Physics (10th Edition)

In a rolling motion, like the case of a wheel, the distance traveled by the axle of the wheel - which is at its center - equals the circular arc length at the edge of the wheel. A similar situation here: the distance traveled by the center of the rolling quarter - the circumference of the black circle - equals the total circular arc length at the edge of the quarter. If we take the radius of a quarter to be $R$, the black circle's radius is $R+R=2R$, so its circumference is $C=2R\times2\pi=4\pi R$ Therefore, the total circular arc length traveled at the outer edge of the quarter is $4\pi R$ Now 1 revolution of the rolling quarter equals a circular arc length of $2\pi R$. Therefore, the number of revolutions the quarter makes is $$\frac{4\pi R}{2\pi R}=2$$