Rotational frequency

<h2 id="related-quantities">Related quantities</h2>
<p><a href="/facts/Tangential_speed/I3qgMPfs">Tangential speed</a> 
  
    
      
        v
      
    
    {\displaystyle v}
  
 (Latin letter <a href="/facts/V/41AF4mKp">v</a>), rotational frequency 
  
    
      
        ν
      
    
    {\displaystyle \nu }
  
, and <a href="/facts/Polar_coordinate_system/HBko4YoN">radial distance</a> 
  
    
      
        r
      
    
    {\displaystyle r}
  
, are related by the following equation:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>

v
              
              
                
                =
                2
                π
                r
                ν
              
            
            
              
                v
              
              
                
                =
                r
                ω
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}v&=2\pi r\nu \\v&=r\omega .\end{aligned}}}

</p><p>An algebraic rearrangement of this equation allows us to solve for rotational frequency:

ν
              
              
                
                =
                v
                
                  /
                
                2
                π
                r
              
            
            
              
                ω
              
              
                
                =
                v
                
                  /
                
                r
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\nu &=v/2\pi r\\\omega &=v/r.\end{aligned}}}

</p><p>Thus, the tangential speed will be directly proportional to 
  
    
      
        r
      
    
    {\displaystyle r}
  
 when all parts of a system simultaneously have the same 
  
    
      
        ω
      
    
    {\displaystyle \omega }
  
, as for a wheel, disk, or rigid wand. The direct proportionality of 
  
    
      
        v
      
    
    {\displaystyle v}
  
 to 
  
    
      
        r
      
    
    {\displaystyle r}
  
 is not valid for the <a href="/facts/Planet/rbPjE6Bx">planets</a>, because the planets have different rotational frequencies.
</p>
<h2 id="regression-analysis">Regression analysis</h2>
<p>Rotational frequency can measure, for example, how fast a motor is running. <i>Rotational speed</i> is sometimes used to mean <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> rather than the quantity defined in this article. Angular frequency gives the change in <a href="/facts/Angle/gFUTEQYq">angle</a> per time unit, which is given with the unit <a href="/facts/Radian_per_second/cG0plKHD">radian per second</a> in the SI system. Since 2π radians or 360 degrees correspond to a cycle, we can convert angular frequency to rotational frequency by

ν
        =
        ω
        
          /
        
        2
        π
        ,
      
    
    {\displaystyle \nu =\omega /2\pi ,}

where
</p>
<ul><li>
  
    
      
        ν
        
      
    
    {\displaystyle \nu \,}
  
 is rotational frequency, with unit cycles per second</li>
<li>
  
    
      
        ω
        
      
    
    {\displaystyle \omega \,}
  
 is angular frequency, with unit radian per second or degree per second</li></ul>
<p>For example, a <a href="/facts/Stepper_motor/sksWdPsj">stepper motor</a> might turn exactly one complete revolution each second.
Its angular frequency is 360 <a href="/facts/Degree_(angle)/f48Ns8SX">degrees</a> per second (360°/s), or 2π <a href="/facts/Radian/Srd38Sxa">radians</a> per second (2π rad/s), while the rotational frequency is 60 rpm.
</p><p>Rotational frequency is not to be confused with <a href="/facts/Speed/I3qgMPfs">tangential speed</a>, despite some relation between the two concepts. Imagine a merry-go-round with a constant rate of rotation.  No matter how close to or far from the axis of rotation you stand, your rotational frequency will remain constant. However, your tangential speed does not remain constant.  If you stand two meters from the axis of rotation, your tangential speed will be double the amount if you were standing only one meter from the axis of rotation.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Angular_velocity/2n8JljAT">Angular velocity</a></li>
<li><a href="/facts/Radial_velocity/B3NmbVLP">Radial velocity</a></li>
<li><a href="/facts/Rotation_period/vHcCt6cu">Rotation period</a></li>
<li><a href="/facts/Rotational_spectrum/QakXReKL">Rotational spectrum</a></li>
<li><a href="/facts/Tachometer/AwwaYKW9">Tachometer</a></li></ul>
<h2 id="notes">Notes</h2>

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

<ol>
<li id="fn:1"><p>Atkins, Tony; Escudier, Marcel (2013). A Dictionary of Mechanical Engineering. Oxford University Press. ISBN 9780199587438. <a href="9780199587438" target="_blank">9780199587438</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>"The rotational frequency n of a rotating body is defined to be the number of revolutions it makes in a time interval divided by that time interval [4: ISO 80000-3]. The SI unit of this quantity is thus the reciprocal second (s−1). However, as pointed out in Ref. [4: ISO 80000-3], the designations “revolutions per second” (r/s) and “revolutions per minute” (r/min) are widely used as units for rotational frequency in specifications on rotating machinery."[2] <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"The SI unit of frequency is hertz, the SI unit of angular velocity and angular frequency is radian per second, and the SI unit of activity is becquerel, implying counts per second. Although it is formally correct to write all three of these units as the reciprocal second, the use of the different names emphasizes the different nature of the quantities concerned. It is especially important to carefully distinguish frequencies from angular frequencies, because by definition their numerical values differ by a factor [see ISO 80000-3 for details] of 2π. Ignoring this fact may cause an error of 2π. Note that in some countries, frequency values are conventionally expressed using “cycle/s” or “cps” instead of the SI unit Hz, although “cycle” and “cps” are not units in the SI. Note also that it is common, although not recommended, to use the term frequency for quantities expressed in rad/s. Because of this, it is recommended that quantities called “frequency”, “angular frequency”, and “angular velocity” always be given explicit units of Hz or rad/s and not s−1."[3] <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>"ISO 80000-3:2019 Quantities and units — Part 3: Space and time" (2 ed.). International Organization for Standardization. 2019. Retrieved 2019-10-23. [2] (11 pages) <a href="https://www.iso.org/standard/64974.html" target="_blank">https://www.iso.org/standard/64974.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>"Rotational Quantities". <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html" target="_blank">http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Rotational frequency open-in-new

Rotational frequency