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?

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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). Number of rotations = (R + r/r) = (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 cm • Radius of rolling ring (r) = 2 cm • Distance traveled by the center of the ring: Distance = 2\pi(R + r) = 2\pi(10 + 2) = 24\pi cm • Circumference of the rolling ring: C = 2\pi r = 2\pi(2) = 4\pi cm • Number of rotations: n = (24\pi/4\pi) = 6 Exam Strategy & Shortcut: Use the direct rolling paradox formula: n = (R/r) + 1 = (10/2) + 1 = 5 + 1 = 6 Common Pitfall: Simply calculating (R/r) = (10/2) = 5 without accounting for the extra rotation resulting from traveling in a circular loop. Final Answer: Therefore, the correct answer is 6.

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