Relating arc lengths between a circle within a larger circle (no-slip)...

Hi, I recently had a vibrations problem in my college course that dealt with a small "pipe" inside a larger pipe. The pipe would roll within the larger pipe with no slip (see page 1 and page 2 of the solution).

Thus, a relationship could be "derived," if you will, between the arc lengths of the two surfaces.

I thought this was a simple ratio relationship (like gear ratios), in which the small pipe's rotation about the center of the large pipe would be $\displaystyle \theta$ and the small pipe's rotation about its own center would be $\displaystyle \theta_{S}$ for "small pipe."

The relationship I found was that $\displaystyle \theta_{S}=\frac{R}{R_1}\theta$, where $\displaystyle R_1$ is the small pipe's radius and $\displaystyle R$ is the large pipe's.

However! The book's solution (the one in the links) introduces a third angle and ends up saying that $\displaystyle \theta_{S}=\frac{R-R_1}{R_1}\theta$. Meaning, the arc length between the two pipes is related by $\displaystyle R_{1}\theta_{S}=(R-R_1)\theta$ whereas before it was $\displaystyle R_{1}\theta_{S}=R\theta$

Could anyone tell me why this is and what is wrong with assuming that there is a direct relationship between the product of both radii and angles?

Thanks!