Now roll the wheel. Clearly, the drive shaft will roll but not enough to travel from R to S. The rest comes from slipping. Not sure how to correctly describe the direction of slippage – I’d call a “braking” slippage as the drive shaft or inner circle is “constrained” from rolling more rapidly – a kind of partly rolling skid.

Building on SteveT above, shrinking the inner circle/shaft reduces the amount of roll and increases the amount of slippage or skidding.

These are fun – I can think about them visually and physically but should learn some of the math.

]]>Imagine the two circles are gears, still firmly attached. For convenience, let the smaller circle be 1/2 the diameter of the larger (this argument will be similar for any ratio of diameters).

Now imagine two more gears, both the same size as the larger circle. One gear meshes with the larger circle, the other with the smaller. Obviously, they can’t both be on the same shaft, so they can rotate at different speeds.

Since the first gear is the same size as the larger circle, it rotates at the same speed as the larger circle.

But the second gear, meshing with the smaller circle, will be a “lower” gear, and will rotate at 1/2 the speed as the first gear.

So after one revolution of the two original circles, a point on the circumference of the first gear will have travelled the distance PQ. But a similar point on the second gear will have travelled 1/2 as far.

So, since the two circles are concentric, when the larger circle moves forward the distance PQ, so must the smaller circle. But only 1/2 that distance (in this example) is due to the smaller circle’s rolling, and the rest must result from slipping. Galileo is spoofing us.

]]>So, and correct me if I’m wrong, the problem with the argument lies in this part: Similarly, therefore, the distance RS must be the circumference of the small circle.

The phrasing makes it seem obvious, but the problem, I think, lies in the fact that the small circle is not the one rolling. The distance traveled depends solely upon the outer circumference and the velocity of the wheel.