Angular velocity
Angular velocity
- What it is: A measure of how fast an object rotates around an axis and in what direction. It tells you how quickly the angular position or orientation changes with time.
- Symbol and units: Represented by the vector ω (omega). The magnitude |ω| is the angular speed (how fast the rotation is), and the direction is perpendicular to the plane of rotation, given by the right-hand rule. SI unit is radians per second (rad/s). The radian is dimensionless, so angular velocity is effectively s−1; using rad/s helps avoid confusion with other rotation measures.
- 2D case (motion in a plane): Angular velocity is a signed scalar. Positive means counterclockwise rotation, negative means clockwise.
- 3D case (space): Angular velocity is a pseudovector. Its magnitude is the rate at which the orientation sweeps out angle, and its direction is along the axis perpendicular to the instantaneous plane of rotation. The right-hand rule fixes the orientation.
- Relation to linear velocity: For circular motion, the tangential (linear) speed v is related to angular velocity by v = r ω, where r is the radius. Equivalently, ω = v / r. If a point has radial and tangential velocity components, only the tangential component v⊥ contributes to the angular velocity: ω = v⊥ / r.
- How angular velocity is determined in general motion: In a plane, ω = dφ/dt, where φ is the angle of the radius vector from a fixed origin. The linear velocity can be split into radial and tangential parts, and the tangential part determines how fast the angle changes. In 3D, the instantaneous axis of rotation is the direction of ω, with the plane of rotation perpendicular to ω.
- Rigid bodies and spin: For a rigid body rotating about a fixed point, all parts rotate with the same angular velocity ω about a single axis (Euler’s rotation theorem). This ω is called the spin angular velocity. Its direction follows the right-hand rule: counterclockwise rotation around the axis means the vector points along the axis in the “upward” sense; clockwise rotation points in the opposite direction.
- Adding rotations: Angular velocities from successive rotations add together just like ordinary vectors for small rotations.
- Higher dimensions: In more than three spatial dimensions, angular velocity cannot be fully captured by a simple vector. It is described more generally by a skew-symmetric tensor, but many practical ideas (rates of turning, axes of rotation) still come from the vector view in three dimensions.
- Quick takeaways: Angular velocity tells you how fast something is turning and about which axis. Its magnitude is the angular speed, and its direction (in 3D) is the rotation axis given by the right-hand rule. For circular motion, it links directly to linear speed via v = r ω.
This page was last edited on 3 February 2026, at 13:17 (CET).