Final answer:
The fraction of a cylinder's kinetic energy at the bottom associated with rotation about an axis through its center of mass is 1/2, which means option B) 1/2 is the correct answer.
Step-by-step explanation:
When determining what fraction of a cylinder's kinetic energy at the bottom is associated with rotation about an axis through its center of mass, we refer to the distribution of energy between translational and rotational motion. For a solid cylinder rolling without slipping, the translational kinetic energy is given by ½mv2 and the rotational kinetic energy by ½Iω2, where I is the moment of inertia and ω (omega) is the angular speed. Since I for a solid cylinder is ½mr2 and ω can be replaced by v/r for rolling without slipping, we find that the rotational kinetic energy is ½ of the translational kinetic energy. Therefore, the fraction of the cylinder's total kinetic energy that is associated with rotation is 1/2 of the kinetic energy, making option B) 1/2 the correct answer.