Where the energy in a spinning object is kept

Moment of inertia, and why mass at the rim counts for far more.

The energy in a spinning object is stored in every one of its moving parts, but not equally, since a piece of mass sitting far from the spin axis is moving much faster than an equal piece sitting close to it for the same rate of turn, and because kinetic energy grows with the square of speed, that outer piece is holding several times more of the total energy than the inner one despite weighing exactly the same.

The US Navy's Basic Machines and How They Work was written in 1971 to teach mechanics from scratch and is still the plainest explanation in print, and one of its diagrams, the same weight moved between a wheel's hub and its rim, was the first place the idea properly made sense.

The physics of moment of inertia

Every point on a spinning object travels around the same centre once per turn, but points further from that centre have further to travel to complete the same single turn, so they move faster than points close in, even though the whole object completes each rotation together. Kinetic energy depends on the square of speed, not speed itself, so a piece of mass moving twice as fast as another piece of the same weight is not carrying twice the energy, it is carrying four times as much. A spinning object's total stored energy is really the sum of every one of its pieces' individual energies, and because the outer pieces are moving fastest, they dominate that sum even when they make up only a modest share of the object's total weight, which is why where mass sits matters at least as much as how much of it there is.

The hammer comparison

Gripping a claw hammer near the very end of its handle and swinging it in a circle from the wrist takes a real, felt effort both to bring up to speed and to stop again, because the heavy steel head is travelling in the widest possible circle, moving fast for the modest turning speed of the wrist. Flip the grip around, holding the hammer near the head instead so the light end of the handle is what swings out furthest, and the same motion at the same wrist speed feels almost effortless to start and to stop, even though it is the same hammer with exactly the same total weight. Nothing about the hammer changed except where its mass sits relative to the pivot, and that alone accounts for the entire difference in how much effort the swing demands, whether starting it or stopping it again.

The one number worth remembering

Moving a given lump of mass from halfway out on a spinning wheel to the very rim, doubling its distance from the axle, does not merely double the energy it stores at a given spin speed, it roughly quadruples it, because that energy depends on the square of how fast the mass is travelling and doubling its distance from the centre doubles its speed for the same rate of turn.

What follows from this

A flywheel designed to store as much energy as possible for its weight puts almost all of its mass out at the rim rather than near the hub, often as a thick heavy ring connected to the axle by comparatively light spokes, because a given amount of material does dramatically more work storing energy out at the rim than the same material would do sitting close to the centre, doing little more than adding weight without adding much speed. The same reasoning explains why a bicycle wheel feels far harder to spin up when its weight sits mostly in a heavy tyre and rim than when the same total weight is concentrated closer to the hub, a difference riders notice in how the wheel accelerates long before anyone explains to them why, and a difference they feel again the moment they try to slow back down, since a wheel that was hard to spin up is equally hard to slow, for exactly the same reason.