Editing Angular velocity (section)

== Orbital angular velocity of a point particle {{anchor|Orbital}} ==

=== Particle in two dimensions ===
[[Image:Angular velocity1.svg|class=skin-invert-image|right|256 px|thumb|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'''.]]

In the simplest case of circular motion at radius <math>r</math>, with position given by the angular displacement <math>\phi(t)</math> from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: <math display="inline">\omega = \frac{d\phi}{dt}</math>. If <math>\phi</math> is measured in [[radian]]s, the arc-length from the positive x-axis around the circle to the particle is <math>\ell=r\phi</math>, and the linear velocity is <math display="inline">v(t) = \frac{d\ell}{dt} = r\omega(t)</math>, so that <math display="inline">\omega = \frac{v}{r}</math>.

In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin "sweeps out" angle. The diagram shows the position vector <math>\mathbf{r}</math> from the origin <math>O</math> to a particle <math>P</math>, with its [[polar coordinates]] <math>(r, \phi)</math>. (All variables are functions of time <math>t</math>.) The particle has linear velocity splitting as <math>\mathbf{v} = \mathbf{v}_\|+\mathbf{v}_\perp</math>,  with the radial component <math>\mathbf{v}_\|</math> parallel to the radius, and the cross-radial (or tangential) component <math>\mathbf{v}_\perp</math> perpendicular to the radius. When there is no radial component, the particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity.

The angular velocity ''ω'' is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:

<math display=block qid=Q240105>\omega = \frac{d\phi}{dt} = \frac{v_\perp}{r}.</math>

Here the cross-radial speed <math>v_\perp</math> is the signed magnitude of <math>\mathbf{v}_\perp</math>, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity <math>\mathbf{v}</math> gives magnitude <math>v</math> (linear speed) and angle <math>\theta</math> relative to the radius vector; in these terms, <math>v_\perp = v\sin(\theta)</math>, so that

<math display=block qid=Q161635>\omega = \frac{v\sin(\theta)}{r}.</math>

These formulas may be derived doing <math>\mathbf{r}=(r\cos(\varphi),r\sin(\varphi))</math>, being <math>r</math> a function of the distance to the origin with respect to time, and <math>\varphi</math> a function of the angle between the vector and the x axis. Then:
<math display=block>\frac{d\mathbf{r}}{dt} = (\dot{r}\cos(\varphi) - r\dot{\varphi}\sin(\varphi), \dot{r}\sin(\varphi) + r\dot{\varphi}\cos(\varphi)),</math>
which is equal to:
<math display=block>\dot{r}(\cos(\varphi), \sin(\varphi)) + r\dot{\varphi}(-\sin(\varphi), \cos(\varphi)) = \dot{r}\hat{r} + r\dot{\varphi}\hat{\varphi}</math>
(see [[Unit vector]] in cylindrical coordinates). 

Knowing <math display="inline">\frac{d\mathbf{r}}{dt} = \mathbf{v}</math>, we conclude that the radial component of the velocity is given by <math>\dot{r}</math>, because <math>\hat{r}</math> is a radial unit vector; and the perpendicular component is given by <math>r\dot{\varphi}</math> because <math>\hat{\varphi}</math> is a perpendicular unit vector.

In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the radius vector turns counter-clockwise, and negative if clockwise. Angular velocity then may be termed a [[pseudoscalar]], a numerical quantity which changes sign under a [[parity (physics)|parity inversion]], such as inverting one axis or switching the two axes.

=== Particle in three dimensions ===
[[Image:Angular velocity.svg|class=skin-invert-image|thumb|250px|The orbital angular velocity vector encodes the time rate of change of angular position, as well as the instantaneous plane of angular displacement. In this case (counter-clockwise circular motion) the vector points up.]]

In  [[three-dimensional space]], we again have the position vector '''r''' of a moving particle. Here, orbital angular velocity is a [[pseudovector]] whose magnitude is the rate at which '''r''' sweeps out angle (in radians per unit of time), and whose direction is perpendicular to the instantaneous plane in which '''r''' sweeps out angle (i.e. the plane spanned by '''r''' and '''v'''). However, as there are ''two'' directions perpendicular to any plane, an additional condition is necessary to uniquely specify the direction of the angular velocity; conventionally, the [[right-hand rule]] is used.

Let the pseudovector <math>\mathbf{u}</math> be the unit vector perpendicular to the plane spanned by '''r''' and '''v''', so that the right-hand rule is satisfied (i.e. the instantaneous direction of angular displacement is counter-clockwise looking from the top of <math>\mathbf{u}</math>). Taking polar coordinates <math>(r,\phi)</math> in this plane, as in the two-dimensional case above, one may define the orbital angular velocity vector as:

: <math>\boldsymbol\omega =\omega \mathbf u = \frac{d\phi}{dt}\mathbf u=\frac{v \sin(\theta)}{r}\mathbf u,</math>

where ''θ'' is the angle between '''r''' and '''v'''. In terms of the cross product, this is:

: <math>\boldsymbol\omega
=\frac{\mathbf r\times\mathbf v}{r^2}.</math><ref>{{cite book |last1=Singh |first1=Sunil K. |title=Angular Velocity |url=https://cnx.org/contents/MymQBhVV@175.14:51fg7QFb@14/Angular-velocity |via=OpenStax |publisher=Rice University |access-date=21 May 2021 |ref=1}}</ref>

From the above equation, one can recover the tangential velocity as:

:<math>\mathbf{v}_{\perp} =\boldsymbol{\omega} \times\mathbf{r}</math>