Explanation of the Coin Rotation Paradox

A moving coin completes one full revolution after only going half the way around the stationary coin.

Start with two identical coins touching each other on a table, and oriented the same way.

Keep one coin stationary (the eagle) and rotate the other coin (the head) around the eagle coin, keeping a point of contact with no slippage. As the head coin reaches the opposite side, the two coins will again be oriented the same way; The head coin has made one full revolution, while it has only gone half the way around the eagle coin.

Explanation:

This apparent paradox is a useful example for explaining simple relativity.

Movement (in our case rotation) has meaning only when defined with respect to a system of reference.

We spontaneously tend to choose ourselves, the room, the table, our eyes, or the stationary coin, as the system of reference. This is case a), below. However, this is only one of the possible choices:

a) If the system of reference chosen is the eagle coin, then the head coin rotates a complete turn, from facing the eagle to facing away from the eagle.

b) If the system of reference chosen is the head coin, then the eagle coin rotates a complete revolution, from the left of the head coin, to its right, while it has only gone half the way around the head coin.

This can be better verified if you place the coins in a small table and rotate the head coin, while you are moving around the table one complete turn, synchronously with the head coin.

c) Finally, if the system of reference chosen is the contact point, then both coins rotate half a turn, both ending upside-down. In this case, if you keep rotating in synch with the contact point, then you will end on the other side of the table.

What remains constant, in all cases, is the apparent movement of the contact point: it moves along both coins along the top half of the edge of both coins (i.e. half the circumference of each coin).

If you continued the rotation for an additional equal amount, the contact point would reach the original position, having traveled one full circumference in both coins. At that point both coins would be in the original relative position.