Angular velocity

In physics, the angular velocity is defined as the rates of change of angular displacement and is a vector quantity (more precisely, a pseudovector) that specifies the angular speed (rotational speed) of an object and the axis about which the object is rotating. This speed can be measured in the SI unit of angular velocity, radians per second, or in terms of degrees per second, degrees per hour, etc. Angular velocity is usually represented by the symbol omega (ω, rarely Ω).

The direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction that is usually specified by the right-hand rule.

The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram (with angles ɸ and θ in radians), if a line is drawn from the origin (O) to the particle (P), then the velocity (v) of the particle has a component along the radius (radial component, v_‖) and a component perpendicular to the radius (cross-radial component, v_⊥). If there is no radial component, then the particle moves in a circle. On the other hand, if there is no cross-radial component, then the particle moves along a straight line from the origin.

A radial motion produces no change in the direction of the particle relative to the origin, so, for the purpose of finding the angular velocity, the radial component can be ignored. Therefore, the rotation is completely produced by the perpendicular motion around the origin, and the angular velocity is completely determined by this component.

In two dimensions, the angular velocity ω is given by

This is related to the cross-radial (tangential) velocity by:

An explicit formula for v_⊥ in terms of v and θ is:

Combining the above equations gives a formula for ω:

In two dimensions, the angular velocity is a single number with an orientation but no direction. The angular velocity in two dimensions is a pseudoscalar, a quantity that changes its sign under a parity inversion (for example if one of the axes is inverted or if axes are swapped). The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis. If the parity is inverted, but the orientation of a rotation is not, then the sign of the angular velocity changes.

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