Well, you mention centrifugal force, a force exerted radially outwards. But my understanding is that there is only centripetal force, Fc, a force directed toward the center, not away. Can you please explain the discrepancy here. My physics book does not define a centrigual force.

Well, this doesn't help. The problems I'm trying to solve are all viewed from a non-accelerated frame of reference. From the point of view of an observer outside the system. This is fairly basic stuff. So it's annoying to have to reinterpret your solution in the context of centripetal force. Please redraw the diagrams using centripetal rather than centrifugal forces.

OK, now I understand that Fc is not so much an independent force as it is the sum (at the top) or difference (at the bottom) of the normal and gravitational forces. good. thx for the extra diagram. One last question: if the pail is being swung so that it almost stops at the top (but doesn't actually stop) does that mean that there is virtually no tension in the string (and therefore virtually no Fn) and that the Fc is therefore almost entirely = to Fg?

right but I didn't intend for the pail to actually stop at the top, just slightly make it over the top and continue to swing. What are the force vectors in this case, where the pail just barely makes it over the top but continues swinging