[Physics] Can center of mass move without any force

Question

For instance, consider a weight on one end of the ring. Assume that the ring has negligible mass compared to the weight. When the weight splits into two, moves around the ring and recombines at the opposite end of the ring, is it true that the center of mass moves without the ring moving in opposite direction?

Answer 1

No. The center of mass will not move, the ring will.