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Rotational frequency

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From Wikipedia, the free encyclopedia

Number of rotations per unit time

Not to be confused withCircular motion.

"Rotation speed" redirects here. For the speed at which aircraft begin torotate, seeV speeds § Regulatory V-speeds.

Rotational frequency
Angular speedω (in radians per second), is greater than rotational frequencyν (inHz), by a factor of 2π.
Other names	rotational speed, rate of rotation
Common symbols	$\nu$ ,n
SI unit	Hz
Other units	rpm,cps
InSI base units	s⁻¹
Derivations from other quantities	ν=ω/(2π rad),n=dN/dt
Dimension	${\mathsf {T}}^{-1}$

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Rotational frequency, also known asrotational speed orrate of rotation (symbolsν, lowercase Greeknu, and alson), is thefrequency ofrotation of an objectaround an axis.ItsSI unit is thereciprocal seconds (s⁻¹); other commonunits of measurement include thehertz (Hz),cycles per second (cps), andrevolutions per minute (rpm).^[1]^[a]^[b]

Rotational frequency can be obtained dividingangular frequency, ω, by a fullturn (2π radians):ν=ω/(2π rad).It can also be formulated as theinstantaneous rate of change of thenumber of rotations,N, with respect to time,t:n=dN/dt (as perInternational System of Quantities).^[4]Similar to ordinaryperiod, the reciprocal of rotational frequency is therotation period orperiod of rotation,T=ν⁻¹=n⁻¹, with dimension of time (SI unitseconds).

Rotational velocity is thevector quantity whose magnitude equals thescalar rotational speed. In the special cases ofspin (around an axis internal to the body) andrevolution (external axis), the rotation speed may be calledspin speed andrevolution speed, respectively.

Rotational acceleration is the rate of change of rotational velocity; it has dimension of squared reciprocal time and SI units of squared reciprocal seconds (s⁻²); thus, it is a normalized version ofangular acceleration and it is analogous tochirpyness.

Related quantities

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Tangential speed $v {\displaystyle v}$ (Latin letterv), rotational frequency $\nu$ , andradial distance $r {\displaystyle r}$ , are related by the following equation:^[5] ${\begin{aligned}v&=2\pi r\nu \\v&=r\omega .\end{aligned}}$

An algebraic rearrangement of this equation allows us to solve for rotational frequency: ${\begin{aligned}\nu &=v/2\pi r\\\omega &=v/r.\end{aligned}}$

Thus, the tangential speed will be directly proportional to $r {\displaystyle r}$ when all parts of a system simultaneously have the same $\omega$ , as for a wheel, disk, or rigid wand. The direct proportionality of $v {\displaystyle v}$ to $r {\displaystyle r}$ is not valid for theplanets, because the planets have different rotational frequencies.^{[citation needed]}

Regression analysis

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Rotational frequency can measure, for example, how fast a motor is running.Rotational speed is sometimes used to meanangular frequency rather than the quantity defined in this article. Angular frequency gives the change inangle per time unit, which is given with the unitradian per second in the SI system. Since 2π radians or 360 degrees correspond to a cycle, we can convert angular frequency to rotational frequency by $\nu =\omega /2\pi ,$ where

$\nu \,$ is rotational frequency, with unit cycles per second
$\omega \,$ is angular frequency, with unit radian per second or degree per second

For example, astepper motor might rotate exactly once per second so that its angular frequency is 360degrees per second (360°/s), or 2πradians per second (2π rad/s), while the rotational frequency is 60 rpm.

Rotational frequency is not to be confused withtangential speed, 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.^{[citation needed]}

Notes

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^"The rotational frequencyn 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]
^"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]