Physics
Physics
Problem
An engineer is choosing among three designs for a circular turntable required to have mass and outer radius . In each design the material of the turntable is assumed to be of uniform density, and any mass removed from one region is relocated to another region so that total mass stays the same. Design 1 – a solid uniform disk. Design 2 – a thick rim: the inner half of the disk’s radius is removed, leaving only the outer half of the original material. Design 3 – an annular removal: the outer half of the disk’s radius is removed and that mass is compressed into a small solid cylinder located symmetrically at the center. A vertical axis passes through the geometric center and is perpendicular to the disk plane. a. Part 1: (a) Rank the three designs according to their rotational inertia about the stated axis, from greatest to least. b. Part 2: (b) For each pair of consecutive designs in your ranking, explain qualitatively why one has a larger rotational inertia than the other, referring specifically to mass distribution. c. Part 3: (c) Propose one additional qualitative modification (not listed above) that would further decrease the rotational inertia while keeping the total mass and outer radius unchanged, and justify how your modification achieves this.Intuitive explanation
Rotational inertia depends on how far mass sits from the rotation axis—material at larger radii contributes much more because . The key insight is that concentrating mass near the center minimizes rotational inertia, while spreading it outward maximizes it. You'll rank these designs by imagining where their mass lives: a solid disk has mass everywhere, a thick rim pushes mass outward, and a central cylinder pulls it inward. Think about which distribution makes the axis "easiest" to spin.
Intuitive explanation
Rotational inertia depends on how far mass sits from the rotation axis—material at larger radii contributes much more because . The key insight is that concentrating mass near the center minimizes rotational inertia, while spreading it outward maximizes it. You'll rank these designs by imagining where their mass lives: a solid disk has mass everywhere, a thick rim pushes mass outward, and a central cylinder pulls it inward. Think about which distribution makes the axis "easiest" to spin.