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	${\displaystyle {\mathsf {T}}^{-1}}$

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In physics, angular velocity (symbol ω or ${\displaystyle {\vec {\omega }}}$, the lowercase Greek letter omega), also known as angular frequency vector,^[1] is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates (spins or revolves) around an axis of rotation and how fast the axis itself changes direction.^[2]

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. ^{[citation needed]}
• Spin angular velocity refers to how fast a rigid body rotates with respect to its center 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,^[3] 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 rotation velocity in units of hertz (also equivalent to s⁻¹).^[4]

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.^[5]

For example, a geostationary satellite completes one orbit per day above the equator (360 degrees per 24 hours)^a 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, ${\displaystyle {\boldsymbol {v}}=r{\boldsymbol {\omega }}}$. With orbital radius 42,000 km from the Earth's center, the satellite's tangential speed through space is thus v = 42,000 km × 0.26/h ≈ 11,000 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).

^a Geosynchronous satellites actually orbit based on a sidereal day which is 23h 56m 04s, but 24h is assumed in this example for simplicity.