Dynamics of Uniform Circular Motion and Centripetal Force

A body moving in a circular path at a constant speed is said to undergo uniform circular motion. Even though its speed never changes, such a body is not moving with constant velocity, because velocity is a vector and the direction of its motion is continuously changing as it goes around the circle -- and a changing velocity means the body has a nonzero acceleration, even at constant speed.

This acceleration, called the centripetal acceleration, can be shown (from the geometry of the changing velocity vector over a small time interval) to have magnitude

where $v$ is the (constant) speed, $r$ is the radius of the circular path, and $\omega = v/r$ is the angular speed; and, importantly, this acceleration is directed, at every instant, radially inward, straight toward the centre of the circle -- never along the direction of motion itself, and never outward.

By Newton's second law, $F = ma$, a body undergoing uniform circular motion must therefore be experiencing a net force of magnitude

$F_c = \frac{mv^2}{r} = m\omega^2 r$ …