Please refer to the figure below. The center of mass of the rod under the net force will accelerate. At the same time, the whole rod will also rotate around its center of mass. In his lectures, Feynman explained mathematically the reason why the center of mass translates according to Newton's law as well as why the rod also rotates around it, which I understand very well.
However, he didn't explain why the same rod (or any other object) will rotate around the fulcrum instead of its center of mass, as in the second case in the figure. My questions are:
Without the fulcrum, the center of mass is the center of rotation. What is the physical law/principle/reason for this no longer being the case when the rod is pivoted at the fulcrum? In other word, what makes the fulcrum the preferred center of rotation over the rod's center of mass? Please note that I am looking for a quantitative explanation.
Intuitively, I can see that if the rod is pivoted at two different points, it can neither rotate nor translate. However, again, what is the physical law for this behavior?