# Angular velocity

File:Angularvelocity.png
Angular velocity describes the speed of rotation. The direction of the angular velocity vector will be along the axis of rotation and In this case (counter-clockwise rotation) toward the viewer.

The angular velocity of a point particle or rigid body describes the rate at which its orientation changes. It is analogous to translational velocity, and is defined in terms of the derivative of orientation with respect to time, just as translational velocity is the derivative of displacement with respect to time. It is customary to introduce the concept of velocity by first defining average velocity as displacement divided by time. There the analogy with angular velocity is less useful: for example, if a body is rotating at a constant angular velocity of one revolution per minute, then over a one-minute period the 'average angular velocity' of the body is zero, because the orientation is exactly the same at the beginning of the time period as it is at the end.

More precisely, if $\displaystyle A(t)$ is the special orthogonal linear transformation which describes the orientation, the angular velocity is defined as $\displaystyle A(t)^{-1}{d\over dt}A(t)$ . It follows that angular velocity is a skew-adjoint linear transformation . It is useful to restrict attention to two or three dimensions and represent the three-dimensional Lie algebra of skew-adjoint linear transformations of V$\displaystyle {}_3$ (R) by R³. The commutator operation, which is the Lie product of the algebra, is represented by the cross product in R³. The rest of this article is devoted to a discussion in that style.

## Vector angular velocity.

Angular velocity is the vector physical quantity that represents the process of rotation (change of orientation) that occurs at an instant of time. For a rigid body it supplements translational velocity of the center of mass to describe the full motion. It is usually represented by the symbol omega (Ω or ω). The magnitude of the angular velocity is the angular speed (or angular frequency) and is denoted by ω. The line of direction of the angular velocity 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.

In SI units, angular velocity is measured in radians per second, (rad/s), although a direction must also be given. The dimensions of angular velocity are T -1, since radians are dimensionless.

For any particle of a moving and spinning body we have

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

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

When the cross product is written in matrix form we have a skew-symmetric matrix with zeros on the main diagonal and plus and minus the components of the angular velocity as the other elements; see also above.

With constant angular acceleration, the angular velocity conforms to the rotational equations of motion, equivalent to the standard linear equations of motion under constant linear acceleration.

Angular frequency is also used instead of normal frequency in some situations that don't actually involve rotation especially in electronics as it makes the expression of sinusoids and various equations that are obtained by calculus on sinusoids simpler. (ωt rather than 2πft).

## The non-circular motion case

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

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

where

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

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

### Derivation

Vector v can be resolved into a pair of components: $\displaystyle \mathbf{v}_\perp$ which is perpendicular to r, and $\displaystyle \mathbf{v}_\|$ 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 is

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

where the vector $\displaystyle \mathbf{r} \times \mathbf{v}$ 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.

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

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

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

Equation (4) gives the magnitude of the angular velocity vector. The vector's direction is given by its normalized version:

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

Then the entire angular velocity vector is given by putting together its magnitude and its direction:

$\displaystyle \boldsymbol\omega = \omega \hat\boldsymbol\omega$

which, due to equations (4) and (5), is equal to

$\displaystyle \boldsymbol\omega = {\mathbf{r} \times \mathbf{v} \over |\mathbf{r}|^2},$

which was to be demonstrated.