What Rotational Inertia Measures

Rotational inertia, called I, is the rotational version of mass. Mass tells you how hard it is to change an object's velocity; I tells you how hard it is to change its angular velocity. But unlike mass, I is not just about how much stuff there is. It is about where that stuff is relative to the axis it spins around. For a point mass, I = m r², where r is the distance from the axis. For a collection of masses, you add up m r² for every piece: I = sum of m_i r_i^2.

That squared r is the whole story. Doubling the mass doubles I, but doubling the distance quadruples it. This is why a long pole spins easily about its center but feels sluggish about one end: around the center, every bit of mass is close to the axis, but around the end, mass that used to sit near the axis now sits far from it, and that distance gets squared. Spreading mass outward is a far more expensive way to resist rotation than piling it up near the axis.

Because I depends on where the axis is, not just on the object, a single rigid body has a different rotational inertia for every axis you could choose through it. A rod spun about its center and the same rod spun about its end are the same object, but different I values, because the distances r_i change when the axis moves. This sets up the next idea: once you know how I is built from mass distribution and axis choice, you can compute it properly by integration for continuous objects, then plug it into the rotational form of Newton's second law, torque = I alpha.