A coin rolling around a coin

Question

Take two identical coins. Hold one fixed on the table. Roll the second coin around the edge of the fixed one, without slipping, until it returns to its starting position.

How many full rotations does the rolling coin make about its own center during one trip around? (It's not 1, that's the trap.)

Accepted Answer

The rolling coin makes two full rotations, not one. Why one seems right. The two coins have equal circumference, so as the rolling coin travels the distance equal to the fixed coin's circumference, its contact point rolls through exactly one circumference's worth of arc. That accounts for one rotation from rolling. It's tempting to stop there. The missing rotation. But the rolling coin isn't traveling in a straight line, its center travels around a circle (a circle of radius equal to two coin-radii). Going once around a closed loop adds one extra full turn to any object's orientation, purely …

A coin rolling around a coin

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