Write the equation for centripetal acceleration in uniform circular motion. Obtain these equations in terms of angular velocity $(\omega)$ and frequency $(\nu)$.

(N/A) In uniform circular motion,an object moves along a circular path of radius $R$ with constant speed $v$.
As the object moves from point $P$ to $P'$ in time interval $\Delta t$,the position vector turns through an angle $\Delta \theta$.
The angular velocity $\omega$ is defined as the rate of change of angular displacement: $\omega = \frac{\Delta \theta}{\Delta t}$.
Since the arc length $\Delta S = R \Delta \theta$,the linear speed is $v = \frac{\Delta S}{\Delta t} = R \frac{\Delta \theta}{\Delta t} = R \omega$.
The centripetal acceleration $a_c$ is given by $a_c = \frac{v^2}{R}$.
Substituting $v = R \omega$,we get $a_c = \frac{(R \omega)^2}{R} = R \omega^2$.
Since angular velocity $\omega = 2 \pi \nu$,where $\nu$ is the frequency,we can also express centripetal acceleration as:
$a_c = R (2 \pi \nu)^2 = 4 \pi^2 \nu^2 R$.