Consider a bead sliding without friction on a wire placed in a horizontal plane and forming a closed loop. The force acting on the bead is always perpendicular to the direction of its motion, to the right or to the left depending the direction the wire is curving. A force to the right or to the left, relative to the direction of the bead's motion, is defined as positive or negative, respectively. Let the mass of the particle be m and its (constant) velocity v, while the length of the wire loop is L.

(a) If the loop doesn't intersect itself, what is the time average of the perpendicular force, and how does it depend on the shape of the loop?

(b) What if the wire does cross itself? (This means that parts of the loop are slightly out of horizontal plane.)

This question was suggested by Y. Kantor. It was inspired by Newton's first attempt to calculate centripetal force as it appears in his "Waste Book" (dated 1664-1665).