Angular velocity

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Angular velocity describes the speed of rotation. The direction of the angular velocity vector will be along the axis of rotation, in this case (counter-clockwise rotation) toward the viewer.

In physics angular velocity is the speed at which something rotates (its angular speed or angular frequency) together with the direction it rotates in. The SI unit of angular velocity is radians per second. Angular velocity is related to rotational speed, which is measured in units such as revolutions per minute. Angular velocity is usually represented by the symbol omega (Ω or ω). The angular speed is denoted by ω.

1 Magnitude and direction
2 Circular motion
3 Non-circular motion in a plane
- 3.1 Derivation
4 See also
5 References

[edit] Magnitude and direction

Angular velocity is a vector quantity, meaning that it has both a magnitude (or length) and a direction. The magnitude is the angular speed, and the direction describes the axis of rotation.

In two dimensions, angular velocity is a one-dimensional vector. It can be represented by a scalar which is positive for an anticlockwise rotation and negative for a clockwise rotation.

In three dimensions, the angular velocity vector is three dimensional. The line of direction is given by the axis of rotation, and the right hand rule indicates the positive direction, 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.

Generally, to describe the angular velocity of a rotation in n-dimensional space requires a $\begin{matrix}{1\over2}\end{matrix} n(n-1)$ -dimensional vector. This number is the dimension (as a real vector space) of the Lie algebra of the Lie group of rotations of an n-dimensional inner product space. ^[1]

[edit] Circular motion

For any particle of a moving and spinning rigid body we have

$\mathbf{v} = \mathbf{v}_t + \boldsymbol\omega \times (\mathbf{r} - \mathbf{r}_c)$

where $\mathbf{v}$ is the total velocity of the particle, $\mathbf{v}_t$ the translational velocity of the body, $\mathbf{r}$ the position of the particle, and $\mathbf{r}_c$ the position of the center of the body.

To describe the motion the "center" can be any particle of the body or imaginary point that is rigidly connected to the body (the translation vector depends on the choice) but typically the center of mass is chosen, because it simplifies some formulas.

[edit] Non-circular motion in a plane

If the motion of a particle is contained in a plane and is described by a position vector-valued function r(t) — with respect to a fixed origin lying in that plane — then the angular velocity vector is

$\boldsymbol\omega = {\mathbf{r} \times \mathbf{v} \over |\mathbf{r}|^2} \qquad \qquad (1)$

where

$\mathbf{v}(t) = \mathbf{r'}(t)$

is the linear velocity vector. Equation (1) is applicable to non-circular motions, e.g. elliptic orbits.

[edit] Derivation

Vector v can be resolved into a pair of components: $\mathbf{v}_\perp$ which is perpendicular to r, and $\mathbf{v}_\|$ which is parallel to r. The motion of the parallel component is completely linear and produces no rotation of the particle (with regard to the origin), so for purposes of finding the angular velocity it can be ignored. The motion of the perpendicular component is completely circular, since it is perpendicular to the radial vector, just like any tangent to a point on a circle.

The perpendicular component has magnitude

$|\mathbf{v}_\perp| = {|\mathbf{r} \times \mathbf{v}| \over |\mathbf{r}|} \qquad \qquad (2)$

where the vector $\mathbf{r} \times \mathbf{v}$ represents the area of the parallelogram two of whose sides are the vectors r and v. Dividing this area by the magnitude of r yields the height of this parallelogram between r and the side of the parallelogram parallel to r. This height is equal to the component of v which is perpendicular to r.

In the case of pure circular motion, the angular velocity is equal to linear velocity divided by the radius. In the case of generalized motion, the linear velocity is replaced by its component perpendicular to r, viz.

$\omega = {|\mathbf{v}_\perp| \over |\mathbf{r}|} \qquad \qquad (3)$

therefore, putting equations (2) and (3) together yields

$\omega = {|\mathbf{r} \times \mathbf{v}| \over |\mathbf{r}|^2} = |\boldsymbol\omega|. \qquad \qquad (4)$

Equation (4) gives the magnitude of the angular velocity vector. The vector's direction — being orthogonal to both r and v — is given by its normalized version:

$\hat\boldsymbol\omega = {\mathbf{r} \times \mathbf{v} \over |\mathbf{r} \times \mathbf{v}|}. \qquad \qquad (5)$