Suppose a ball moves horizontally to the right and hits a vertical stick at a position closer to its upper end. I understand intuitively, the stick will have linear and angular momentum, and that it will spin around its center of mass.
But why must the mass rotate around its center of mass? Why couldn't the stick simply move horizontally to the right (along with the ball) and not spin at all since the stick's center of mass isn't pinned to anything? It seems like there exists an imaginary pin at the center of mass causing it to rotate around that point regardless of where the ball hit.
And my second question is can the point of reference be chosen at a point other than the center of mass (but still be on the stick), and calculate the torque about that point and still be able to calculate the stick's motion after collision? I've thought about picking the point the ball collides with the stick as the point of reference. Then by the definition of torque, the torque will be 0, which implies there will be no rotation at all? Why is this reasoning incorrect?