If a yo-yo's string is not attached to anything and the yo-yo is dropped, it is obvious it will fall at g. In this scenario, Tension of string = 0.

If a yo-yo's string is attached to a cieling and it is dropped, what is R (path of center of mass)? Well, we know from experience that the yo-yo is going to fall at least, but is it going to fall at g? We know it is rotating as well. What would T (tension of string) be? Would T = yo-yo's mass * gravity?

Assume that there is no friction.

A related question is: if there is a stick in space and a force is applied to the center of mass, and for another stick in space an equivalent force is applied to the edge of the stick, will both sticks' center of mass move the same? Would the 2nd stick be rotating?