The Center of Gravity (CG), more properly called the Center of Mass (COM), is a basic concept in classical mechanics. A lesser known and somewhat less basic concept is the Center of Percussion (CP). This is the momentary center of rotation after a kick, or impulse.
The usual examples are the hand reaction of a baseball bat or tennis racket, or of a hammer on an anvil. But the concept is also useful in understanding airplane dynamic modes.
The first animation shows a tumbling dumbbell. It is set in motion by an impulse. An impulse is a force of such short duration that the object receiving it does not have a chance to change its position during the "kick".
The impulse is in the horizontal direction, but it is not applied at the CG of the dumbbell. The first thing to notice is that the CG travels at a constant velocity in a horizontal straight line, i.e. in the direction of the impulse. A force anywhere on an object will always accelerate its CG, and after the impulse there are no further forces on the object.
The next thing to notice is that the dumbbell rotates at a constant rate. The impulse applies a momentary torque to the object, and this causes it to start rotating. After the impulse there are no further torques on the object, so it keeps rotating at a constant rate.
The initial impulse is applied directly to the top dumbbell weight, and not to the bottom one. The pulse will only accelerate the top mass, and not the bottom one.
Since both masses are equal and half that of the total dumbbell, it stands to reason that the top mass starts moving with twice the speed of the dumbbell's CG. The difference in the initial horizontal velocity between the two masses also immediately gives the rate of rotation.
After the impulse, the dumbbell translates and rotates at a constant rate. This makes the motion identical to that of a rolling wheel. The animation shows this imaginary wheel, running without slipping on an imaginary floor.
At any given time, the wheel rotates around the point of contact with the floor, as if it were momentarily nailed there. In kinematics, this point is called the instantaneous "pole" of the motion.
The location of the pole moves ( in a circle ) in the frame of reference translating and rotating with the wheel. It also moves ( in a straight line ) in the inertial frame of reference connected to the floor. These pole paths have nice names in kinematics ( "polhode" etc. ), and there is an enormous body of theory attached to it which is very important in designing linkages, and which we will not go into here.
The above is only true of course, if we take as an inertial reference the fixed world as it was before the dumbbell started moving. There will be those who say that the wheel is "really" rotating around its center of mass, and that it is somehow wrong to say that it rotates around some other "pole".
Taking the COM as the inertial reference is sometimes convenient, and sometimes not, but it is certainly not the "real truth". It would be silly to maintain that a pendulum rotates about its COM, instead of about its point of suspension. It makes the mathematics needlessly awkward.
In a pendulum, the motion is simple ( it has only one degree of freedom ), and the forces in the pivot point vary. This is the reverse of a free tumbling mass or gyroscope, where the forces are constant or zero, but the motions are complex ( like in precession and nutation ). Horses for courses.
The animation shows a little bicycle valve on the inside of the rim. There is a moment when the bicycle valve stands still, and the wheel rotates around it. Moments later, the wheel rotates around the mass that got the initial kick. If we block the motion of the upper mass at that moment in time by a reverse pulse, then the lower mass becomes the momentary center of percussion for that impulse.
We can stretch this moment in time indefinitely by blocking the upper mass with a stiff spring. The lower mass will stay still. It is a permanent center of percussion, with no force and no motion. It is like a pivot, but there is no need to fix it mechanically.
We will meet this "free floating" center of percussion in the motions of an airplane in some of its eigenmodes.