A particle starting from rest, moves in a circle of radius ' r '....

From third equation of motion for circular motion \omega^2-\omega_0^2=2 $\alpha \theta$where, $\omega=$ final angular velocity of particle $\omega_0=$ initial angular velocity $\alpha=$ angular acceleration and $\theta=$ angular displacement Here, $\omega=\frac{\mathrm{v}_0}{\mathrm{r}} \mathrm{rad} / \mathrm{s}$ (where, $\mathrm{r}$ radius of the circle) $\omega_0=0$ (initially particle is at rest) $\theta=2 \pi n$ (for $n$ rounds) Substituting these values in Eq. (i), we get \left(\frac{\mathrm{v}_0}{\mathrm{r}}\right)^2-0=2 $\alpha($2 $\pi \mathrm{n}) \Rightarrow \alpha=\frac{\mathrm{v}_0^2}{4 \pi \mathrm{nr}^2} \mathrm{rad}$/ \mathrm{s}^2