Solution to the gear rotation question

This Meccano model has a closed loop of four different gears. No two of them have the same diameter, and no two have the same number of teeth. They turn smoothly. Any even number of gears, with parallel axles, or even smooth wheels, can be put into such a loop, and they will turn freely, no matter what their tooth count or size. There's a simple way to prove this, using elementary geometry.

The underlying principle isn't physics, but geometry, and it applies not only to gears, but to smooth friction wheels as well. The figure shows two friction wheels of different diameter. When they turn without slipping, the two circles must turn through the same arc, but in opposite sense of rotation. That is, if the left wheel turns clockwise through arc A, the right wheel turns counter-clockwise through arc B, and B = - A.

If these were gears, then if one gear turns through N teeth, the other gear turns through N teeth in the other direction. It follows that if you had a string of an odd number of gears and tried to make a loop by meshing the gears at the end of the loop, their points of contact would be moving in opposite directions. If you closed a loop of an even number of gears, they would all turn quite smoothly through the same arc.

There are even exotic gears that have non-circular perimteters, such as oval or elliptical. You could make a model with an even number of these in a loop, providing spring loaded axles so that the gears were continually meshed, and the gears in the model would turn smoothly. Some toy construction sets have gears that have the same diameter and the same tooth size, but differ in tooth count by just one tooth. It wouldn't matter in these models. Steel-Tech © sets had such gears, for what reason I have no idea.

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