newtonian mechanics - Does a body always rotate purely about its center of mass?

For example : A rod is held stationary in vertical position on a smooth horizontal ground and then released. Now the center of mass has velocity and acceleration and every point of the rod has some acceleration and velocity which are different from center of mass'. How can one prove that the rod is rotatating purely about centre of mass?

Answer

Does a body always rotate purely about its center of mass?

Well, that depends. The first assumption you need is that the body is rigid. Violate this assumption and all bets are off the table because you can't even necessarily classify all motions as "rotations": for example if a long thin board starts twisting sinusoidally into/out-of a helix shape, back and forth, is that vibration mode a "rotation" about some "axis?" Positive, or negative? And what axis?

The formal way to make an object rigid is to fix all of the distances which means any motion of the object is, technically speaking, an isometry. Isometries come in three flavors: an isometry is a composition of reflections, translations, and rotations. Reflections are not valid for motion because they are not "continuous" isometries. Every other motion can indeed be viewed as a translation of a point and a rotation about that point, for any point on (or off!) the object. So, choose the point to be the center of mass: then any motion must be a translation of the center of mass plus a rotation about that center of mass.

Notice that there's a lingering nasty problem: Suppose we have a ball on a tether, which we swing around our head. Assume that the ball is not spherically symmetric: then yeah, we can describe the ball's motion between two points as a translation followed by a rotation about the center of mass: however, this is not what we mean when we talk about what the ball is "rotating around", which is us, holding the tether.

For that we have to turn not just to the abstract mathematics of the groups $E^+(3)$ and $O(3)$ and $SO(3),$ but also the dynamics of the system.

In these cases we can say that if all forces are central (the system can be modeled as a bunch of point masses which have forces directed at each other, satisfying Newton's third law), then as long as there are no external forces, the dynamics decouples into a continuous translation of the center of mass in a straight line (the center of mass does not accelerate) and a continuous rotation about some axis $\vec \omega$.