# Angular velocity **Angular velocity** is the rate at which an object's angular position changes with time, denoted by the Greek letter omega, \( \omega \). For a particle moving on a circular path, its magnitude is the time derivative of the rotation angle \( \theta \), and its SI unit is the [radian per second](https://www.edgepedia.org/radian-per-second) (rad/s)[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). As a vector, angular velocity points along the axis of rotation and encodes both how fast the body spins and in which sense, clockwise or counterclockwise, it spins about that axis[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). | Key fact | Detail | |---|---| | Definition | \( \omega = \dfrac{d\theta}{dt} \), the instantaneous time rate of change of the angle \( \theta \)[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables) | | SI unit | Radians per second (rad/s); 1 revolution per second equals \( 2\pi \) rad/s[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables) | | Vector direction | Along the axis of rotation, given by the right-hand rule: fingers curl in the direction of rotation, thumb points along \( \vec{\omega} \)[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables) | | Link to linear speed | \( v = r\omega \), so tangential speed grows with distance \( r \) from the axis at fixed \( \omega \)[2](https://openstax.org/books/college-physics/pages/6-1-rotation-angle-and-angular-velocity?modal=MH) | | Rigid-body property | Every point of a rotating rigid body shares the same spin angular velocity, independent of which points are chosen[3](https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Introductory_Dynamics%3A_2D_Kinematics_and_Kinetics_of_Point_Masses_and_Rigid_Bodies_%28Steeneken%29/03%3A_Rigid_Body_Dynamics/09%3A_Kinematics_of_Rigid_Bodies/9.03%3A_Velocities_in_a_rigid_body) | | Related quantity | Angular acceleration \( \alpha = d\omega/dt \), in rad/s², describes how \( \omega \) itself changes[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables) | ## Definition and units The average angular velocity over an interval is \( \bar{\omega} = \Delta\theta/\Delta t \), and the instantaneous angular velocity is the limit of this ratio as \( \Delta t \to 0 \)[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables): \[ \omega = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}. \] Because one complete revolution corresponds to \( 2\pi \) radians, a rotation rate quoted in revolutions per second must be multiplied by \( 2\pi \) to obtain radians per second; a rate in revolutions per minute (rpm) is converted by multiplying by \( 2\pi/60 \)[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables) • [4](https://www.phys.hawaii.edu/~morse/P170Fa15-14.pdf). The sign convention follows the angle: counterclockwise rotation is taken as positive and clockwise as negative[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). ## The angular velocity vector The scalar \( \omega = d\theta/dt \) gives only the rotation rate. The vector \( \vec{\omega} \) adds the axis and sense of rotation: it points along the instantaneous axis of rotation, with its direction fixed by the right-hand rule. If the fingers of the right hand wrap counterclockwise from the x-axis toward the y-axis, the thumb points along the positive z-axis, so a \( \vec{\omega} \) along +z corresponds to counterclockwise rotation and one along −z to clockwise rotation[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). This is useful for a spinning rigid body, where a reader wants to know both the axis and the direction of spin about it[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). **A vector with a caveat.** The instantaneous angular velocity can be treated as a vector even though a finite rotation generally cannot be represented by ordinary vector addition; a small rotation can be written as a vector of magnitude \( d\theta \) directed by the right-hand rule, and the angular velocity follows as the limit[5](https://physicslibrary.org/encyclopedia/AngularVelocityAndAngularAccelerationAsVectors.html). Consistently with this, the instantaneous \( \omega \) and \( \alpha \) can be represented by vectors, but their average values cannot[6](https://link.springer.com/chapter/10.1007/978-3-030-15195-9_7). ## Relation to linear velocity A point at distance \( r \) from the axis of rotation moves with tangential (linear) speed[2](https://openstax.org/books/college-physics/pages/6-1-rotation-angle-and-angular-velocity?modal=MH) \[ v = r\omega \qquad \text{or} \qquad \omega = \frac{v}{r}. \] The linear speed is proportional to the distance from the center of rotation, so it is largest for a point on the rim of a wheel[2](https://openstax.org/books/college-physics/pages/6-1-rotation-angle-and-angular-velocity?modal=MH). In vector form, the tangential velocity is the cross product of the angular velocity and the position vector, \( \vec{v} = \vec{\omega} \times \vec{r} \)[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). A car tire illustrates the relationship: the speed of the tread relative to the axle equals the car's forward speed \( v \), so the car moves at \( v = r\omega \) with \( r \) the tire radius. A faster car means a faster-spinning tire, and a larger-radius tire at the same \( \omega \) produces a greater forward speed[2](https://openstax.org/books/college-physics/pages/6-1-rotation-angle-and-angular-velocity?modal=MH). ## Rigid bodies versus orbiting points For a rigid body in pure rotation, every point traces a circular path in the same time, so the angular velocity is the same for all points and independent of which two points on the body are used to measure it; this quantity is sometimes called the spin angular velocity[3](https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Introductory_Dynamics%3A_2D_Kinematics_and_Kinetics_of_Point_Masses_and_Rigid_Bodies_%28Steeneken%29/03%3A_Rigid_Body_Dynamics/09%3A_Kinematics_of_Rigid_Bodies/9.03%3A_Velocities_in_a_rigid_body). It differs from the orbital angular velocity of a single point mass moving around an axis, which can depend on the position of the rotation axis or origin[3](https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Introductory_Dynamics%3A_2D_Kinematics_and_Kinetics_of_Point_Masses_and_Rigid_Bodies_%28Steeneken%29/03%3A_Rigid_Body_Dynamics/09%3A_Kinematics_of_Rigid_Bodies/9.03%3A_Velocities_in_a_rigid_body). ## Angular acceleration When \( \omega \) changes, as when a skater pulls in their arms and spins faster or a hard disk slows to a halt, the change is described by the angular acceleration \( \alpha \), the derivative of angular velocity with respect to time[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables): \[ \alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}, \] with units of rad/s²[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). The vector \( \vec{\alpha} \) points along +z when \( \vec{\omega} \) points along +z and \( d\omega/dt \) is positive, and along −z when \( d\omega/dt \) is negative[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). Differentiating \( \vec{v} = \vec{\omega} \times \vec{r} \) with constant \( r \) gives the tangential acceleration \( a_t = r\alpha \), and in vector form \( \vec{a} = \vec{\alpha} \times \vec{r} \)[1](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables). ## References 1. [10.1 Rotational Variables, University Physics Volume 1, OpenStax](https://openstax.org/books/university-physics-volume-1/pages/10-1-rotational-variables) 2. [6.1 Rotation Angle and Angular Velocity, College Physics, OpenStax](https://openstax.org/books/college-physics/pages/6-1-rotation-angle-and-angular-velocity?modal=MH) 3. [9.3: Velocities in a rigid body, Engineering LibreTexts](https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Introductory_Dynamics%3A_2D_Kinematics_and_Kinetics_of_Point_Masses_and_Rigid_Bodies_%28Steeneken%29/03%3A_Rigid_Body_Dynamics/09%3A_Kinematics_of_Rigid_Bodies/9.03%3A_Velocities_in_a_rigid_body) 4. [Physics 121C Mechanics course notes, University of Hawaii](https://www.phys.hawaii.edu/~morse/P170Fa15-14.pdf) 5. [Angular Velocity and Angular Acceleration as Vectors, Physics Library](https://physicslibrary.org/encyclopedia/AngularVelocityAndAngularAccelerationAsVectors.html) 6. [Rotation of Rigid Bodies, Springer Nature Link](https://link.springer.com/chapter/10.1007/978-3-030-15195-9_7) --- *Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Motion, forces, and dynamics › Rigid-body rotation › Angular kinematics* *Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026* *Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.* License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license