Period, Radius Vector and Angular Speed

Consider an object of mass $m$ moving with uniform speed $v$ around a circle of radius $r$. The period $T$ is the time taken to complete one full revolution — equivalently, the time to travel a distance equal to the circle's circumference, $2\pi r$. Since speed is distance divided by time, $v = \dfrac{2\pi r}{T}$, i.e. $T = \dfrac{2\pi r}{v}$ (Eq. 3.48).

During circular motion, the object's position relative to the centre of the circle is described by its radius vector $\vec{r}$ — a vector of constant magnitude $r$, always pointing outward from the centre to the particle's current position. Because the object performs UCM, this radius vector sweeps out equal angles in equal time intervals, exactly analogous to how a rectilinearly-moving object covers equal distances in equal time intervals when its speed is constant. This motivates defining angular speed $\omega$ as the angle swept by the radius vector per unit time — the rotational counterpart of ordinary (linear) speed. …