@elemtilas: It is a wheel. One half of the wheel has its normal weight, one half is 90% lighter. No inertia required; the heavy part produces a torque. (Edited to replaced "mill" with "wheel", to make it clearer.)

@elemtilas: Picture added. The left half is always heavier than the right half. The difference in weight produces a torque, which makes the wheel rotate. Do not ever allow the engine to run without a load!

Clever. The original post mentions the inverse square law, so the wheel would have to be big enough, though, for the gravity differential to be significant across both halves, possibly up to 10m radius, and heavy/massive enough to get useful output out of it. It also needs to be borne in mind that the energy isn't free; it is drawn from the heat energy stored in the Heavenium and would run down as the Heavenium becomes discharged. All of those things combined could limit its use as an aircraft engine.

Re: the diagram, you'd want the Heavenium as a vertical bar along the right-hand side of the wheel to minimize the field extending into the left hand side. Remember that the field fades as the inverse square of distance, so the field from the left end of the bar (as currently drawn) extends into the "heavy" side, reducing efficiency.

@GrumpyYoungMan: Mechanical engineers are welcome to draw a better picture. (This picture I remember from my early teens, when I discovered cavorite.)

@AlexP What would happen if you allowed the engine to run without a load?

@TheDaleks the wheel (a perpetual motion machine) without load would spin faster and faster (since Heavenium continues to draw energy into the system) until it tears itself apart spectacularly and everybody who survives is horribly splintered.

Ah, so the heavenium isn't actually on the wheel in any way. That explains!

All very interesting - perhaps 2 wheels spinning in opposite directions? Also I would assume that an unloaded "engine" would indeed run fast but frictional forces on the axel could prevent disaster (depending on the details)

@user535733 Yes you are correct definitely preferable and no doubt would be used, however given that this is pre-industrial some of the kit pressed into use might be a little rudimentary so I would expect a little bit of friction might be unavoidable.

@AlexP re "the society won't remain preindustrial for long". A good point, however heavenium is rare so by adjusting how rare it is I can adjust what it is used for. If its rarer than gold then it might only be used by the military. If it is as rare as silver then you are probably correct.

@Slarty one step farther: You might lend it to the army one year during a campaign, then rescind it and lend it to airship merchants another year to increase tax revenue. The definition of a 'sovereign' in that world might be "the person who owns all of the heavenium and decides whom to lend it to." Wars and empires may not be for control of land and resources...but for control of the one resource that leads to control over all the others.

Instead of a water wheel, how about a paddle wheel? The paddles extending over the heavenium are lighter than the ones on the other side. If they're long, you get a great moment-arm lever effect.

This is a clever idea, however, I am not certain that the math/physics of it works out. Despite the water being heavier on the left side than the right side, the mass, inertia and possibly pressure on both sides would still be the same and I’m not sure that there would be an actual flow of water. Consider for instance a similar water wheel with a magical material dividing it in half vertically, that permits solid paddles to pass through but not liquid water, with all of the water on the left. Now the left side is clearly heavier too, but with no water flow the wheel will not move.

@RBarryYoung: There is no water to flow. The second sentence in the answer. The wheel is dry. That's the whole point -- how make a water wheel rotate and generate power without water.

Ah, yes. My bad.

Even without the water, I need to think this through. I am not convinced that moving the paddles from normal gravity to lighter gravity would not count as “work” that would consume as much energy as would be gained by the converse.

@RBarryYoung: Maybe it would count. It doesn't matter. The point was how to make an engine powered by the heavenium. This is an engine powered by the heavenium.