Angular frequency

<h2 id="unit">Unit</h2>
<p>In <a href="/facts/SI/NShmfzlD">SI</a> <a href="/facts/Units_of_measurement/pvTpWbmC">units</a>, angular frequency is normally presented in the unit <a href="/facts/Radian/Srd38Sxa">radian</a> per <a href="/facts/Second/jtFxnJXx">second</a>. The unit <a href="/facts/Hertz/6Wshj5qm">hertz</a> (Hz) is dimensionally equivalent, but by convention it is only used for frequency <i>f</i>, never for angular frequency <i>ω</i>. This convention is used to help avoid the confusion<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> that arises when dealing with quantities such as frequency and angular quantities because the units of measure (such as cycle or radian) are considered to be one and hence may be omitted when expressing quantities in terms of SI units.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p><p>In <a href="/facts/Digital_signal_processing/PEG7azr8">digital signal processing</a>, the frequency may be normalized by the <a href="/facts/Sampling_rate/xxck7aLN">sampling rate</a>, yielding the <a href="/facts/Normalized_frequency_(digital_signal_processing)/hsY0aY3E">normalized frequency</a>.
</p>
<h2 id="examples">Examples</h2>
<h3>Circular motion</h3>
<p class="note">Main article: <a href="/facts/Circular_motion/r4YaTspL">Circular motion</a></p>
<p>In a rotating or orbiting object, there is a relation between distance from the axis, 
  
    
      
        r
      
    
    {\displaystyle r}
  
, <a href="/facts/Tangential_speed/I3qgMPfs">tangential speed</a>, 
  
    
      
        v
      
    
    {\displaystyle v}
  
, and the angular frequency of the rotation. During one period, 
  
    
      
        T
      
    
    {\displaystyle T}
  
, a body in circular motion travels a distance 
  
    
      
        v
        T
      
    
    {\displaystyle vT}
  
. This distance is also equal to the circumference of the path traced out by the body, 
  
    
      
        2
        π
        r
      
    
    {\displaystyle 2\pi r}
  
. Setting these two quantities equal, and recalling the link between period and angular frequency we obtain: 
  
    
      
        ω
        =
        v
        
          /
        
        r
        .
      
    
    {\displaystyle \omega =v/r.}
  
 Circular motion on the unit circle is given by

ω
        =
        
          
            
              2
              π
            
            T
          
        
        =
        
          2
          π
          f
        
        ,
      
    
    {\displaystyle \omega ={\frac {2\pi }{T}}={2\pi f},}

where:
</p>
<ul><li><i>ω</i> is the angular frequency (SI unit: <a href="/facts/Radians_per_second/cG0plKHD">radians per second</a>),</li>
<li><i>T</i> is the <a href="/facts/Frequency/QUSbPaW8">period</a> (SI unit: <a href="/facts/Second/jtFxnJXx">seconds</a>),</li>
<li><i>f</i> is the <a href="/facts/Ordinary_frequency/QUSbPaW8">ordinary frequency</a> (SI unit: <a href="/facts/Hertz/6Wshj5qm">hertz</a>).</li></ul>
<h3>Oscillations of a spring</h3>

<p>An object attached to a spring can <a href="/facts/Oscillation/6vSrm84a">oscillate</a>. If the spring is assumed to be ideal and massless with no damping, then the motion is <a href="/facts/Harmonic_oscillator/7IQNToGt">simple and harmonic</a> with an angular frequency given by<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>

ω
        =
        
          
            
              k
              m
            
          
        
        ,
      
    
    {\displaystyle \omega ={\sqrt {\frac {k}{m}}},}

where
</p>
<ul><li><i>k</i> is the <a href="/facts/Spring_constant/7hTUcKUm">spring constant</a>,</li>
<li><i>m</i> is the mass of the object.</li></ul>
<p><i>ω</i> is referred to as the natural angular frequency (sometimes be denoted as <i>ω</i>0).
</p><p>As the object oscillates, its acceleration can be calculated by

a
        =
        −
        
          ω
          
            2
          
        
        x
        ,
      
    
    {\displaystyle a=-\omega ^{2}x,}

where <i>x</i> is displacement from an equilibrium position.
</p><p>Using standard frequency <i>f</i>, this equation would be

a
        =
        −
        (
        2
        π
        f
        
          )
          
            2
          
        
        x
        .
      
    
    {\displaystyle a=-(2\pi f)^{2}x.}

</p>
<h3>LC circuits</h3>
<p>The resonant angular frequency in a series <a href="/facts/LC_circuit/tzEfI7Hf">LC circuit</a> equals the square root of the <a href="/facts/Multiplicative_inverse/yczdYyCw">reciprocal</a> of the product of the <a href="/facts/Capacitance/0pH046TL">capacitance</a> (<i>C</i>, with SI unit <a href="/facts/Farad/gPAYcVoE">farad</a>) and the <a href="/facts/Inductance/oIxj7JxX">inductance</a> of the circuit (<i>L</i>, with SI unit <a href="/facts/Henry_(unit)/uwVv9zue">henry</a>):<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>

ω
        =
        
          
            
              1
              
                L
                C
              
            
          
        
        .
      
    
    {\displaystyle \omega ={\sqrt {\frac {1}{LC}}}.}

</p><p>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.
</p>
<h2 id="terminology">Terminology</h2>
<p>Although angular frequency is often loosely referred to as frequency, it differs from frequency by a factor of 2π, which potentially leads confusion when the distinction is not made clear.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Cycle_per_second/WxtGGRFx">Cycle per second</a></li>
<li><a href="/facts/Radian_per_second/cG0plKHD">Radian per second</a></li>
<li><a href="/facts/Degree_(angle)/f48Ns8SX">Degree (angle)</a></li>
<li><a href="/facts/Mean_motion/ks7h1nX5">Mean motion</a></li>
<li><a href="/facts/Rotational_frequency/VqLm3L3R">Rotational frequency</a></li>
<li><a href="/facts/Simple_harmonic_motion/u2cGAd91">Simple harmonic motion</a></li></ul>
<h2 id="references-and-notes">References and notes</h2>

<p>Related Reading:
</p>
<ul><li>Olenick, Richard P.; Apostol, Tom M.; Goodstein, David L. (2007). <a href="https://books.google.com/books?id=xMWwTpn53KsC&q=angular+frequency&pg=RA1-PA383"><i>The Mechanical Universe</i></a>. New York City: Cambridge University Press. pp. 383–385, 391–395. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-71592-8.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Cummings, Karen; Halliday, David (2007). Understanding physics. New Delhi: John Wiley & Sons, authorized reprint to Wiley – India. pp. 449, 484, 485, 487. ISBN 978-81-265-0882-2.(UP1) <a href="978-81-265-0882-2" target="_blank">978-81-265-0882-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"ISO 80000-3:2019 Quantities and units — Part 3: Space and time" (2 ed.). International Organization for Standardization. 2019. Retrieved 2019-10-23. [1] (11 pages) <a href="https://www.iso.org/standard/64974.html" target="_blank">https://www.iso.org/standard/64974.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Holzner, Steven (2006). Physics for Dummies. Hoboken, New Jersey: Wiley Publishing. pp. 201. ISBN 978-0-7645-5433-9. angular frequency. <a href="978-0-7645-5433-9" target="_blank">978-0-7645-5433-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Lerner, Lawrence S. (1996-01-01). Physics for scientists and engineers. Jones & Bartlett Learning. p. 145. ISBN 978-0-86720-479-7. <a href="978-0-86720-479-7" target="_blank">978-0-86720-479-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>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. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>"SI units need reform to avoid confusion". Editorial. Nature. 548 (7666): 135. 7 August 2011. doi:10.1038/548135b. PMID 28796224. <a href="https://doi.org/10.1038%2F548135b" target="_blank">https://doi.org/10.1038%2F548135b</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>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. <a href="978-0-534-46479-0" target="_blank">978-0-534-46479-0</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>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) <a href="0-07-139307-2" target="_blank">0-07-139307-2</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Angular frequency open-in-new

Angular frequency