Suppose you put two weights at each end of a string or rope and then throw it away. As we all know the weights will circulate around the center of mass. Now the problem is what is the tension of the string. I am confused about this since there seem to be two contributions to the tension. First the first weight's centripetal force around the center of mass and secondly the second weight's centripetal force around the very same center of mass. But it turns out the two are equal and that you should only count one of them, not add both. My question is, what physical law says the forces should not be added? After all, at the center of mass, there will be two forces in each direction.