Final answer:
The moment of inertia of a hoop is greater than that of a disk with the same mass and radius because the hoop has all its mass at a distance R from the axis, leading to a larger value of MR² versus (1/2)MR² for the disk.
Step-by-step explanation:
Moment of Inertia of Hoop vs Disk
The question asks why the moment of inertia of a hoop with mass M and radius R is greater than that of a disk with the same mass and radius. The key concept here is how the mass distribution affects rotational inertia. Since the moment of inertia (I) depends on the distribution of mass around the axis of rotation, and is mathematically defined as I = Σmr², where m is the mass and r is the distance from the axis of rotation. For a hoop, all the mass is concentrated at a distance R from the center, giving it a moment of inertia of MR². On the other hand, a disk's mass is distributed from the center all the way out to its edge, resulting in a smaller average distance for each mass element to the axis, which gives a moment of inertia of (1/2)MR².
The correct answer is b) The disk has a higher mass concentration at the center, which results in a smaller average distance r for the mass elements to the axis of rotation, hence a smaller moment of inertia compared to the hoop where the entire mass is at radius R.