity is positive since the satellite travels [[Retrograde and prograde motion|prograde]] with the Earth's rotation (the same direction as the rotation of Earth).

{{Note|sidereal|a||}} Geosynchronous satellites actually orbit based on a sidereal day which is 23h 56m 04s, but 24h is assumed in this example for simplicity.