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 pseudovector that is equal to the derivative, with respect to time, of angular position with respect to a fixed half-line (angular position also being a pseudovector, which is angular equivalent of position vector).

The magnitude of the pseudovector, ${\displaystyle \omega =\|{\boldsymbol {\omega }}\|}$, represents the angular speed (or angular frequency), the angular rate at which the object rotates (spins or revolves). The pseudovector direction ${\displaystyle {\hat {\boldsymbol {\omega }}}={\boldsymbol {\omega }}/\omega }$ is normal to the instantaneous plane of rotation or angular displacement.

There are two types of angular velocity:

• Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin.
• Spin angular velocity refers to how fast a rigid body rotates around a fixed axis of rotation, and is independent of the choice of origin, in contrast to orbital angular velocity.

Angular velocity has dimension of angle per unit time; this is analogous to linear velocity, with angle replacing distance, with time in common. The SI unit of angular velocity is radians per second, although degrees per second (°/s) is also common. 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).

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.

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).