A coin rolling around a coin
A coin rolls without slipping around the rim of an identical coin that is held fixed, until it comes back to its starting position. How many full turns has the rolling coin made?
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Follow the centre of the rolling coin. What path does it trace?
Rolling without slipping on a curve adds the curvature of the path to the turning.
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Idea. Count the turns from the motion of the centre. The centre of the rolling coin travels a circle of radius , twice the length of the rim it rolls along, and the coin turns once for each length that its centre travels.
Step 1: the centre's path. The centres of the two coins stay apart, so the rolling coin's centre goes around a circle of radius , of length .
Step 2: the turns. Rolling without slipping, the coin turns by the distance its centre travels divided by its radius: , that is 2 full turns.
The trap. The rim of the fixed coin has length , equal to the rolling coin's circumference, which suggests 1 turn. The extra turn comes from going around the fixed coin: even sliding around it without rolling, always touching at the same point, the coin would turn once.
In general. Rolling around a coin of radius gives turns; rolling inside a ring gives .