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Consider a spring which is elongated in outer space and left to oscillate. As we know from intuition the spring will oscillate such that the middle point on it stands still. My question is:

Is this because the middle point is the center of mass?

At first I thought well of course but then I became doubtful. This is because from my understanding, the center of mass is not really a physical point on the spring - that is you can't take a spring and then point on the atom in the middle of it and say that it's the center of mass. The center of mass is a weighted average of all the position vectors of each atom in the spring. Thus it is a more abstract point which is associated with the coordinate frame in which you represent the spring, not the physical spring itself. Nonetheless you can actually point to a physical part of the spring and say: This stands still. So why is that?

I can try explain in another way: I understand the center of mass and its applications when dealing with point charges in vacuum. Then the center of mass is just an abstract mathematical point. But when you are dealing with a continuous body it suddenly seems to become a physical point on that body.