SAT Question Everyone Got Wrong

[Jonathan Busby]

I think the confusion that arises over this question is due to the accidental and almost subconscious conflation of the rotational behavior of a disc on a perfectly flat, level surface with the rotational behavior of a disc over a *curved* surface.

When a disc rotates over any surface, the number of rotations is equal to the distance traced out by the center of the disc divided by the circumference of the disc. Now, on a perfectly flat surface, the distance traced out by the center of the disc is equal to the distance the disc has traveled. So, in this case, one can divide the distance traveled by the disc by its circumference to arrive at the number of rotations. So, on a curved surface, such as another disc in this case, one must be able to find the number of rotations by dividing the stationary disc's circumference by the circumference of a moving disc, right? Right? Nope The key mistake in this reasoning is ignoring the distance the center of the peripheral disc has to traverse to make a full circuit around the stationary disc. Since the number of rotations the peripheral disc makes is equal to the distanced traversed by its center divided by its circumference, we arrive at the following :

Size of circle traversed by center of moving disc :

\[ \displaystyle (r_{m} + r_{s}) \cdot 2 \cdot \pi \]

Where \( \displaystyle r_{m} \) is radius of the moving disc and \( \displaystyle r_{s} \) is the radius of the stationary disc.

Now, since the number of rotations the disc makes is equal to the distance traveled by its center divided by its circumference, we have :

\[ \displaystyle \dfrac{((r_{m} + r_{s}) \cdot 2 \cdot \pi )}{r_{m} \cdot 2 \cdot \pi} = \]


\[ \displaystyle 1 + \dfrac{r_{s}}{r_{m}} \]


Hope that clears it up for somebody

Regards,

Jonathan