Rotational inertia measures a rigid system's resistance to changes in its rotation — the rotational analog of ordinary mass, which measures resistance to changes in linear motion (Unit 2). But rotational inertia has an extra layer of complexity that ordinary mass doesn't: it depends not just on how much mass a system has, but on how that mass is distributed relative to the axis of rotation.
For a single point mass rotating a perpendicular distance r from an axis:
For a system made of several point masses, total rotational inertia is simply the sum of each individual contribution:
Explore this directly below — the exact three-point-mass system from this unit's header art.
Three point masses on a rotating disk, matching this unit's header art exactly. Adjust each mass and radius — I(total) = Σmᵢrᵢ² updates live.
Notice how much more sensitive I is to radius than to mass — doubling a mass doubles its contribution, but doubling its radius quadruples it.
Three point masses sit on a light rod: 2 kg at 0.3 m from the axis, 1 kg at 0.5 m, and 3 kg at 0.15 m. Find the system's total rotational inertia.
For a solid object made of continuous, spread-out mass rather than discrete points, the sum becomes an integral over every differential mass element dm:
Here, r is each differential mass element's own perpendicular distance from the axis of rotation — exactly the same idea as the discrete sum above, just handled with calculus instead of simple addition.
Derive the rotational inertia of a uniform rod of mass M and length L, rotating about an axis through one end, perpendicular to the rod.
This same integration process — done for different shapes — produces the standard rotational inertia formulas you'll see referenced throughout the rest of this unit. Compare several of them directly below, using the same mass and characteristic size for each.
Same mass, same characteristic size — four very different rotational inertias. This is the classic "why does a hoop resist spinning up more than a solid disk" comparison.
All four shapes share the same mass and characteristic radius/length — the hoop wins every time, since every bit of its mass sits at the maximum possible distance r = R from the axis, while the disk and sphere have mass spread closer to the center too.