Angular Displacement

For a particle moving on a circle of radius $r$ about a centre O, let its position be at A at time $t=0$ and at B at a later time $t$. The angle $\angle AOB=\theta$ swept out at the centre during this interval is called the angular displacement. It relates to the arc length $s=AB$ travelled along the circle by exactly the radian-measure relation of §2.11.4:

Angular displacement is measured in radians.

Angular velocity $\vec\omega$ is the rate of change of angular displacement:

with SI unit radian per second (rad s$^{-1}$). Angular velocity is a vector, directed along the axis of rotation, with its sense fixed by the right-hand rule (curl the fingers in the sense of the rotation; the thumb gives the direction of $\vec\omega$).

Angular acceleration $\vec\alpha$ is the rate of change of angular velocity, $\vec\alpha = d\vec\omega/dt$. It too is a vector, and — importantly — it need not point along the same direction as $\vec\omega$ itself (e.g. if only the magnitude of the angular velocity is changing while the axis stays fixed, $\vec\alpha$ is along the axis same as $\vec\omega$; but in general the two can differ in direction).

Linking angular and linear (tangential) quantities. In a small time $\Delta t$, an object moving on a circle of radius $r$ sweeps a small arc $\Delta s = r\Delta\theta$. Dividing by $\Delta t$ and taking the limit $\Delta t\to0$:

since $ds/dt$ is exactly the linear (tangential) speed $v$, directed tangent to the circle at the particle's location, while $\omega$ is the angular speed. (More generally, without restricting to circular motion, the vector relation is $\vec v = \vec\omega\times\vec r$; for circular motion $\vec\omega$ and $\vec r$ are perpendicular, so this reduces to the scalar form $v=r\omega$ above.)

Differentiating $v=r\omega$ with respect to time (with $r$ constant for motion on a fixed circle), …