Moment of Inertia of Different Rigid Bodies

The two integration-based derivations of §5.4.1–§5.4.3 (rod, ring, disc), together with the two theorems of §5.4.5, are enough in principle to derive the moment of inertia of any standard regularly-shaped rigid body about any of its physically meaningful axes. Rather than repeat every such derivation, the results for eight of the most commonly encountered shapes — thin rod, thin rectangular sheet, thin ring, thin disc, thin hollow cylinder, uniform solid cylinder, thin hollow (spherical shell) sphere, and uniform solid sphere — are collected together in the master reference table below, for every geometrically natural axis of each shape (through the center, through one end, along a diameter, tangent to the surface, and so on).

A few patterns worth noting when using this table: for any given shape and mass, an axis passing through the center of mass always gives the smallest moment of inertia among all the parallel axes for that shape (a direct consequence of the parallel axis theorem, since $I=I_C+Md^2\ge I_C$ for any $d$, with equality only at $d=0$); and, comparing across different shapes of the same mass $M$ and outer radius $R$, moment of inertia increases as more of the mass sits farther from the axis — a solid sphere ($\tfrac25MR^2$) has less than a solid disc/cylinder ($\tfrac12MR^2$), which has less than a hollow (shell) sphere ($\tfrac23MR^2$), which has less than a ring/hollow cylinder ($MR^2$), for rotation about the natural central axis of each. This ta …