Who can answer this one correctly? (coins rotating about other coins)

Steven G

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Suppose you have a quarter (coin) and let's say its circumference is c.
If you rotate one quarter around another, where you start and finish at the same place on the stationary coin, then how many revolutions did the rotating quarter make?
 
Suppose you have a quarter (coin) and let's say its circumference is c.
If you rotate one quarter around another, where you start and finish at the same place on the stationary coin, then how many revolutions did the rotating quarter make?
Do I only get one chance at this too?

The answer is two. ?
 
The radius of each coin is the same (r) so, as the rotating coin travels around the stationary coin, its centre must follow a circular path (of radius 2r) therefore the distance the rotating coin travels is 2π(2r) = 4πr. (See Fig. 1.)

Centres.jpg

Now consider that circular path straightened out into a line (4πr long) as in Fig. 2. (q.v.)

Rolling Coin.png

A coin rolling along that line will cover a distance of 2πr (its circumference) in one rotation, therefore it will require two full rotations to cover the distance 4πr.
QED. ?
 
Last edited:
Suppose you have a quarter (coin) and let's say its circumference is c.
If you rotate one quarter around another, where you start and finish at the same place on the stationary coin, then how many revolutions did the rotating quarter make?
Extension of the problem:
Suppose you have a dollar (coin) and let's say its diameter is 4d and a dime (coin) and let's say its diameter is d.
If you rotate one dime around the dollar, where you start and finish at the same place on the stationary coin, then how many revolutions did the rotating dime make?
 
General case: rotating small coin A of diameter [imath]d[/imath] once around another, stationary coin B of diameter [imath]ad[/imath]. Consider an equivalent motion done in 2 stages:
1) A and B rotate together in place (so that centers of both coins stay in place) and B gets rotated exactly once. Coin A here makes [imath]ad[/imath] revolutions.
2) Now glue A to B at the touching point and rotate A one revolution in the opposite direction bring both coins to the same state as in the original problem. Here coin A makes one more revolution, bringing the total to [imath](a+1)[/imath].
 
You guys are good!
This was an SAT problem and the authors said that it would be one revolution. However a few bright test takers challenged that answer and were correct.
 
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