Final answer:
The rotational inertia of the rod is greater when spun around the end because the mass is further from the rotation axis, significantly increasing the rotational inertia due to the square dependence of mass distribution distance. The hoop and spherical shell have greater rotational inertia than the disk and solid sphere respectively, due to the entirety of mass being located at the maximum distance from the center.
Step-by-step explanation:
The rotational inertia of a long rod spun around an axis through one end perpendicular to its length (ML²/3) is greater than the rotational inertia of the same rod spun around an axis through its center (ML²/12). This is because rotational inertia depends not just on the total mass but also on the distribution of that mass relative to the rotation axis. When the mass is distributed farther away from the axis, as in the case of the rod spun from one end, each mass element contributes more to the rotational inertia because the moment of inertia varies as the square of the distance to the rotation axis.
Comparing the rotational inertia of a hoop to a disk, or a spherical shell to a solid sphere, involves a similar principle. A hoop has all of its mass at the maximum radius from the center, leading to a larger moment of inertia (MR²) compared to a disk, which has more mass distributed closer to the center, thus giving it a smaller rotational inertia (½MR²). Similarly, a spherical shell's mass is all at the radius, thus having greater rotational inertia (¾MR²) than a solid sphere (½MR²).