The Parallel Axis Theorem

Mass far from the spin axis resists angular acceleration more than mass close to it, because moment of inertia weights each bit of mass by the square of its distance from the axis. The center-of-mass axis is special: it's the axis about which mass is distributed as close as possible, so it gives the smallest possible moment of inertia for that shape. Move the axis anywhere else, parallel to the original, and every piece of mass is now farther out on average, so I can only go up.

The theorem makes that increase exact: I = I_cm + M d², where I_cm is the moment of inertia about the axis through the center of mass, M is the total mass, and d is the perpendicular distance between the new axis and the center-of-mass axis. You never need to re-integrate rho r² dV for the new axis. Instead you find I_cm once, usually from a table or a quick integral, then just add one term that depends only on the total mass and how far you shifted.

This is why Unit 5 builds moments of inertia by integration first, then hands you this shortcut: once you know I_cm for a rod, disk, or sphere, every off-center axis parallel to it is one algebra step away. It also feeds directly into torque and angular acceleration problems, since tau = I alpha needs the correct I for whatever axis the object actually rotates about, like a pivot at one end instead of the middle.