Final answer:
The moment of inertia of a hoop is greater than that of a disk due to the hoop's more distributed mass. The moment of inertia of a spherical shell is greater than that of a solid sphere due to the shell's less concentrated mass.
Step-by-step explanation:
The moment of inertia of a hoop is greater than that of a disk with the same mass and radius because a hoop has more distributed mass compared to a disk. This means that the mass is spread out further from the axis of rotation, resulting in a larger moment of inertia.
Similarly, the moment of inertia of a spherical shell is greater than that of a solid sphere with the same mass and radius because the mass in a spherical shell is less concentrated towards the center compared to a solid sphere. The mass in a solid sphere is uniformly distributed, whereas the mass in a spherical shell is concentrated towards the outer surface, resulting in a larger moment of inertia.
The moment of inertia of an object depends on how its mass is distributed relative to the axis of rotation. For a hoop with mass M and radius R, all the mass is located at a distance R from the axis, so the moment of inertia is MR2. In contrast, a disk of the same mass and radius has its mass more evenly distributed, with a significant portion closer to the axis, leading to a smaller moment of inertia.
Similarly, for a spherical shell versus a solid sphere with identical mass M and radius R, the mass of the spherical shell is all at the surface, at distance R from the center. However, a solid sphere has mass throughout its volume, much of which is closer to the center, again resulting in a smaller moment of inertia for the solid sphere compared to the shell.