A circular coin of radius 1 cm is allowed to roll freely on the periphery over a circular disc of radius 10 cm. If the disc has no movement and the coin completes one revolution rolling on the periphery over the disc and without slipping , then what is the number of times the coin rotated about its centre?

The problem describes the 'coin rotation paradox.' When a coin of radius r rolls around the periphery of a fixed disc of radius R, the number of rotations it makes about its own center is given by the formula (R/r) + 1 [2]. In this case, the radius of the coin (r) is 1 cm and the radius of the disc (R) is 10 cm. The ratio of the circumferences is 10/1 = 10, which represents the rotations due to the distance traveled along the path [2]. However, because the path itself is circular rather than a straight line, the coin must complete one additional rotation to return to its original orientation. Thus, the total number of rotations is 10 + 1 = 11. This occurs because the center of the rolling coin travels a larger circular path of radius R + r.

Sources

1. [2] https://www.scientificamerican.com/article/the-sat-problem-that-everybody-got-wrong/