Talk:Angular velocity

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Confusing Tensor/Matrix multiplication notation! and a call to reorganisation[edit]

Can anyone explain to me what the centered dot being used in the following expression stands for:

This tensor W(t) will act as if it were a operator :
Given the orientation matrix A(t) of a frame, we can obtain its instant angular velocity tensor W as follows. We know that:

Not to mention:

Therefore is a rotation matrix and in a time dt is an infinitesimal rotation matrix. Therefore it can be expanded as

What really got me fired up was this bit:

Because of W is the derivative of an orthogonal transformation, the
bilinear form is skew-symmetric[disambiguation needed]. (Here stands for the scalar product).

I am inclined to think that outside of this last case, the centered dot either stands for matrix multiplication or for scalar multiplication. Whoever wrote these parts has no idea of the mathematical notation in use.

Moving to the big picture, I am in favor of a complete reorganisation and rewrite of the article. I humbly invite all who are interested to contact me on my talk page. Hayazin (talk) 22:33, 6 February 2015 (UTC)

Angular Velcity of a Particle[edit]

I don't have time to fix the article at the moment, but I just wanted to point out that it is fundamentally wrong to talk about a particle having angular velocity. Particles are infintesimally small and thus it is impossible for them to have an angular velocity. It is also very incorrect to declare that something has an angular velocity about a point. Angular velocity is point independent. The first section (that's all I've read) is propogating incorrect and misinformation. Any reputable dynamics text will verify this. Feel free to contact me for more details. Moorepants Jan 21, 2013 —Preceding undated comment added 07:00, 22 January 2013 (UTC)


I think the article should have something like the introduction I've given it. As I see it, Lie algebras provide the explanation of the reason the angular velocity vector behaves as it does. I wish I had found out about them a lot sooner. However, the definition I have given is not correct. I'm trying to think of the right words, but would anyone care to jump in and help?Buster79 09:00, 1 October 2005 (UTC)


I put a context tag on this page because the introduction launches right into formal mathematical functions without explaining what will happen in non-expert terms or mentioning (anywhere in the article, really) how angular velocity is used in the real world. --Craig Stuntz 20:25, 19 May 2006 (UTC)

Perhaps the introduction should just be scrapped (I won't be offended...). What about "Angular velocity is a quantity which represents the angular speed of a rotation together with its direction, that is, its axis and sense (whether the rotation is clockwise or anticlockwise).", something like the introductory sentence in velocity. Then jump in to the school textbook stuff about vectors. I had hoped my introduction would enlighten; from a certain viewpoint, it's more intuitive to derive it using a derivative than simply to make the definition that the direction of the vector gives the axis of rotation. Still, if it doesn't help, it doesn't help. As to 'used in the real world'? Yep. The article needs a simple example or two. Any physical situation that's at all complicated is probably better treated in angular momentum.Buster79 17:25, 21 May 2006 (UTC)

I mostly agree with this. I personally find the notion of using a derivative for velocity pretty intuitive, but I have no idea what a "special orthogonal linear transformation" or a "skew-adjoint linear transformation" is. Nothing wrong with very technical terms but I don't think they belong in the intro. I suspect a number of people -- middle schoolers and most high schoolers -- who might be interested in the article wouldn't know what a derivative is, either, though, so a "non-math" explanation is also necessary. --Craig Stuntz 02:30, 22 May 2006 (UTC)

Your current revision is much improved; thanks. I think some high schoolers (and below) would still have a problem understanding it, but it's at least understandable to non-mathematics specialists now. Good work, Buster79! --Craig Stuntz 14:19, 2 August 2006 (UTC)


In my opinion, revolutions per minute is not a unit of angular velocity or angular frequency, but of frequency. 1 rpm equals 60 Hz.-- 17:06, 20 November 2006 (UTC)

I think that's incorrect. All units used for vector quantities deal only with the magnitude of the measure in question, so "rpm" is a unit for both angular speed and angular velocity. 1 rpm is not the same as 1 Hz. The units of rpm are angle over time. The units of Hz are simply 1 over time, without reference to the angle. -- Slowmover 16:57, 22 November 2006 (UTC)
Where does "rpm" refer to angle? In my opiniion, "revolution" is not a measure of angle but is simply used to specify the kind of things that are counted. -- 17:07, 24 November 2006 (UTC)
One revolution is 360°, or 2π radians, which is a measurement of angle. It is not unit-less. So, 1 rpm is equal to 2π radians per minute, which is possibly easier to recognize as units of angle over time. 1 radian per minute is 1/(2π) rpm, and so on. It's important to see that a revolution is a divisible quantity, like meters or miles. It is measured quantity which can have any real number value, not just integers. -- Slowmover 22:09, 28 November 2006 (UTC)
One thing that just occurred to me is that this is really a semantic argument. The same thing applies to frequency. The units of angle are actually "dimension-less" units, whether radians or revolutions. Circular motion is inherent in the definiton of frequency (since each tick of a clock or the frequency of sound refers to a cycling through a full "revolution" of the available motion). Thus, it probably makes no difference if you talk of angular frequency or angular velocity, except for the vector component. -- Slowmover 22:20, 28 November 2006 (UTC)
Just have a look at the article revolutions per minute. There they say "Revolutions per minute (abbreviated rpm, RPM, r/min, or min−1) is a unit of frequency, commonly used to measure rotational speed, in particular in the case of rotation around a fixed axis." The article about rotational speed says: "Rotational speed tells how many complete rotations there are per time unit, and it is measured in revolutions per second (1/s or Hertz) in the SI System. " (Neither of the articles has been written by me.) -- 09:43, 2 December 2006 (UTC)
This whole issue of units of angular velocity and differences between angular velocity and frequency is a common problem (and angular frequency, which we define for talking about oscillations only adds to the confusion). First of all, although it's not how we normally think of it, a "revolution" is a unit of angle. This is increasingly stated explicitly, for example, in introductory physics textbooks (see, for example, R. D. Knight, Physics for Scientists and Engineers: A Strategic Approach, 2nd ed., (2008)). A revolution is just a unit of angle where once around the circle is defined as 1 unit (as opposed to 360 units for degrees, etc.). So, a frequency stated in revolutions per minute is really an angle per unit time. In this sense it is an angular velocity (rate of change of angle with respect to time). So, in a very real sense when refering to circular motion, frequency and angular velocity are the same thing. But in common usage by physicists (mathematicians may differ) the term "angular velocity" always refers to this quantity quoted in radians/second, and "frequency" is used for this same quantity quoted in Hz or rpm. When another "oddball" unit (e.g. degrees per second, gradians per fortnight...) is used then this is, or should be, explicitly defined. So, frequency and angular velocity are essentially the same thing, and the terms are used to clarify what units are being used. --GLeeDads (talk) 15:15, 12 November 2008 (UTC)

Non-circular motion[edit]

is not true in general but only if the motion of the partical is contained in a plane and if that plane contains the origin. In general, ω is neither orthogonal to the position vector r nor to the velocity vector v, so equation (5) is not true. -- 17:24, 20 November 2006 (UTC)

Time derivative of the anomalies[edit]

Are there three types of angular velocity? in motion on a ellipse, each corresponding to the time derivative of mean anomaly, true anomaly and eccentric anomaly?-- (talk) 12:39, 2 October 2012 (UTC)

Right hand rule/direction of angular velocity[edit]

While the article describes the direction of angular velocity as being the axis of spinning, it doesn't say anything about what this represents, and what difference it makes whether it is up/down. Perhaps a real world analogy like hula hooping could be used to describe the relevance of this vector. Richard001 06:35, 7 December 2006 (UTC)


I have resectioned things to make clear the differences between the angular velocity of a particle and the spin angular velocity of a rigid body. Almost all of the contents of the previous page have been included, except the "derivation" section has been shortened. PAR 02:39, 22 January 2007 (UTC)

Physics/Medicine Perspective[edit]

Medical Physics Query: I am not a physicist, just one very curious how the heart works so well for so long. Discovery of Angular Velocity in a living organism was an unexpected find in reading of other areas in this pursuit. 1. Application of living physiology to Angular Velocity is perhaps better appreciated in time phases of a coiled snake readying and striking prey or perceived danger. 2. The snake is a compliant rudimentary tubular organism comprised of many layers of neuromuscle, connective tissue, bone and gas. To most effectively strike prey or respond to threat, the snake recoils itself into the smallest sphere possible. 3. Recoil of the snake enables increasingly favorable numbers of interfaces between the physiologic layers to impart maximum angular velocity towards the target in coil. 4. Sensory input from the snake allows expedient recoil when prey is sighted and suggests this is by no means a passive/relaxation phase. 5. The smallest unit of the heart [muscle] is a [cell] called a [cardiomyocyte]. 6. [Cardiomyocyte]s are decidedly not [particle]s but tend to exhibit some astonishingly similar behavior around a long and short axis. 7. Theory in this area suggests divergent strings of living cardiomyocytes [myocardial muscle mass] grouped and electrically herded around a center liquid [mass] tidally pushed in and out. 8. String theory of myocardial tissue implies anchored and fixed vectors matched 180 degrees to the connective tissue of the heart, namely the heart valves, skeleton of the heart and central body of the heart. 9. Demand driven tidal forward and backward flow begins to describe the number of potential vectors arranged in superimposed angles in solid muscle mass opposed against a viscous, living blood mass. 10. Posit that the [heart/myocardium/cardiac muscle mass] rotates on a long axis clockwise and counterclockwise in layers of muscular sheets arranged one on another from the internal/blood exposed surfaces [endocardium] to the outer/[pericardial space] surfaces [epicardium]. 11. Posit that no single layer turns back or forth on the long axis more than 30 degrees, but all layers in summation turn at least 360 degrees in response to physiologic demand. 12. Thus understood, revolutions per minute around the long axis are readily posited as and extrapolated to time variables [heart rate]. 13. Posit the distal muscular layers of the heart fold and unspread inward to expel blood [systole/pump/ejection fraction= End Systolic Volume/End Diastolic Volume]triggered by sequential concurrent electrical polarization. 14. Volumetric derivation of Systole is perhaps best mathematically stated as Ejection Fraction [EF]. 15. EF is readily extrapolated to [Cardiac Output/CO], first defined by [Adolph Fick]. 16. Posit the proximal muscular layers of the heart unfold and spread outward to impel blood [diastole/sump/] as defined by [Arthur Guyton] guided by sequential electrical depolarization. 17. Posit that superimposed muscular layers engage one another on a plurality of angles around a well defined (mostly plasma) long axis suggesting shear and countershear of angular velocity. 18. Suggest most pragmatic (inexpensive and noninvasive) readily imaged solid long axis definition as Cardiac Apex (south myocardial pole)-Central Body of the Heart [collagen]-(Left AV collagen ring)/ (Right PV collagen ring). 19. Long axis performance physics of the heart are best and first attributed to Robert Hooke. 20. Short axis determinants are most expediently attributed to muscle mass with scarce influence of blood mass. Short axis performance physics of the heart are best and first attributed to Pierre LaPlace. 21. Posit that myocardial layering back and forth of available subsets of cardiomyocytes illustrates principles of shear and subsequent torque generation at work impelling and expelling blood in and out 60-80 times/minute for 60-80 years. 22. Crossing/uncrossing of multiple muscular bands suggests that Francisco Torrent-Guasp was looking ahead and further implies a force multiplier such as Angular Velocity at work within bands of a dedicated systolic heart and a dedicated diastolic heart.--Lbeben (talk) 03:00, 27 May 2008 (UTC)

Angular Frequency![edit]

The page describes angular frequency, not angular velocity!

Angular frequency has units Hz or s^-1 (or rad*s^-1 if you prefer) and is a measure of the rate of change of the angle. Angular velocity has units m^2*s^-1 and is the tangential velocity × radius. (× = cross product).

Just like angular momentum = tangential momentum × radius and torque (i.e. angular foce) = tangential force × radius.

On the angular momentum page, it says that angular momentum = mass * angular velocity (just like linear momentum = mass * linear velocity), so how can this be if the angular momentum page states that its units are kg m^2 s^-1, and this page says its units are s^-1 ?

-- (talk) 08:33, 5 June 2008 (UTC)

Er, the angular momentum page says = moment of inertia * ang.velocity. Moment of intertia has SI units kg m²; thus is consistent with angular velocity having units s-1. Springy Waterbuffalo (talk) 10:45, 5 June 2008 (UTC)
I've never heard of an angular velocity having units of m^2*s^-1. I'm not sure what the physical significance of such a quantity would be. An angular velocity is most definitely a rate of change of angle and so it should be in some set of units that say angle/time. As physicists normally define it an angular velocity is in rad/s, whereas the same quantity quoted in Hz, rpm, etc. would be called frequency. Your definition of it as is not correct. What is true is that , where r is a position vector with reference to an origin on the axis of rotation (the angular velocity is not uniquely defined since you need to specify what axis you are refering it to).
Angular frequency is something else again. In studying oscillatory motion we find ourselves writing down expressions for position as a function of time that look like where f is the frequency in oscillations per second. These expressions are a little awkward to work with and so it is convenient to define so that . We call this the angular frequency. We often imagine the oscillation as a "projection" of a circular motion onto a plane and this is some of what drives this tendency to treat oscillatory motion using similar language to what we use for circular motion. But this is an abstract construction which turns out to be convenient. —Preceding unsigned comment added by GLeeDads (talkcontribs) 15:53, 12 November 2008 (UTC)

Angular velocity a vector in two dimensions?[edit]

From the article, section Two dimensions: The equation

allows the angular velocity to be found.

... I'm a bit surprised in omega being written in bold (TeX \boldsymbol). As the article cross product states: «The cross product is not defined except in three-dimensions [...]» ? --Abdull (talk) 09:41, 28 June 2008 (UTC)

I removed this junk, but it does feel strange that I had to do it since I'm certainly no expert in the field. If somebody puts it back, please include a better explaination of what the cross product is supposed to mean in 2D and how the choice of axes would influence the angular velocity. - LM —Preceding unsigned comment added by (talk) 22:49, 27 October 2009 (UTC)


Therefore it is a skew symmetric 3x3 matrix. We can therefore take its dual to get a 3 dimensional vector. is called the angular velocity tensor. If we take the dual of this tensor, matrix multiplication is replaced by the cross product. ]...]

What are the dual of a matrix and the dual of a tensor? Thanks, --Abdull (talk) 10:04, 28 June 2008 (UTC)

In this case dual means a vector that represents the skew-symmetric matrix. Any skew symmetric matrix has this form:

The dual vector in this case is defined as (x,y,z). —Preceding unsigned comment added by (talk) 10:18, 20 February 2011 (UTC)

Relation to linear velocity[edit]

What is the relation of this quantity to peripheral speed on an elliptic trajectory? —Preceding unsigned comment added by (talk) 17:35, 2 September 2010 (UTC)

Pseudovectors are vectors[edit]

Just a comment about some recent changes about vectors and pseudovectors.

A vector is a member of a vector space, and the definition of a vector space is vector space definition. Therefore, as an algebraic structure, a pseudovector is a vector. Hence is not necessary to speak about "angular velocity pseudovectors". To say "angular velocity vector" is equally correct.

--Juansempere (talk) 17:45, 12 February 2011 (UTC)

Link loop in first line[edit]

In the beginning of the article, in the first line, the link to angular speed redirects to the angular velocity article (so it's a loop). Quite unhelpful, I'd take away the link but maybe someone has a better proposal for rewording. L.livnev (talk) 15:33, 19 June 2011 (UTC)

An editor changed the redirect recently without explanation from angular frequency, which is a far more appropriate target so I've change it back.--JohnBlackburnewordsdeeds 16:08, 19 June 2011 (UTC)

Inconsistency in the introduction[edit]

The angular velocity, as its name indicates, has to do with angles. Its magnitude is therefore measured in rad/s. But a few line after it is said, that "It is sometimes also called the rotational velocity and its magnitude the rotational speed, typically measured in cycles or rotations per unit time"... Stricly speeking, it can not be the same object. There is : the angular velocity, which magnitude is in rad/s, and another vector, the rotational speed, highly related to the first one but nevertheless different, with unit cycle/s. 2011.10.19 16:30

It is the same. E.g. a rotational speed of 10 cycles/second is the same as angular velocity wit 62.8 rad/s. It's no different to using inches, centimetres or miles to measure distance. It's still the same distance.
It's important as rotational speeds are often given in such units; "RPM" or revolutions per minute for example. But it's easier to do many calculations if rad/s are used instead.--JohnBlackburnewordsdeeds 15:07, 19 October 2011 (UTC)

unclear statement in first section[edit]

In the section "particle in 2 dimensions", it is stated :"However, it must be remembered that the velocity vector can be also decomposed into tangential and normal components." I don't understand to what it refers; tangential/normal to what ? —Preceding undated comment added 19:06, 19 October 2011 (UTC).

You're right: that normally is used for e.g. components relative to motion, but in such the perpendicular component of velocity is zero. I removed it then fixed a few other things in the language and presentation to hopefully good effect.--JohnBlackburnewordsdeeds 20:16, 19 October 2011 (UTC)
Ok. I will adapt it in the french translation i'm currently writing. By the way, for the radial component you write v in the text, but in the figure there is v. I think the latter is better, isn't it ? (talk) 16:13, 20 October 2011 (UTC)
No, it's ‖ in the diagram: it's right up against the right edge of the image so might not be obvious. I had to search for that in a Unicode table, which may be why it wasn't used before.--JohnBlackburnewordsdeeds 16:17, 20 October 2011 (UTC)
I'm confused.. i can't reproduce the "bug" On you page, the "‖" in index in the text appeared (on my screen at least) somehow like a single bar... i can't see it anymore.
There is a mistake in the first formula. It should be . Isn't it ? (talk) 16:37, 20 October 2011 (UTC)
In the last sentence in "dimension 2" : If the axes are reflected, but the sense of a rotation does not, then the sign of the angular velocity changes.. I think you meant "If the axes are swaped". Because one could understand "if each single axe is reflected (inversed)", which would leave the orientation of the plane unchanged, and thus no sign-change in the \omega. (talk) 17:11, 20 October 2011 (UTC)
On '‖' and '∥' I see '‖' both in the text ("radial component v") and in the diagram. The formula was a typo, fixed (feel free to fix them yourself; I do often make silly mistakes which I simply don't spot as I'm focussed on some other part of the text when I hit submit).
The last paragraph was rewritten as it was incorrect: it used to say " the x and y axes are exchanged (or inverted)", but if both axes are inverted it's a rotation through π, which definitely does not change the orientation (in 3D inverting all the axes does change the orientation, but not in 2D). I changed it to 'reflected' as that describes what happens: at least in 2D every improper rotation is a reflection. But I can see how it can be unclear so have changed it to 'parity inversion' as that's what's used at pseudoscalar, and added a link to further improve it.--JohnBlackburnewordsdeeds 18:32, 20 October 2011 (UTC)
i like the new version. :) (talk) —Preceding undated comment added 11:49, 21 October 2011 (UTC).

In the section "Particule in 3 dim", I don't understand the statement "The combination of the origin point and the perpendicular component of the velocity defines a plane of rotation". You meant probably the combination of the radius vector and the velocity vector. Didn't you ?

In the section "Addition of angular velocity vectors", in the line "An internal operation (addition) which is associative, commutative, distributive and with zero and unity elements", i think you meant "inverse elements" and not "unity elements". (talk) 14:51, 25 October 2011 (UTC)

In the section "Addition of angular velocity vectors", you say "The only property that presents difficulties to prove is the commutativity of the addition". But i think the associativity is not much easier to prove, since you have to consider 3 Frames... (talk) 14:51, 25 October 2011 (UTC)

In the section "Angular velocity vector for a frame", i think some words are missing in the sentence :"Any transversal section of a plane perpendicular to this axis has to behave as a two dimensional rotation". (talk) 14:51, 25 October 2011 (UTC)

In the section "Components from Euler angles", i don't understand the sentence "but now the velocity vector can be changed to the fixed frame..." I don't understand the formulation "can be changed to ..." (talk) 10:36, 30 October 2011 (UTC)


"To prove it we start tarivative of \mathcal{R}\mathcal{R}^t being R(t) a rotation matrix:" what is "tarivative"? — Preceding unsigned comment added by (talk) 13:15, 29 November 2012 (UTC)

vandalism... [1] --Steve (talk) 02:40, 30 November 2012 (UTC)

speed vs velocity[edit]

Why is the term 'velocity' used repeatedly instead of 'speed' when referring to a scalar e.g. "In two dimensions the angular velocity ω is given by" ... — Preceding unsigned comment added by (talk) 11:12, 23 April 2016 (UTC)

Assessment comment[edit]

The comment(s) below were originally left at Talk:Angular velocity/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

(Just a few comments, haven't got time for an in-depth review)
  • Needs more links to related concepts such as angular momentum and angular motion in general. A template grouping the most important concepts in rotational kinetics at the bottom would help too.
  • Real world examples of angular acceleration (I made the suggestion of hula hoops) would help, as well as some examples of fields where calculations using angular velocity are regularly used.
  • Mathematical section is covered quite well
    • A diagram showing velocity resolved into parallel and perpendicular components could be useful in the derivation section
Richard001 07:02, 7 December 2006 (UTC)

Last edited at 07:08, 7 December 2006 (UTC). Substituted at 07:50, 29 April 2016 (UTC)

Short description[edit]

The so-called short description of this article is currently: "physical quantity, the rate of change of angular position, and if regarded as a vector, the direction is the axis of rotation". This is 125 characters long — the recommended length is a maximum of 90 characters, and even that's pretty long (about 40 is recommended). The short description is not supposed to fully summarize the article, it's just meant as a concise subtitle to give context at a glance for an article, especially on mobile. I modified it to just "physical quantity" but that got reverted without any explanation. Based on this line, "The short description should focus on distinguishing the subject from similar ones rather than precisely defining it.", the edit I made seemed appropriate. Other similar articles like velocity and angular momentum aren't as wordy in their short descriptions, but those too could use some paring down. Thoughts? --CrocodilesAreForWimps (talk) 03:36, 16 September 2019 (UTC)

"Orders of magnitude (angular speed)" listed at Redirects for discussion[edit]


An editor has asked for a discussion to address the redirect Orders of magnitude (angular speed). Please participate in the redirect discussion if you wish to do so. signed, Rosguill talk 17:23, 8 April 2020 (UTC)

Rigid body rotating about point[edit]

Can anyone explain to me how a rigid body rotates about point? I think it is clear how a rigid body can rotate about an axis, but about a point? I believe all of this is nonsense. If it is not, then a clear picture of a rigid body rotating about point (but not about an axis) should be added. — Preceding unsigned comment added by (talk) 04:27, 21 June 2020 (UTC)