You have two quarters touching each other on a tabletop. If you hold one quarter still and roll a second once around the circumference of the first quarter, how many revolutions will the second quarter make? Try to do this without actually doing it with quarters. You can test your answer once you're done.

Suppose the rolling coin were half the diameter of the coin being held still. How many revolutions would it make in one trip around the still coin?

Solution

The first inclination is to say that the rolling quarter will make one revolution since it has the same diameter as the still quarter. This is true in the frame of reference of the still quarter's circumference, but then you have to remember that this frame of reference also turns 360°. That means that the rolling quarter makes two full revolutions in your (sitting in your chair) frame of reference.

Now the outside coin will make two revolutions (because its circumference is half that of the still coin) plus one more revolution because its frame of reference is also moving.

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