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.
What a learner can do afterwards
- 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
1 · Read
Spinning bodies get their own vocabulary: angle replaces position, angular velocity replaces velocity, and angular acceleration replaces acceleration. 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.
Steady spin-up behaves exactly like steady speeding up. With constant angular acceleration, angle and angular velocity follow the familiar constant acceleration equations with rotational symbols: omega equals omega zero plus alpha t, with a matching quadratic for angle. Spin from rest, coast at a fixed rate, brake to a stop: each phase is one equation.
Every spinning point also moves in a straight line sense, with radius as the bridge. Linear speed is radius times angular speed, so outer rim points race while hub points crawl: at radius 2 with omega 3, the speed is 6. Two accelerations act: tangential from changing the spin rate, equal to radius times alpha, and centripetal from turning toward the centre, equal to radius times omega squared and always inward.
Angle, omega and alpha mirror linear motion, radians rule the maths, and radius turns spin into speed and two accelerations.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.