Angular velocity

Angular velocity is the vector physical quantity that represents the rotation of a spinning body. It is usually represented by the symbol 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 allow 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.

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.

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
:
where
:
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: which is perpendicular to r, and which is parallel to r. The motion of the parallel component is completely linear and produces no rotation of the particle (w.r.t. 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
:
where the vector 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.
:
therefore, putting equations (2) and (3) together yields
:

Equation (4) gives the magnitude of the angular velocity vector. The vector's direction is given by its normalized version:
:
Then the entire angular velocity vector is given by putting together its magnitude and its direction:
:
which, due to equations (4) and (5), is equal to
:
which was to be demonstrated.