Sorry for the vague title, but pls read my question below
Imagine a rigid body b with a point-mass tail t attached to its back at joint J. The tail has 1 DoF and can be actuated by a motor with axis of rotation M (parallel to yaw axis). The whole system is in free space (i.e no effect of gravity and air drag) and is in rest (i.e zero initial angular/linear momentum and no external force/torque acting).
Also, suppose that COM of the system is located at joint J and the tail is aligned with the roll axis of the body.
Edit: The COM of the body is assumed to be at b.
The initial configuration is shown below in the image :
Now, if the tail is rotated, the body will rotate in the opposite direction, since no external torque is being applied and the angular momentum will remain conserved (i.e zero). As there is no external force acting, the position of COM of the system will not change, i.e joint J will be static. This should look like as -
But then, wouldn't the new COM of the system will be at J' (as the COM of the system should be on the line joining the centre of mass of the body and tail). Isn't this incorrect?
My question is, why is this happening? How should the body and tail rotate so as to satisfy both the conservation of angular momentum (no external torque) and centre of mass (no external force).
If possible, answer with relevant equations. Also, pls explain with a diagram, showing the position of body, tail, joint J and the COM of the system assuming the same initial configuration.