Where Does the Centripetal Force Come From?

A.

"When an object is moving in uniform circular motion on the end of a rope, say, and in a vertical plane, where does the cetnripetal force come from? It has a centripetal force directed towards the center, and tension and gravity - do I add all these forces to get the resultant force on the body?"

When a body is undergoing uniform cetripetal (circular) motion, we know it must have the correct centripetal force acting on it - F=mv²/r towards the center. But this centripetal force is not an additional force that magically appears when the body starts going around in a circle. Instead, the centripetal force is just the resultant of the actual forces acting on the body. If the resultant doens't correspond to the required centripetal force, then the circular motion just doen't happen. Instead, so other motion will - it might be a different circular motion, it might be "circle like" (ie, repetitive, but not around an exact circle) or it might be something completely different.

In the example asked about, tension plus gravity have to give exactly the centripetal force - this is easy at the top and the bottom of the swing where they are parallel (or anti parallel), but when the body is out to the side, the tension needs to be directed away from the center slightly (or a lot) to account for the component of gravity that is not in the radial direction. Thus, if you were to try to spin a mass on a string in uniform circular motion, you would need to move your hand around in a circle about the center of rotation to make the tension pull in the required directions.

You probably do this unconsciously already - take a look next time you get the chance!