Dictionary - Definition of ANGULAR VELOCITY

Definition: ANGULAR VELOCITY


Part of Speech	Definition
Noun	1. (physics) the rate of change of the angular position of a rotating body; usually expressed in radians per second or radians per minute.[Wordnet].
Source: WordNet 3.0 Copyright © 2006 by Princeton University. All rights reserved.		Top

Extended Definition: ANGULAR VELOCITY

Angular velocity

Do not confuse with angular frequency

Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. The direction of the angular velocity vector will be along the axis of rotation; in this case (counter-clockwise rotation) the vector points toward the viewer.

In physics, the angular velocity is a vector quantity (more precisely, a pseudovector) which specifies the angular speed, and axis about which an object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, degrees per hour, etc. When measured in cycles or rotations per unit time (e.g. revolutions per minute), it is often called the rotational velocity and its magnitude the rotational speed. Angular velocity is usually represented by the symbol omega (Ω or ω). The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction which is usually specified by the right hand rule.

The angular velocity of a particle

Two dimensions

The angular velocity of the particle at P with respect to the origin O is determined by the perpendicular component of the velocity vector V .

The angular velocity of a particle in a 2-dimensional plane is the easiest to understand. As shown in the figure on the right (typically expressing the angular measures φ and θ in radians), if we draw a line from the origin (O) to the particle (P), then the velocity vector ( $\mathbf{v}$ ) of the particle will have a component along the radius ( $\mathrm{v}_\parallel\,$ - the radial component) and a component perpendicular to the radius ( $\mathrm{v}_\perp$ - the tangential component).

A radial motion produces no rotation of the particle (relative to the origin), so for purposes of finding the angular velocity the parallel (radial) component can be ignored. Therefore, the rotation is completely produced by the tangential motion (like that of a particle moving along a circumference), and the angular velocity is completely determined by the perpendicular (tangential) component.

It can be seen that the rate of change of the angular position of the particle is related to the tangential velocity by:

$\mathrm{v}_\perp=r\,\frac{d\phi}{dt}$

Utilizing θ, the angle between vectors $\mathrm{v}_\parallel\,$ and v, or equivalently as the angle between vectors r and v, gives:

$\mathrm{v}_\perp=|\mathrm{\mathbf{v}}|\,\sin(\theta)$

Combining the above two equations and defining the angular velocity as ω=dφ/dt yields:

$\omega=\frac{|\mathrm{\mathbf{v}}|\sin(\theta)}{|\mathrm{\mathbf{r}}|}$

In two dimensions the angular velocity is a single number which has no direction. A single number which has no direction is either a scalar or a pseudoscalar, the difference being that a scalar does not change its sign when the x and y axes are exchanged (or inverted), while a pseudoscalar does. The angle as well as the angular velocity is a pseudoscalar. The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis. If the axes are inverted, but the sense of a rotation does not, then the sign of the angle of rotation, and therefore the angular velocity as well, will change.

It is important to note that the pseudoscalar angular velocity of a particle depends upon the choice of the origin and upon the orientation of the coordinate axes.

The equation

$\boldsymbol\omega=\frac{\mathbf{r}\times\mathbf{v}}{|\mathrm{\mathbf{r}}|2}$

allows the angular velocity to be found.

Three dimensions

In three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in this case is generally thought of as a vector, or more precisely, a pseudovector. It now has not only a magnitude, but a direction as well. The magnitude is the angular speed, and the direction describes the axis of rotation. The right hand rule indicates the positive direction of the angular velocity pseudovector, namely:

If you curl the fingers of your right hand to follow the direction of the rotation, then the direction of the angular velocity vector is indicated by your right thumb.

Just as in the two dimensional case, a particle will have a component of its velocity along the radius from the origin to the particle, and another component perpendicular to that radius. The combination of the origin point and the perpendicular component of the velocity defines a plane of rotation in which the behavior of the particle (for that instant) appears just as it does in the two dimensional case. The axis of rotation is then a line perpendicular to this plane, and this axis defined the direction of the angular velocity pseudovector, while the magnitude is the same as the pseudoscalar value found in the 2-dimensional case. Define a unit vector $\hat{n}$ which points in the direction of the angular velocity pseudovector. The angular velocity may be written in a manner similar to that for two dimensions:

$\boldsymbol\omega=\frac{|\mathrm{\mathbf{v}}|\sin(\theta)}{|\mathrm{\mathbf{r}}|}\,\hat{n}$

which, by the definition of the cross product, can be written:

$\boldsymbol\omega=\frac{\mathbf{r}\times\mathbf{v}}{|\mathrm{\mathbf{r}}|2}$

Higher dimensions

In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor which is a second rank skew-symmetric tensor. This tensor will have n(n-1)/2 independent components and this number is the dimension of the Lie algebra of the Lie group of rotations of an n-dimensional inner product space. ^[1] It turns out that in three dimensional space angular velocity can be represented by vector because number of independent components is equal to number of dimensions of space.

Angular velocity of a rigid body

Position of point P located in the rigid body (shown in blue). R_i is the position with respect to the lab frame, centered at O and r_i is the position with respect to the rigid body frame, centered at O' . The origin of the rigid body frame is at vector position R from the lab frame.

In order to deal with the motion of a rigid body, it is best to consider a coordinate system that is fixed with respect to the rigid body, and to study the coordinate transformations between this coordinate and the fixed "laboratory" system. As shown in the figure on the right, the lab system's origin is at point O, the rigid body system origin is at O' and the vector from O to O' is R. A particle (i) in the rigid body is located at point P and the vector position of this particle is R_i in the lab frame, and at position r_i in the body frame. It is seen that the position of the particle can be written:

$\mathbf{R}_i=\mathbf{R}+\mathbf{r}_i$

The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time. This means that the length of the vector $\mathbf{r}_i$ is unchanging. By Euler's rotation theorem, we may replace the vector $\mathbf{r}_i$ with $\mathcal{R}\mathbf{r}_{io}$ where $\mathcal{R}$ is a rotation matrix and $\mathbf{r}_{io}$ is the position of the particle at some fixed point in time, say t=0. This replacement is useful, because now it is only the rotation matrix $\mathcal{R}$ which is changing in time and not the reference vector $\mathbf{r}_{io}$ , as the rigid body rotates about point O'. The position of the particle is now written as:

$\mathbf{R}_i=\mathbf{R}+\mathcal{R}\mathbf{r}_{io}$

Taking the time derivative yields the velocity of the particle:

$\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathbf{r}_{io}$

where V_i is the velocity of the particle (in the lab frame) and V is the velocity of O' (the origin of the rigid body frame). Since $\mathcal{R}$ is a rotation matrix its inverse is its transpose. So we substitute $\mathcal{I}=\mathcal{R}T\mathcal{R}$ :

$\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{I}\mathbf{r}_{io}$

$\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{R}T\mathcal{R}\mathbf{r}_{io}$

$\mathbf{V}_i=\mathbf{V}+\frac{d\mathcal{R}}{dt}\mathcal{R}T\mathbf{r}_{i}$

Continue by taking the time derivitve of $\mathcal{R}\mathcal{R}T$ :

$\mathcal{I}=\mathcal{R}\mathcal{R}T$

$0=\frac{d\mathcal{R}}{dt}\mathcal{R}T+\mathcal{R}\frac{d\mathcal{R}T}{dt}$

Applying the formula (AB)^T = B^TA^T:

$0=\frac{d\mathcal{R}}{dt}\mathcal{R}T+(\frac{d\mathcal{R}}{dt}\mathcal{R}T)T$

$\frac{d\mathcal{R}}{dt}\mathcal{R}T$ is the negative of its transpose. Therefore it is a skew symmetric 3x3 matrix. We can therefore take its dual to get a 3 dimensional vector. $\frac{d\mathcal{R}}{dt}\mathcal{R}T$ is called the angular velocity tensor. If we take the dual of this tensor, matrix multiplication is replaced by the cross product. Its dual is called the angular velocity pseudovector, ω.

$\boldsymbol\omega=[\omega_x,\omega_y,\omega_z]$

Substituting ω into the above velocity expression:

$\mathbf{V}_i=\mathbf{V}+\boldsymbol\omega\times\mathbf{r}_i.$

It can be seen that the velocity of a point in a rigid body can be divided into two terms - the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the "spin" angular velocity of the rigid body as opposed to the angular velocity of the reference point O' about the origin O.

It is an important point that the spin angular velocity of every particle in the rigid body is the same, and that the spin angular velocity is independent of the choice of the origin of the rigid body system or of the lab system. In other words, it is a physically real quantity which is a property of the rigid body, independent of one's choice of coordinate system. The angular velocity of the reference point about the origin of the lab frame will, however, depend on these choices of coordinate system. It is often convenient to choose the center of mass of the rigid body as the origin of the rigid body system, since a considerable mathematical simplification occurs in the expression for the angular momentum of the rigid body.

References

Rotations and Angular Momentum on the Classical Mechanics page of the website of John Baez, especially Questions 1 and 2.

Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 0-201-07392-7.

Landau, L.D.; Lifshitz, E.M. (1997). Mechanics. Butterworth-Heinemann. ISBN 0-750-62896-0.

Source: adapted by the editor from Wikipedia, the free encyclopedia; from the article "Angular velocity". Image Credit.
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Topics by Level of Interest: ANGULAR VELOCITY
Topics sorted by level of Interest	Level (1=low, 600=high)	Topics sorted Alphabetically	Level (1=low, 600=high)
Angular velocity	22	Angular velocity	22
Angular velocity tensor	7	Angular velocity tensor	7
Constant angular velocity	3	Constant angular velocity	3
Source: the editor, created by/for EVE to gauge likely levels of human interest in linguistically triggered topics (compiled across various sources, such as Wikipedia and specialty expression glosses).

Translations: ANGULAR VELOCITY

Language	Translations (or nearest inflections or synonyms, in parentheses)
Al Arabiya	السرعة الزاوية (angular velocity). Additional references: Al Arabiya, Saudi Arabia, Algeria, angular velocity. (volunteer & more translations)
Al Fus-Ha	السرعة الزاوية (angular velocity). Additional references: Al Fus-Ha, Saudi Arabia, Algeria, angular velocity. (volunteer & more translations)
Arabic	السرعة الزاوية (angular velocity). Additional references: Arabic, Saudi Arabia, Algeria, angular velocity. (volunteer & more translations)
Bahasa Malaysia	Halaju angular (Angular velocity). Additional references: Bahasa Malaysia, Malaysia, Brunei, angular velocity. (volunteer & more translations)
Bahasa Malayu	Halaju angular (Angular velocity). Additional references: Bahasa Malayu, Malaysia, Brunei, angular velocity. (volunteer & more translations)
Bohemian	úhlová rychlost (angular velocity). Additional references: Bohemian, Czech Republic, angular velocity. (volunteer & more translations)
Brazilian Portuguese	velocidade angular (angular frequency, angular rate, angular speed, angular velocity, radian frequency). Additional references: Brazilian Portuguese, Portugal, Angola, angular velocity. (volunteer & more translations)
Central Danish	vinkelhastighed (angular frequency, angular rate, angular speed, angular velocity, radian frequency). Additional references: Central Danish, Denmark, Germany, angular velocity. (volunteer & more translations)
Cestina	úhlová rychlost (angular velocity). Additional references: Cestina, Czech Republic, angular velocity. (volunteer & more translations)
Chinese Simplified	角速度 (angular velocity). Additional references: Chinese Simplified, China, Brunei, angular velocity. (volunteer & more translations)
Chinese Traditional	角速度 (angular velocity). Additional references: Chinese Traditional, China, Brunei, angular velocity. (volunteer & more translations)
Corse	pulsazione (pulsation, angular frequency, heartbeat, pulse, radian frequency), bàttitu (pulsation, angular frequency, heartbeat, pulse, radian frequency). Additional references: Corse, France, Italy, angular velocity. (volunteer & more translations)
Corsi	pulsazione (pulsation, angular frequency, heartbeat, pulse, radian frequency), bàttitu (pulsation, angular frequency, heartbeat, pulse, radian frequency). Additional references: Corsi, France, Italy, angular velocity. (volunteer & more translations)
Corsican	pulsazione (pulsation, angular frequency, heartbeat, pulse, radian frequency), bàttitu (pulsation, angular frequency, heartbeat, pulse, radian frequency). Additional references: Corsican, France, Italy, angular velocity. (volunteer & more translations)
Corso	pulsazione (pulsation, angular frequency, heartbeat, pulse, radian frequency), bàttitu (pulsation, angular frequency, heartbeat, pulse, radian frequency). Additional references: Corso, France, Italy, angular velocity. (volunteer & more translations)
Corsu	pulsazione (pulsation, angular frequency, heartbeat, pulse, radian frequency), bàttitu (pulsation, angular frequency, heartbeat, pulse, radian frequency). Additional references: Corsu, France, Italy, angular velocity. (volunteer & more translations)
Czech	úhlová rychlost (angular velocity). Additional references: Czech, Czech Republic, angular velocity. (volunteer & more translations)
Danish	vinkelhastighed (angular frequency, angular rate, angular speed, angular velocity, radian frequency). Additional references: Danish, Denmark, Germany, angular velocity. (volunteer & more translations)
Dansk	vinkelhastighed (angular frequency, angular rate, angular speed, angular velocity, radian frequency). Additional references: Dansk, Denmark, Germany, angular velocity. (volunteer & more translations)
Deutsch	Winkelgeschwindigkeit (angular velocity, angular speed), die Winkelgeschwindigkeit (angular rate, angular speed, angular velocity). Additional references: Deutsch, Germany, Austria, angular velocity. (volunteer & more translations)
Dutch	Draaisnelheid (Angular velocity), hoeksnelheid (angular frequency, angular rate, angular speed, angular velocity, radian frequency), hoekfrequentie (angular frequency, angular velocity, radian frequency). Additional references: Dutch, Netherlands, Aruba, angular velocity. (volunteer & more translations)
Finnish	kulmanopeus (angular velocity, angular speed). Additional references: Finnish, Finland, Russia (Europe), angular velocity. (volunteer & more translations)
Français	vitesse angulaire (angular velocity, angular frequency, angular rate, angular speed, radian frequency), pulsation (pulsation, heartbeat, angular frequency, beating, throb). Additional references: Français, France, Algeria, angular velocity. (volunteer & more translations)
French	vitesse angulaire (angular velocity, angular frequency, angular rate, angular speed, radian frequency), pulsation (pulsation, heartbeat, angular frequency, beating, throb). Additional references: French, France, Algeria, angular velocity. (volunteer & more translations)
Gaelg	bieauid uillinagh (angular velocity). Additional references: Gaelg, United Kingdom, angular velocity. (volunteer & more translations)
Gailck	bieauid uillinagh (angular velocity). Additional references: Gailck, United Kingdom, angular velocity. (volunteer & more translations)
German	Winkelgeschwindigkeit (angular velocity, angular speed), die Winkelgeschwindigkeit (angular rate, angular speed, angular velocity). Additional references: German, Germany, Austria, angular velocity. (volunteer & more translations)
Greek	κυκλική συχνότητα (angular frequency, angular velocity, radian frequency), γωνιακή συχνότητα (angular frequency, angular velocity, radian frequency), γωνιακή ταχύτητα (angular frequency, angular speed, angular velocity, radian frequency), μέτρο γωνιακής ταχύτητας (angular rate, angular speed, angular velocity). Additional references: Greek, Greece, Albania, angular velocity. (volunteer & more translations)
Greek (transliteration)	kikliki sikhnotita (angular frequency, angular velocity, radian frequency), goniaki sikhnotita (angular frequency, angular velocity, radian frequency), goniaki takhutita (angular frequency, angular speed, angular velocity, radian frequency), metro goniakis takhutitas (angular rate, angular speed, angular velocity). Additional references: Greek, Greece, Albania, angular velocity. (volunteer & more translations)
Hanguk Mal	각속도 (angular velocity). Additional references: Hanguk Mal, Korea, South, Korea, angular velocity. (volunteer & more translations)
Hanguohua	각속도 (angular velocity). Additional references: Hanguohua, Korea, South, Korea, angular velocity. (volunteer & more translations)
Hebrew	מהירות זוויתית (Angular velocity). Additional references: Hebrew, Israel, angular velocity. (volunteer & more translations)
High Arabic	السرعة الزاوية (angular velocity). Additional references: High Arabic, Saudi Arabia, Algeria, angular velocity. (volunteer & more translations)
High German	Winkelgeschwindigkeit (angular velocity, angular speed), die Winkelgeschwindigkeit (angular rate, angular speed, angular velocity). Additional references: High German, Germany, Austria, angular velocity. (volunteer & more translations)
Hochdeutsch	Winkelgeschwindigkeit (angular velocity, angular speed), die Winkelgeschwindigkeit (angular rate, angular speed, angular velocity). Additional references: Hochdeutsch, Germany, Austria, angular velocity. (volunteer & more translations)
Italian	velocita angolare (angular speed, angular velocity). Additional references: Italian, Italy, Croatia, angular velocity. (volunteer & more translations)
Ivrit	מהירות זוויתית (Angular velocity). Additional references: Ivrit, Israel, angular velocity. (volunteer & more translations)
Japanese	角速度 (angular velocity, angular), かくそくど (angular velocity). Additional references: Japanese, Japan, Taiwan, angular velocity. (volunteer & more translations)
Korean	각속도 (angular velocity). Additional references: Korean, Korea, South, Korea, angular velocity. (volunteer & more translations)
Malay	Halaju angular (Angular velocity). Additional references: Malay, Malaysia, Brunei, angular velocity. (volunteer & more translations)
Malayu	Halaju angular (Angular velocity). Additional references: Malayu, Malaysia, Brunei, angular velocity. (volunteer & more translations)
Manx	bieauid uillinagh (angular velocity). Additional references: Manx, United Kingdom, angular velocity. (volunteer & more translations)
Manx Gaelic	bieauid uillinagh (angular velocity). Additional references: Manx Gaelic, United Kingdom, angular velocity. (volunteer & more translations)
Melaju	Halaju angular (Angular velocity). Additional references: Melaju, Malaysia, Brunei, angular velocity. (volunteer & more translations)
Melayu	Halaju angular (Angular velocity). Additional references: Melayu, Malaysia, Brunei, angular velocity. (volunteer & more translations)
Portuguese	velocidade angular (angular frequency, angular rate, angular speed, angular velocity, radian frequency). Additional references: Portuguese, Portugal, Angola, angular velocity. (volunteer & more translations)
Ruotsi	vinkelhastighet (angular speed, angular velocity). Additional references: Ruotsi, Sweden, Finland, angular velocity. (volunteer & more translations)
Russian	угловая скорость (angular velocity, angular speed). Additional references: Russian, Russia, China, angular velocity. (volunteer & more translations)
Russian (transliteration)	uglovaya skorostʹ (angular velocity, angular speed). Additional references: Russian, Russia, China, angular velocity. (volunteer & more translations)
Russki	угловая скорость (angular velocity, angular speed). Additional references: Russki, Russia, China, angular velocity. (volunteer & more translations)
Russki (transliteration)	uglovaya skorostʹ (angular velocity, angular speed). Additional references: Russki, Russia, China, angular velocity. (volunteer & more translations)
Sjaelland	vinkelhastighed (angular frequency, angular rate, angular speed, angular velocity, radian frequency). Additional references: Sjaelland, Denmark, Germany, angular velocity. (volunteer & more translations)
Slovak	uhlová rýchlosť (angular speed, angular velocity), kruhová rýchlosť (angular velocity). Additional references: Slovak, Slovakia, Hungary, angular velocity. (volunteer & more translations)
Slovakian	uhlová rýchlosť (angular speed, angular velocity), kruhová rýchlosť (angular velocity). Additional references: Slovakian, Slovakia, Hungary, angular velocity. (volunteer & more translations)
Slovene	Kotna hitrost (Angular frequency, Angular velocity). Additional references: Slovene, Slovenia, Austria, angular velocity. (volunteer & more translations)
Slovenian	Kotna hitrost (Angular frequency, Angular velocity). Additional references: Slovenian, Slovenia, Austria, angular velocity. (volunteer & more translations)
Slovenscina	Kotna hitrost (Angular frequency, Angular velocity). Additional references: Slovenscina, Slovenia, Austria, angular velocity. (volunteer & more translations)
Spanish	velocidad angular (angle speed, angular frequency, angular velocity, angular rate, angular speed). Additional references: Spanish, Spain, Mexico, angular velocity. (volunteer & more translations)
Standard Malay	Halaju angular (Angular velocity). Additional references: Standard Malay, Malaysia, Brunei, angular velocity. (volunteer & more translations)
Suomea	kulmanopeus (angular velocity, angular speed). Additional references: Suomea, Finland, Russia (Europe), angular velocity. (volunteer & more translations)
Suomi	kulmanopeus (angular velocity, angular speed). Additional references: Suomi, Finland, Russia (Europe), angular velocity. (volunteer & more translations)
Svenska	vinkelhastighet (angular speed, angular velocity). Additional references: Svenska, Sweden, Finland, angular velocity. (volunteer & more translations)
Swedish	vinkelhastighet (angular speed, angular velocity). Additional references: Swedish, Sweden, Finland, angular velocity. (volunteer & more translations)
Turkish	açısal hız (angular velocity). Additional references: Turkish, Turkey, Bulgaria, angular velocity. (volunteer & more translations)
Source: Eve, based on a combination of meta analysis and graph theory (for near and back translations).		Top