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<title>Radian per second</title>

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</style><table class="infobox ib-unit"><tbody><tr><th colspan="2" class="infobox-above">radian per second</th></tr><tr><td colspan="2" class="infobox-image"><div class="infobox-caption">Angular speed <i><a href="Omega" title="Omega">ω</a></i> (in radians per second), is greater than frequency <i><a href="Nu_(letter)" title="Nu (letter)">ν</a></i> (in <a href="Hertz" title="Hertz">Hz</a>), by a factor of 2π, because 2π rad/s corresponds to 1 Hz.</div></td></tr><tr><th colspan="2" class="infobox-header">General information</th></tr><tr><th scope="row" class="infobox-label"><a href="System_of_units_of_measurement" title="System of units of measurement">Unit system</a></th><td class="infobox-data"><a href="SI" class="mw-redirect" title="SI">SI</a></td></tr><tr><th scope="row" class="infobox-label">Unit of</th><td class="infobox-data"><a href="Angular_speed" class="mw-redirect" title="Angular speed">angular speed</a></td></tr><tr><th scope="row" class="infobox-label">Symbol</th><td class="infobox-data">rad/s, rad⋅s<sup>−1</sup></td></tr></tbody></table><style data-mw-deduplicate="TemplateStyles:r1236303919">

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<p>The <b>radian per second</b> (symbol: <b>rad⋅s<sup>−1</sup></b> or <b>rad/s</b>) is the unit of <a href="Angular_velocity" title="Angular velocity">angular velocity</a> in the <a href="International_System_of_Units" title="International System of Units">International System of Units</a> (SI). The <a href="Radian" title="Radian">radian</a> per <a href="Second" title="Second">second</a> is also the SI unit of <a href="Angular_frequency" title="Angular frequency">angular frequency</a> (symbol <i>ω</i>, omega). The radian per second is defined as the angular frequency that results in the <a href="Angular_displacement" title="Angular displacement">angular displacement</a> increasing by one radian every second.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Relation_to_other_units">Relation to other units</h2></div>

<p>A <a href="Frequency" title="Frequency">frequency</a> of one <a href="Hertz" title="Hertz">hertz</a> (1 Hz), or one <a href="Cycle_per_second" title="Cycle per second">cycle per second</a> (1 cps), corresponds to an angular frequency of 2<span class="texhtml mvar" style="font-style:italic;">π</span> radians per second. This is because one <a href="Cycle_(rotational_unit)" class="mw-redirect" title="Cycle (rotational unit)">cycle of rotation</a> corresponds to an <a href="Angle" title="Angle">angular</a> rotation of 2<span class="texhtml mvar" style="font-style:italic;">π</span> radians.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Since the radian is a <a href="Dimensionless_unit" class="mw-redirect" title="Dimensionless unit">dimensionless unit</a> in the <a href="SI" class="mw-redirect" title="SI">SI</a>, the radian per second is dimensionally equivalent to the hertz—both can be expressed as <a href="Reciprocal_seconds" class="mw-redirect" title="Reciprocal seconds">reciprocal seconds</a>, s<sup>−1</sup>. So, context is necessary to specify which <a href="Kind_of_quantity" class="mw-redirect" title="Kind of quantity">kind of quantity</a> is being expressed, angular frequency or ordinary frequency.

</p><p>One radian per second also corresponds to about 9.55 <a href="Revolutions_per_minute" title="Revolutions per minute">revolutions per minute</a> (rpm).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <i>Degrees per second</i> may also be defined, based on <a href="Degrees_(angle)" class="mw-redirect" title="Degrees (angle)">degree of arc</a>, where 1 degree per second (°/s) is equivalent to <style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml mvar" style="font-style:italic;">π</span></span><span class="sr-only">/</span><span class="den">180</span></span>⁠</span> rad⋅s<sup>−1</sup>.

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<caption>Quantity correspondence

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<th>Angular frequency <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span></th>

<th>Frequency <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =\omega /{2\pi }}">

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<annotation encoding="application/x-tex">{\displaystyle \nu =\omega /{2\pi }}</annotation>

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</math></span><img src="./ad8a9090ead417f082c41bc85c190d6cfcf81721.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.433ex; height:2.843ex;" alt="{\displaystyle \nu =\omega /{2\pi }}" loading="lazy"></span>

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<td>1 Hz

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<td>≈ 0.159155 Hz

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<td>≈ 9.5493 rpm

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<td>≈ 1 rpm

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<div class="mw-heading mw-heading2"><h2 id="Coherent_units">Coherent units</h2></div>

<p>A use of the unit radian per second is in calculation of the power transmitted by a shaft. In the <a href="International_System_of_Quantities" title="International System of Quantities">International System of Quantities</a> (SI) and the <a href="International_System_of_Units" title="International System of Units">International System of Units</a>, widely used in <a href="Physics" title="Physics">physics</a> and <a href="Engineering" title="Engineering">engineering</a>, the power <i>p</i> is equal to the angular speed <i>ω</i> multiplied by the <a href="Torque" title="Torque">torque</a> <i>τ</i> applied to the shaft: <span class="nowrap"><i>p</i> = <i>ω</i> ⋅ <i>τ</i></span>. When <a href="Coherence_(units_of_measurement)" title="Coherence (units of measurement)">coherent units</a> are used for these quantities, which are respectively the <a href="Watt" title="Watt">watt</a>, the radian per second, and the <a href="Newton-metre" title="Newton-metre">newton-metre</a>, and thus <span class="nowrap"><a href="Watt" title="Watt">W</a> = rad/s × <a href="Newton-metre" title="Newton-metre">N·m</a></span>, no numerical factor needed when performing the numerical calculation. When the units are not coherent (e.g. <a href="Horsepower" title="Horsepower">horsepower</a>, <a href="Turn_(angle)" title="Turn (angle)">turn</a>/min, and <a href="Pound-foot_(torque)" title="Pound-foot (torque)">pound-foot</a>), an additional factor will generally be necessary.

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<ul><li><a href="Cycle_per_second" title="Cycle per second">Cycle per second</a></li>

<li><a href="Normalized_frequency_(digital_signal_processing)" class="mw-redirect" title="Normalized frequency (digital signal processing)">Normalized frequency (digital signal processing)</a></li>

<li><a href="Order_of_magnitude_(angular_velocity)" class="mw-redirect" title="Order of magnitude (angular velocity)">Order of magnitude (angular velocity)</a></li>

<li><a href="Radian_per_metre" class="mw-redirect" title="Radian per metre">Radian per metre</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite class="citation book cs1">"6.1: Rotation Angle and Angular Velocity". <a rel="nofollow" class="external text" href="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"><i>College Physics</i></a>. OpenStax<span class="reference-accessdate">. Retrieved <span class="nowrap">20 January</span> 2024</span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://conversion.org/frequency/radians-per-second/">"radians per second :: unit"</a>. <i>conversion.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">28 January</span> 2024</span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation book cs1">"Metric Conversion: Language Resources". <a rel="nofollow" class="external text" href="https://webarchive.nla.gov.au/awa/20160229064041/https://www.macquariedictionary.com.au/resources/view/resource/10/"><i>Macquarie Dictionary</i></a>. <a href="Macquarie_Dictionary" title="Macquarie Dictionary">Macquarie Dictionary Publishers</a>. Archived from <a rel="nofollow" class="external text" href="https://www.macquariedictionary.com.au/resources/view/resource/10/">the original</a> on 29 February 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">27 January</span> 2024</span> – via <a href="National_Library_of_Australia" title="National Library of Australia">National Library of Australia</a>.</cite></span>

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