My name is Craig Ellis and I have a question on the 10-12 grade level. I am a student teacher intern studying to be a high school teacher so I really should be able to figure out this question myself, but for the life of me I have been unable to do so. The problem was posed by my methods teacher.
   We have a circle of radius 3. inside the circle and tangent to the circle of radius 3 at one point is a circleof radius 1. The question is if we could roll the smaller circle around the inside of the larger circle how many revolutions would it take to get around to where we started.
   The professor said that the answer was not three although it seems that it should be to me. I am obviously doing something wrong. I even built a model and it came out as three. Next we are to roll the circle around the outside of our circle of radius three and see how many revolutions that takes. Anyway I hope that you can make better sense of this than I can and thanks in advance for any light you are able to shed on problem.

Thanks Again

Hi Craig

Your answers are both correct -- it's three revolutions or two revolutions depending on what you want to call a revolution. Before explaining this mysterious situation, I should point out that the same sort of question arises concerning our moon as it moves about the earth once each month: the same face of the moon always points toward the earth, so to us on earth it looks as if the moon does not rotate -- it just slides around the earth on its 29.5-day journey. For an astronaut who is far enough out in space to see both the earth and the moon, he would see the moon rotate once about its axis as it makes one revolution in its orbit about the earth. (To keep the two motions straight astronomers use the special terminology that that a body ROTATES about its axis and REVOLVES in an orbit about a body fixed in space. For the rest of us these words are interchangable, hence the ambiguity.)
   For the situation with one circle rolling about the circumference of a fixed circle, let's call the rolling circle R and its radius r, while the fixed circle is F with radius f. In your problem, r = 1, f = 3, and R is inside F, tangent at the top, say 12:00 (thinking of F as a clock).
   Mark a point P at the top of the rolling circle where at the start it is tangent to F. As R moves about F in the clockwise direction, the point P moves counterclockwise about the centre of R. Since f/r = 3/1 = 3, the circumference of the rolling circle is 1/3 the fixed circumference, so P comes back to touch the big circle 3 times as R makes one orbit -- at 4:00, 8:00, then back to 12:00. Thus THREE is the answer to the question,
   How often does a point P fixed to the circumference of the rolling circle return to the fixed circle as R rolls once about F? Note that you get the same answer if R rolls about the outside of F, but in the outside case R orbits in the counterclockwise direction as it rotates counterclockwise.
   Returning to the original question,

How many revolutions does the rolling circle make ABOUT ITS CENTER during one orbit about the fixed circle? the answer is TWO if it moves about the inside of F, and FOUR about the outside. To see this you draw an arrow from the center of R pointing to P. As R rolls about the inside of F the arrow points up at 12:00 and at 6:00; about the outside it points down at 12, 9, 6, and 3 (in that order).
   If you are still stuck, you could try a simpler example where the ratio of the radii is, for example, 2:1. Still stuck? Try the ratio being essentially 1:1, with the inside circle just pressing lightly on the outside circle as the 'point of pressure' moves around the outside circle. The situation with the ratio 1:1 and R rolling about the outside of F is probably easier to simulate with, say, a couple of quarters.
   The story is clearly explained by Jack Eidswick in his lovely little article, "Two Trisectrices for the Price of One Rolling Coin," in THE COLLEGE MATHEMATICS JOURNAL, 24:5 (November, 1993) pages 422-430. For the general answer we adopt the convention that r > 0 if R is OUTSIDE F, and r < 0 INSIDE F: the number of revolutions (of the rolling circle about its center) is then f/r - 1 (where we interpret a "negative revolution" to be clockwise).

Chris and Walter


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