Rotational Inertia of Common Shapes

Every uniform shape spinning about a natural axis has a rotational inertia of the form I = k m r^2. The mass m and radius r set the scale, but the coefficient k carries the actual shape information: it is a number between 0 and 1 that tells you how spread out the mass is relative to the axis. A point mass or thin hoop, where every bit of material sits at the same distance r, has k = 1, the maximum possible. Anything with mass closer to the axis has a smaller k, because that inner mass contributes less to resisting rotation.

Walking the shapes in order of how hollow they are: a thin hoop or ring has k = 1, since all its mass is at radius r. A solid disk has k = 1/2, because its mass is spread continuously from the center out to r, with plenty of mass close to the axis pulling the average down. A solid sphere has k = 2/5, even lower than the disk, since a sphere packs relatively more of its mass close to the center compared to a flat disk. A thin spherical shell has k = 2/3, higher than the solid sphere for the same reason the hoop beats the disk: pushing mass to the surface raises it.

This coefficient is what turns the generic rotational inertia idea from the previous lesson into something you can actually calculate for real objects, and it feeds directly into torque and angular acceleration problems where I shows up in tau = I alpha. Knowing the ranking of k values also lets you reason qualitatively, without recomputing anything, whenever a problem changes a shape's mass distribution.