Angular velocity

In kinematics, angular velocity (symbol ω or ⁠

ω

→

{\displaystyle {\vec {\omega }}}

⁠, the lowercase Greek letter omega), also known as the angular frequency vector, is a three-dimensional Euclidean vector that uniquely identifies the plane, direction and angular speed of rotation of a particle rotating in a circle at constant speed in three dimensions.

The direction

ω

^

=

ω

/

‖

ω

‖

{\displaystyle {\hat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}/\|{\boldsymbol {\omega }}\|}

is normal to the instantaneous plane of rotation. The sense of angular velocity is conventionally specified by the right-hand rule, implying counterclockwise rotations (as viewed from "above" the plane of rotation); negation (multiplication by −1) leaves the magnitude unchanged but flips the axis in the opposite direction.

The magnitude of this vector,

ω

=

‖

ω

‖

{\displaystyle \omega =\|{\boldsymbol {\omega }}\|}

, represents the angular speed, the angular rate at which the object rotates (spins or revolves).

The angular velocity as given above for point particles, is called orbital angular velocity. A rigid body rotating about a fixed axis has each point of the body having the same orbital angular velocity. Hence such a rigid body can be given an angular velocity (called spin angular velocity) equal to the orbital angular velocity of each point in the body.

The angular velocity, as given above for rotation in a fixed circle at constant speed, can be generalized to more general motion in three dimensions. More specifically, given that the angular velocity of a particle rotating in a fixed circle in three dimensions at constant speed can be determined by its position with respect to the center of the circle and its velocity, the angular velocity of a particle whose position in three dimensions is twice-continuously differentiable with respect to time is determined in the same way by its position from the center of curvature and its velocity.

Angular velocity has dimension of per unit time.