A student investigates how the distribution of kinetic energy between translation and rotation depends on the geometry of a rigid object that rolls without slipping.
A solid uniform sphere, a solid uniform cylinder, and a thin-walled hollow sphere each have the same mass M and radius R. All three are set spinning about fixed horizontal axes through their centers on a laboratory table so that the speed of their centers of mass is v0 once rolling without slipping is achieved.
(a) Rank the three objects from greatest to least according to the fraction of their total kinetic energy that is in rotational form. Clearly indicate any ties and justify your ranking qualitatively in terms of the distribution of mass.
(b) For the solid uniform sphere, calculate the numerical value of the fraction of its total kinetic energy that is translational when it rolls without slipping.
(c) Describe how your answer to part (b) would change, if at all, for the hollow sphere and for the solid cylinder. Support your answer with reasoning that refers to each object’s mass distribution.