Problem 5 (Rolling Circles)

An interesting, but not so easy problem, is to determine the number of turns a circle will make when rolled around the outside of a fixed circle. For example, suppose you place two pennies, side by side, face up. Now hold one penny and rotate the other around it in the counterclockwise direction. It is not easy to visualize the number of turns the rolling penny makes as it rotates around the outside of the fixed penny. We have two questions, one you can do by physical experimentation, the other with your mind.

• How many turns does the rolling penny make around the outside of the fixed penny ?

• What if the diameter of the rolling circle is 1 and the diameter of the fixed circle is F. How many turns will the rolling circle make around the outside of the fixed circle ?

Solution

By experimentation you discover that the rolling penny makes a complete revolution when it rolls only half way around on itself, and two complete revolutions when it rolls all the way around. (For no specific reason, we roll the outside penny around the fixed penny in the counterclockwise direction.)

(b) To determine the number of turns a rolling circle with radius R makes as it rolls around a fixed circle with radius F, we think of unrolling the fixed circle (as if it were a hooly hoop) on a flat surface, which would have length of 2 pi F.

Hence, if we roll the outside circle of radius R (circumference 2 pi R) on this line, the number of turns it will make will be

# ROTATIONS OF A CIRCLE ON A LINE OF LENGTH 2 pi F = (2 pi F)/(2 pi R) = F/R

But, the rolling circle in our problem does not roll along a line, but around a circle and hence the number of rotations the rolling circle makes as it rolls around a fixed circle of radius F is one more than F/R, or

T = F/R + 1

You can see this by imagining sliding the outside circle around the fixed circle (keeping the same point of the sliding circle in contact with the fixed circle), and noting it rotates once as a result of going around the circle. The following table gives the number of turns for different values of R and F.

R/F	1	2	3	4
1	2	3	4	5
2	1 1/2	2	2 1/2	3
3	1 1/3	1 2/3	2	2 1/3
4	1 1/4	1 1/2	1 3/4	2

For example, if the radius of the fixed circle was F = 3 inches, and the radius of the rolling circle was R = 4 inches, then the rolling circle would undergo T = 1 3/4 turns as it rolled around the fixed circle.
It is interesting to note that if the rolling circle were inside the fixed circle as shown in the picture below, then (by a similar argument) the number of rotations of the rolling circle will be R/F - 1. In this case, if we slide the inside circle counterclockwise inside the outer fixed circle, keeing the same point of the inner circle in contact with the outer circle, then the rotation of the inner circle is clockwise, but when we roll the inner circle counterclockwise inside the outer circle, the rotation is clockwise, and hence we subtract 1 from F/R instead of adding it.

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Last modified on Wednesday, March 17, 1999