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. 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.
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