Uniform circular motion.

Explanation

When a point object is moving on a circular path with a constant speed, it covers equal distances on the circumference of the circle in equal intervals of time. Then the object is said to be in uniform circular motion. This is shown in Figure

In uniform circular motion, the velocity is always changing but speed remains the same. Physically it implies that magnitude of velocity vector remains constant and only the direction changes continuously.
If the velocity changes in both speed and direction during the circular motion, we get non uniform circular motion.

As seen already, in uniform circular motion the velocity vector turns continuously without changing its magnitude (speed), as shown in Figure.
Note that the length of the velocity vector(blue) is not changed during the motion, implying that the speed remains constant. Even though the velocity is tangential at every point in the circle, the acceleration is acting towards the center of the circle. This is called centripetal acceleration. It always points towards towards the center of the circle. This is shown in the Figure.

The centripetal acceleration is derived from a simple geometrical relationship between position and velocity vectors

(i) Let the directions of position and velocity vectors shift through the same angle `theta` in a small interval of time `trianglet`.
(ii) For uniform circular motion, `r=|vecr_1|=|vecr_2|" and " v=|vecv_1|=|vecv_2|`. If the particle moves from position vector `vecr_1 " to " vecr_2`, the displacement is given by `triangle vecr = vecr_2 - vecr_1` and the change in velocity from `vecv_1" to "vecv_2` is given by `triangle vecv= vecv_2 - vecv_1`.
(iii) The magnitude of the displacement `triangle r" and of "triangle v` satisfy the following relation.
`(triangle r)/(r )= -v((triangle r)/(r ))`
Then, `a=(triangle v)/(triangle t)=(v)/(r )((triangle v)/(triangle t))= -(v^2)/(r )`
For uniform circular motion `v= omega r`, where `omega` is the angular velocity of the particle about the center. Then centipetal acceleration can be written as
`a= -omega^2 r`.