I have two coins - one has $N$ times the diameter of the other. I roll the smaller coin one full lap around the circumference of the larger coin like a gear, keeping contact it in with the edge of the larger coin and keeping the larger coin still.
How many rotations does the smaller coin make around its own center in the rest frame? Equivalently, if you drew an arrow on the small coin which was initially pointing East, how many times during the process would the arrow point North?
Answer
Answer :
$N+1$
Explanation :
Circumference of bigger coin $= 2\pi N r$
Circumference of smaller coin $= 2\pi r$
Number of revolutions $= \frac{2\pi Nr}{2\pi r}=N$
But the smaller coin has also revolved about the center of the whole configuration, which increases the number of revolutions by 1
So total $= N+1$
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