Given the mass and the velocity of an object in circular motion with radius #r#, how do we calculate the magnitude of the centripetal force, #F#?

Acceleration is defined as the rate of change of velocity, and velocity has a direction.

Since the direction of something in circular motion is always changing, it requires a constant acceleration, and Newton's Second Law indicates that constant acceleration requires a constant force.

#F=ma#

If we look at the instantaneous linear velocity of the object at any moment as #v# #ms^-1#, the expression for the centripetal acceleration is:

#a=v^2/r#

So the force acting is simply #F=(mv^2)/r#

Let's look at where that expression for the acceleration comes from.

The speed of the object is constant: it is not accelerating because its speed increases (speeding up) or decreases (slowing down). It is accelerating because its direction is changing constantly.

If we imagine a direction vector, which is a tangent to the circle in which the object is moving, at each moment it points in a slightly different direction.

Acceleration is just change in velocity divided by change in time:

#a=(Delta v)/(Delta t)#

This Khan Academy video offers a more-detailed explanation of how we get from there to #a=v^2/r#: https://www.khanacademy.org/science/physics/centripetal-force-and-gravitation/centripetal-acceleration-tutoria/v/visual-understanding-of-centripetal-acceleration-formula

If, instead, we measure the radial velocity of the object in radians per second #(rads^-1)#, the expression is:

#F=m omega^2 r#