Suppose I have a perfectly cylindrical pencil and set it in motion on a perfectly smooth and horizontal table, in such a way that the axis of the pencil is in translation.
Is the state of this object , after having been pushed by me, an example of inertial motion?
I mean can I predict that , in case the table were actually perfectly smooth, the pencil would continue to move in straight line with constant velocity ( under the hypothesis that the surface of the table has no bound)?
The problem I see is that the material points that do not belong to the axis of rotation are accelerated ( since they follow not a straight path, but a cycloidal one). But acceleration implies force, so inertia does not apply to these points, but only to those which are located on the axis.
Other problem, it seems that gravity does some work on the points that are not located on the axis. This work seems to be conpensated by the work done on each symmetric point, but is this enough to say that the sum of the forces acting in the direction of motion is equall to $0$?