--- title: "Rotational Kinematics and the Angular Velocity Vector" description: "Angular displacement, velocity and acceleration are defined by the same derivatives as their linear partners, and angular velocity points along the axis. Every point of a rigid body then has a linear " canonical: https://lightmysky.com/learn/science/rotational-kinematics-and-the-angular-velocity-vector-mt_2UqK-zViqf source: https://lightmysky.com/learn/science/rotational-kinematics-and-the-angular-velocity-vector-mt_2UqK-zViqf.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Rotational Kinematics and the Angular Velocity Vector Angular displacement, velocity and acceleration are defined by the same derivatives as their linear partners, and angular velocity points along the axis. Every point of a rigid body then has a linear speed set by its distance from that axis. Subject: Science · Area: Forces & Motion · Ages 20 to 21 Page: https://lightmysky.com/learn/science/rotational-kinematics-and-the-angular-velocity-vector-mt_2UqK-zViqf ## Ready when they can - Relates angular and linear speed, tangential acceleration and centripetal acceleration for a point on a rotating body - Uses the right-hand rule to give an angular velocity its direction - Solves constant angular acceleration problems with the rotational analogues of the constant-acceleration equations ## Lesson: How spin is measured Spinning bodies get their own vocabulary: angle replaces position, angular velocity replaces velocity, and angular acceleration replaces acceleration. Angular velocity is the rate of turn, omega equals d theta over d t, and angular acceleration is its rate of change, alpha equals d omega over d t. Both are signed components about the chosen fixed axis, with units radians per second and radians per second squared. Angular velocity is a vector along the axis: curl the fingers of your right hand with the spin and your thumb points along it. Flip the spin and the arrow flips too. Rotation maths needs radians, since arc length equals radius times angle only in radians. These formulas assume fixed-axis rotation, with r the perpendicular distance from the axis to the point. Steady spin-up behaves exactly like steady speeding up. With constant angular acceleration about a fixed axis, the rotational constant acceleration equations are omega equals omega zero plus alpha t, angle change equals omega zero t plus half alpha t squared, and omega squared equals omega zero squared plus 2 alpha times the angle change. The angle change is how much the body has turned, not the final orientation, which also needs the starting angle. Spin from rest, coast at a fixed rate, brake to a stop: each phase is one equation. *(drawing: Angle, angular velocity, and angular acceleration replace position, velocity, and acceleration.)* Every spinning point also has linear motion, with the perpendicular radius as the bridge. The linear velocity is omega cross r, where r runs from the axis to the point, so it points tangentially by the right-hand rule. Its size is radius times the magnitude of omega, so outer rim points race while hub points crawl. Two accelerations act, and they are perpendicular. The tangential part comes from changing the spin rate, with signed size radius times alpha, and the centripetal part comes from turning toward the centre, equal to radius times omega squared, always inward. Total acceleration is the root of the sum of their squares. **Example.** Every spinning point also has linear motion. Worked example: at radius 3 m with alpha 2 radians per second squared and omega 2 radians per second, the tangential part is radius times alpha, which is 6 metres per second squared, and the centripetal part is radius times omega squared, which is 12. The total is root of 36 plus 144, about 13.4 metres per second squared. **Recap.** Angle, omega and alpha mirror linear motion, radians rule the maths, and radius turns spin into speed and two accelerations. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Circular Motion: Angular Speed and Centripetal Force](https://lightmysky.com/learn/science/circular-motion-angular-speed-and-centripetal-force-mt_PW1G6nYStq) - [Radian Measure and Sectors](https://lightmysky.com/learn/mathematics/radian-measure-and-sectors-mt_SvAYAu6mQ4) ## Opens up - [Moment of Inertia by Integration and the Parallel-Axis Theorem](https://lightmysky.com/learn/science/moment-of-inertia-by-integration-and-the-parallel-axis-theorem-mt_VFRk1WIzAK)