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((Short description|SI unit of angular velocity or angular frequency))
((More citations needed|date=January 2024))
((infobox unit
| name     = radian per second
| image    = AngularFrequency.gif
| caption  = Angular speed ''[[omega|ω]]'' (in radians per second), is greater than frequency ''[[nu (letter)|ν]]'' (in [[Hertz|Hz]]), by a factor of 2π, because 2π&nbsp;rad/s corresponds to 1&nbsp;Hz.
| standard = [[SI]]
| quantity = [[angular speed]]
| symbol   = rad/s
| symbol2  = rad⋅s<sup>−1</sup>
))

The '''radian per second''' (symbol: '''rad⋅s<sup>−1</sup>''' or '''rad/s''') is the unit of [[angular velocity]] in the [[International System of Units]] (SI). The [[radian]] per [[second]] is also the SI unit of [[angular frequency]] (symbol ''ω'', omega). The radian per second is defined as the angular frequency that results in the [[angular displacement]] increasing by one radian every second.<ref>((cite book |title=College Physics |publisher=OpenStax |url=https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/06%3A_Uniform_Circular_Motion_and_Gravitation/6.01%3A_Rotation_Angle_and_Angular_Velocity |language=en |chapter=6.1: Rotation Angle and Angular Velocity|access-date=20 January 2024))</ref>

==Relation to other units==
A [[frequency]] of one [[hertz]] (1&nbsp;Hz), or one [[cycle per second]] (1&nbsp;cps), corresponds to an angular frequency of 2((pi)) radians per second. This is because one [[cycle (rotational unit)|cycle of rotation]] corresponds to an [[angle|angular]] rotation of 2((pi)) radians.<ref>((cite web |title=radians per second :: unit |url=http://conversion.org/frequency/radians-per-second/ |website=conversion.org |access-date=28 January 2024))</ref> 

Since the radian is a [[dimensionless unit]] in the [[SI]], the radian per second is dimensionally equivalent to the hertz—both can be expressed as [[reciprocal seconds]], s<sup>−1</sup>. So, context is necessary to specify which [[kind of quantity]] is being expressed, angular frequency or ordinary frequency. 

One radian per second also corresponds to about 9.55 [[revolutions per minute]] (rpm).<ref>((cite book |url=https://www.macquariedictionary.com.au/resources/view/resource/10/ | chapter=Metric Conversion: Language Resources |publisher=[[Macquarie Dictionary|Macquarie Dictionary Publishers]] |title=Macquarie Dictionary |via=[[National Library of Australia]] |archive-url=https://webarchive.nla.gov.au/awa/20160229064041/https://www.macquariedictionary.com.au/resources/view/resource/10/ |archive-date=29 February 2016 |access-date=27 January 2024 ))</ref> ''Degrees per second'' may also be defined, based on [[degrees (angle)|degree of arc]], where 1 degree per second (°/s) is equivalent to ((sfrac|((pi))|180)) rad⋅s<sup>−1</sup>.

:{| class="wikitable"
|+ Quantity correspondence
|-
! Angular frequency <math>\omega</math> !! Frequency <math>\nu = \omega/{2\pi}</math>
|-
|| 2π&nbsp;rad/s ||1&nbsp;Hz
|-
|| 1&nbsp;rad/s || ≈ 0.159155&nbsp;Hz
|-
|| 1&nbsp;rad/s || ≈ 9.5493&nbsp;rpm
|-
|| 0.1047&nbsp;rad/s || ≈ 1&nbsp;rpm
|-
|}

== Coherent units ==
A use of the unit radian per second is in calculation of the power transmitted by a shaft. In the [[International System of Quantities]] (SI) and the [[International System of Units]], widely used in [[physics]] and [[engineering]], the power ''p'' is equal to the angular speed ''ω'' multiplied by the [[torque]] ''τ'' applied to the shaft: ((nowrap|1=''p'' = ''ω'' ⋅ ''τ'')).  When [[Coherence (units of measurement)|coherent units]] are used for these quantities, which are respectively the [[watt]], the radian per second, and the [[newton-metre]], and thus ((nowrap|1=[[watt|W]] = rad/s × [[newton-metre|N·m]])), no numerical factor needed when performing the numerical calculation. When the units are not coherent (e.g. [[horsepower]], [[turn (angle)|turn]]/min, and [[pound-foot (torque)|pound-foot]]), an additional factor will generally be necessary.

== See also ==
* [[Cycle per second]]
* [[Normalized frequency (digital signal processing)]]
* [[Order of magnitude (angular velocity)]]
* [[Radian per metre]]

== References ==
((Reflist))

[[Category:SI derived units]]
[[Category:Units of angular velocity]]


((physics-stub))

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