I am having trouble wrapping my head around a physics concept. If we roll solid sphere down an inclined plane it will reach the bottom at a different time than if we were to say, roll a hoop down the same inclined plane. This is because they have different rotational inertias, and so more of the potential energy will go into rotational energy. But doesn't this conflict with the law that the acceleration of the center of mass of a system of particles depends only on the net force acting on the system of particles, and therefore the center of mass of a solid sphere and a hoop should reach the bottom of the inclined plane at the same time? Another thing that is confusing me. In a physics class we were shown an example where a rod experienced an impulse either at the center of mass of the rod, or at the top end of the rod. The question was which will have more kinetic energy. The answer was that because the acceleration of the center of mass was the same for both, and the rod pushed at the top also had some rotational energy, the kinetic energy of the rod pushed at the top (and not directly in the center) was greater. The extra energy came from the fact that although the impulse was the same, the work done was not the same because the rod which was pushed from the top had its force act over a greater distance. But doesn't this above scenario seem to conflict with the first scenario presented? Would appreciate any help, Thank You.