Maxwell's Wheel: Energy Conservation in a Falling Yo-Yo

A yo-yo that fights its own moment of inertia

Maxwell’s wheel is a heavy disc mounted on a thin axle, with two strings wound around the axle and anchored above. Released from rest, it unwinds and descends, but far more slowly than free fall — and the reason is entirely about where the falling gravitational energy is allowed to go. Every joule of potential energy lost has to become kinetic energy, but here the disc can only move downward at the same rate the axle unwinds, which ties the linear speed to the spin rate through the axle radius r: v = ωr.

m g h  =  ½ m v²  +  ½ I ω²           energy conservation
v = ω r                                  rolling constraint (r = axle radius, small)
 
a = g / (1 + I / (m r²))                 resulting downward acceleration

Why a thin axle changes everything

Because the disc’s moment of inertia I is dominated by mass sitting far from the axis (I ≈ ½ M R² for a uniform disc of radius R), while the constraint uses a much smaller axle radius r, the ratio I / (m r²) can be enormous — tens or hundreds. Plugging that into the acceleration formula shows most of the falling energy is forced into spin rather than downward motion, so the wheel descends at a small fraction of g. The same physics, scaled up, is why a solid ball rolling down a ramp accelerates more slowly than a frictionless block sliding down the same ramp: some of the available energy has nowhere to go but into rotation.

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Bottoming out without stopping

When the string is fully unwound, the wheel is still spinning fast — it doesn’t stop, because there is nothing at that instant to remove its angular momentum. Instead the string begins winding back up in the opposite sense, and the same energy balance now runs in reverse: rotational and translational kinetic energy convert back into height as the wheel climbs. In an idealised, frictionless model with an inextensible string, the wheel returns to exactly its starting height, then falls again — a perpetual up-and-down oscillation, entirely powered by gravity and the interplay between linear and rotational inertia, with no external energy input required by the model.

Where the real device loses energy

Real wheels don’t oscillate forever. Air resistance drains a little kinetic energy on every pass, the string isn’t perfectly inextensible, and friction at the string-axle contact converts a small amount of mechanical energy into heat every time the wheel changes rotational direction at the bottom of its travel. Each bounce therefore reaches a slightly lower peak than the one before, an exponentially decaying envelope superimposed on the same energy-trading oscillation the ideal model predicts.

Frequently asked questions

Why does the wheel climb back up instead of just stopping at the bottom?

At the lowest point the wheel is still spinning — nothing removes its angular momentum. As the string winds back the other way, that rotational and translational kinetic energy convert back into height, so an ideal frictionless wheel returns to (very close to) its starting position and the cycle repeats.

Why does it fall so much slower than something in free fall?

Its downward speed is locked to its spin rate by the string, and its moment of inertia is set mostly by mass sitting far from the thin axle. That mismatch means most of the released gravitational energy is forced into spin rather than downward motion, cutting the effective acceleration to a small fraction of g.

What role does the axle radius specifically play?

The acceleration formula divides by 1 plus I/(m r²), where r is the axle radius. A thinner axle makes that ratio larger and the acceleration smaller — physically, a thin axle forces a large amount of spin for a small amount of descent, funnelling more energy into rotation.

Try it live

Everything above runs in your browser — open Maxwell's Wheel and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

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