A teaser. For the setup, go to the link.

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In each case the rolling coin has made one complete rotation. But the red arc at the top is half the length of the red line at the bottom. Why?

I have a more physics-y than a formal math-y explanation of why, which I will post soon.

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OK, here’s my answer.

In the rolling case, all you have is rotation. On rotation gives you 2*pi, so it rolls one circumference.

But in the other case you have rotation and revolution (spin and also orbital motion). Going halfway around the coin gives you an equal contribution of each, so the amount of spin only requires pi rotation, and it rolls half of the circumference. If the coin’s point of contact never changed, it would still do a rotation over the course of its revolution. If the orientation stayed fixed, the point of contact would make a complete trip around the coin.

A related example of this is the moon. If viewed from an external inertial frame (where the distant stars appear to be fixed), the moon rotates around the earth every ~4 weeks. But since it’s tidally locked and always has the same part facing the earth, it also rotates once about its axis.