Rotational Inertia of Rings, Disks and Shells

Moment of inertia measures how mass is spread out from a rotation axis, weighted by distance squared: I = integral of r² dm. For a thin hoop or ring, every single bit of mass is the same distance R from the central axis. That means r² is a constant, R², and pulls straight out of the integral. What's left is just the integral of dm, which is the total mass M. So I_hoop = MR², no calculus needed once you see why.

A disk is different because its mass is spread from the center out to R, so pieces at different radii contribute different amounts. The fix is to slice the disk into a series of thin concentric rings, each of radius r and thickness dr. Each ring has its own tiny mass dm, and since the disk has uniform surface density sigma = M/(π R²), the ring's mass is dm = sigma times its area, 2 π r dr. Plug that into dI = r² dm and integrate r from 0 to R. The r³ integral gives R⁴/4, and after substituting sigma back in, the R² terms cancel neatly to leave I_disk = (1/2) M R^2.

This ring-slicing method is the template for every solid shape in this unit: pick a slice where you already know the inertia (a ring, a rod), write its mass in terms of its geometry, and integrate. It also sets up why a solid sphere or cone needs slicing into disks instead of rings, and why a hoop, disk, and sphere all spinning with the same mass and radius land on different fractions of MR^2.