In physics, angular velocity (symbol ω or ⁠${\displaystyle {\vec {\omega }}}$⁠, the lowercase Greek letter omega), also known as the angular frequency vector, 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.

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:

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 rotation 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).

In the simplest case of circular motion at radius ⁠${\displaystyle r}$⁠, with position given by the angular displacement ${\displaystyle \phi (t)}$ from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: ⁠${\displaystyle \textstyle \omega ={\frac {d\phi }{dt}}}$⁠. If ${\displaystyle \phi }$ is measured in radians, the arc-length from the positive x-axis around the circle to the particle is ⁠${\displaystyle \ell =r\phi }$⁠, and the linear velocity is ⁠${\displaystyle \textstyle v(t)={\frac {d\ell }{dt}}=r\omega (t)}$⁠, so that ⁠${\displaystyle \textstyle \omega ={\frac {v}{r}}}$⁠.

In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin "sweeps out" angle. The diagram shows the position vector ${\displaystyle \mathbf {r} }$ from the origin ${\displaystyle O}$ to a particle ⁠${\displaystyle P}$⁠, with its polar coordinates ⁠${\displaystyle (r,\phi )}$⁠. (All variables are functions of time ⁠${\displaystyle t}$⁠.) The particle has linear velocity splitting as ⁠${\displaystyle \mathbf {v} =\mathbf {v} _{\Vert }+\mathbf {v} _{\perp }}$⁠, with the radial component ${\displaystyle \mathbf {v} _{\|}}$ parallel to the radius, and the cross-radial (or tangential) component ${\displaystyle \mathbf {v} _{\perp }}$ perpendicular to the radius. When there is no radial component, the particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity.