A circle of radius 1 is inside a circle of radius 5. The smaller circle is rotated clockwise on the larger circle without slipping until it arrives back at its starting point P. How
many revolutions (turns) will the small circle make? Be careful. The answer is not 5.

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  4

Step-by-step explanation:

Consider the arcs PQ and P'Q. When turns are made without slipping, these have the same measure. The length of arc PQ is ...

  PQ = CP·θ

The length of arc P'Q is ...

  P'Q = O₂Q·(∠PO₂Q)

Segment O₂A is parallel to CP, so angle AO₂Q is θ. Then ∠PO₂Q = θ+α, where α is the angle by which P' has rotated counterclockwise from A.

We are given that ...

  CP = 5, O₂Q = 1

Using the above relations, we can write ...

  5θ = 1(θ+α)

  α = 4θ

That is, for every full turn θ = 2π, the smaller circle makes turns totalling 4(2π) = 8π, or 4 full turns.

The small circle makes 4 revolutions to arrive at its starting point.

each side has a different length.  

step-by-step explanation: