“Write equation of centripetal acceleration for uniform circular motion. Obtain this equations in terms of angular velocity $(\omega )$ and frequency  $(v)$ .”

As the object moves from P to P' in time $\Delta t\left(=t^{\prime}-t\right)$, the line turns through an angle $\Delta \theta$ as shown in the figure.

$\Delta \theta$ is called angular distance.

The angular speed $\omega$ as the time rate of change of angular displacement.

$\therefore \omega=\frac{\Delta \theta}{\Delta t}$

Now, if the distance travelled by the object during the time $\Delta t$ is $\Delta \mathrm{S}$, i.e. PP' is $\Delta \mathrm{S}$, then

$\therefore v=\frac{\Delta \mathrm{S}}{\Delta t}$

$\therefore \Delta \mathrm{S}=\mathrm{R} \Delta \theta$

$v=\frac{\mathrm{R} \Delta \theta}{\Delta t}$

$\therefore v=\mathrm{R} \omega$

Centripetal acceleration $a_{\mathrm{C}}$

$a_{c}=\frac{(\mathrm{R} \omega)^{2}}{\mathrm{R}}=\frac{\mathrm{R}^{2} \omega^{2}}{\mathrm{R}}$

$\therefore a_{c}=\mathrm{R} \omega^{2}$