Centripetal acceleration ($\mathbf{a}=-{\omega }^2\mathbf{r}$)

For uniform circular motion of radius $R$ and angular speed $\omega$, the acceleration is directed toward the centre and has magnitude ${\omega }^{2}R=v^2/R$. Writing $\mathbf{r}$ for the position relative to the centre gives $\ddot{\mathbf{r}}=-{\omega }^2\mathbf{r}$. The required resultant force is $m\ddot{\mathbf{r}}$; it may be supplied by a normal force, gravity, or another physical interaction.

A centripetal force is the inward resultant force needed for circular motion. Newton's second law and centripetal acceleration give magnitude $m{v}^2/r$ for a particle of mass $m$, speed $v$, and path radius $r$. It is a role played by an actual interaction, such as tension or an electric force, rather than an additional interaction to add to a free-body diagram. A purely radial centripetal force is perpendicular to the velocity, so it does no work and leaves the speed unchanged.