How much work is done by the centripetal force on an object moving in uniform circular motion?

For a particle in uniform circular motion,the acceleration $\vec{a}$ at a point $P(R, \theta)$ on the circle of radius $R$ is (Here $\theta$ is measured from the $x$-axis):

$A$ particle is performing $U.C.M.$ along the circumference of a circle of diameter $50 \ cm$ with a frequency of $2 \ Hz$. The acceleration of the particle in $m/s^2$ is:

$A$ body is whirled in a horizontal circle of radius $20 \,cm$. It has an angular velocity of $10 \,rad/s$. What is its linear velocity at any point on the circular path in $m/s$?

Two cars $A$ and $B$ are moving along concentric circular paths of radii $R_A$ and $R_B$. If both cars complete their respective circular paths in the same time,what is the ratio of the angular speed of $A$ to that of $B$?

What is uniform circular motion? By using a proper figure,derive the equation for centripetal acceleration $a_c = \frac{v^2}{r}$ for uniform circular motion. Show that its direction is towards the center.