Physics

Physics

Problem

A solid disk of mass M and radius R can spin about its central axis. Case 1: a small mass m is rigidly attached to the rim. Case 2: the same mass m is attached halfway between the center and the rim. Determine which case results in the greater rotational inertia about the central axis and give one sentence explaining why.
Intuitive explanation

Rotational inertia depends on how mass is distributed relative to the axis—the farther from the axis, the harder it is to spin up. Since the moment of inertia is I=∑miri2I = \sum m_i r_i^2, the distance from the axis matters quadratically. Case 1 places all of mass mm at radius RR, while Case 2 places it at R/2R/2, so Case 1 has the greater rotational inertia because the mass is twice as far away, contributing four times as much to II.

Intuitive explanation

Rotational inertia depends on how mass is distributed relative to the axis—the farther from the axis, the harder it is to spin up. Since the moment of inertia is I=∑miri2I = \sum m_i r_i^2, the distance from the axis matters quadratically. Case 1 places all of mass mm at radius RR, while Case 2 places it at R/2R/2, so Case 1 has the greater rotational inertia because the mass is twice as far away, contributing four times as much to II.

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