Centripetal acceleration and force

Why there is an acceleration

Even though the magnitude of the velocity is constant in UCM, the velocity vector changes direction at every instant. Now, acceleration measures the change in the velocity vector (magnitude AND direction). There is therefore an acceleration, called centripetal acceleration, directed continuously towards the center of the circle.

Expression for centripetal acceleration

a = v^2 / R = omega^2 * R

This acceleration does not change the speed (in magnitude): it only continually deflects the trajectory to keep it on the circle.

Centripetal force

According to Newton's second law (F = m*a), for an object of mass m to follow a circle at constant speed, a resultant force, called the centripetal force, must be directed towards the center:

F = mv^2/R = momega^2*R

This force is not a new type of force: it is simply the name given to the resultant of the real forces (tension in a string, gravitation, reaction from the ground, etc.) when it produces circular motion.

Classic pitfall: the "centrifugal force" (which seems to push outward) does not exist in a Galilean (fixed) frame of reference. It only appears if one places oneself in the rotating frame of reference of the object itself; it is a so-called fictitious force, useful for calculations in that frame but absent if the motion is observed from outside, for example from the ground.