Every time I draw a free body diagram for something moving in a circle I get stuck on whether to add an inward arrow labelled centripetal force. Some of my notes draw it, others say it is already accounted for.
When I draw both the tension and a centripetal force I get an answer that is too big by a factor of two, so clearly one of them should not be there. What is the actual rule?
Accepted Answer
Centripetal is a job, not a force. Nothing is ever "the centripetal force" in the way something is a tension or a weight.
Draw only the real forces: tension, gravity, normal, friction. Then set the net inward component of those equal to mv²/r. That equation is Newton's second law applied in the radial direction, and mv²/r belongs on the ma side of it, never on the force side.
Your factor of two is exactly what happens when the tension is doing the job and you add an arrow for the job as well.
The word describes a direction. "Centripetal" means pointing at the centre, the same way "downward" means pointing down. A downward force is not a kind of force either.
One place people still get caught after learning the above: in a rotating frame you do add an outward term, and it is tempting to call that the missing arrow. It is not. That term appears because the frame is accelerating, and it is not a real force either.
Pick a frame and stay in it. Almost every sign error I have seen in this topic is an inertial diagram with a centrifugal term quietly added to it.
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