A small ring of negligible thickness and radius 2 cm moves on a bigger rung of radius 10 cm. How many rotations will the small ring take on the bigger ring to make a complete round?

### Concept & Strategy When a circle of radius $r$ rolls without slipping around the outside of a fixed circular path of radius $R$, the center of the rolling circle traces a circle of radius $(R + r)$. $$\text{Number of rotations} = \frac{R + r}{r} = \frac{R}{r} + 1$$ *(Note: The extra $1$ rotation accounts for the full $360^\circ$ rotation acquired purely by orbiting the larger center, known as the Coin Rotation Paradox.)* ### Step-by-Step Solution * **Given:** * Radius of fixed rung ($R$) = $10\text{ cm}$ * Radius of rolling ring ($r$) = $2\text{ cm}$ * **Distance traveled by the center of the ring:** $$\text{Distance} = 2\pi(R + r) = 2\pi(10 + 2) = 24\pi\text{ cm}$$ * **Circumference of the rolling ring:** $$C = 2\pi r = 2\pi(2) = 4\pi\text{ cm}$$ * **Number of rotations:** $$n = \frac{24\pi}{4\pi} = 6$$ ### Exam Strategy & Shortcut Use the direct rolling paradox formula: $$n = \frac{R}{r} + 1 = \frac{10}{2} + 1 = 5 + 1 = 6$$ ### Common Pitfall Simply calculating $\frac{R}{r} = \frac{10}{2} = 5$ without accounting for the extra rotation resulting from traveling in a circular loop. ### Final Answer Therefore, the correct answer is **6**.