Angular velocity

Angular velocity
Common symbols	ω
SI unit	rad⋅s−1
In SI base units	s−1
Extensive?	yes
Intensive?	yes (for rigid body only)
Conserved?	no
Behaviour under coord transformation	pseudovector
Derivations from other quantities	ω = dθ / dt
Dimension	${\displaystyle {\mathsf {T}}^{-1}}$

In kinematics, angular velocity (symbol ω or ⁠${\displaystyle {\vec {\omega }}}$⁠, 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 ${\displaystyle {\hat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}/\|{\boldsymbol {\omega }}\|}$ is normal to the instantaneous plane of rotation. The sense of angular velocity is conventionally specified by the right-hand rule, implying clockwise rotations (as viewed on 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, ${\displaystyle \omega =\|{\boldsymbol {\omega }}\|}$, 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. The SI unit of angular velocity is radians per second,. The radian is a dimensionless quantity, thus the SI units of angular velocity are dimensionally equivalent to reciprocal seconds, s⁻¹, although rad/s is preferable to avoid confusion with rotational velocity in units of hertz (also equivalent to s⁻¹).

For example, a geostationary satellite completes one orbit per sidereal day above the equator (approximately 360 degrees per 24 hours) has angular velocity magnitude (angular speed) ω = 360°/24 h = 15°/h (or 2π rad/24 h ≈ 0.26 rad/h) and angular velocity direction (a unit vector) parallel to Earth's rotation axis (⁠${\displaystyle {\hat {\omega }}={\hat {Z}}}$⁠, in the geocentric coordinate system). If angle is measured in radians, the linear velocity is the radius times the angular velocity, ⁠${\displaystyle v=r\omega }$⁠. With orbital radius 42000 km from the Earth's center, the satellite's tangential speed through space is thus v = 42000 km × 0.26/h ≈ 11000 km/h. The angular velocity is positive since the satellite travels prograde with the Earth's rotation (the same direction as the rotation of Earth).