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How many times does a circle revolve as it rolls along the circumference of another circle?

Tomas Garza

Author

Tomas Garza

Title

How many times does a circle revolve as it rolls along the circumference of another circle?

Description

Understanding an apparent paradox about rolling circles

Category

Essays, Posts & Presentations

Keywords

Rolling circles, counting revolutions

URL

http://www.notebookarchive.org/2024-04-5ki5bol/

DOI

https://notebookarchive.org/2024-04-5ki5bol

Date Added

2024-04-12

Date Last Modified

2024-04-12

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112.32 kilobytes

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CC0 1.0

How many times does a circle revolve as it rolls along the circumference of another circle?

Tomas Garza

Description

In Figure 1 below, observe the two circles A and B (white and green, respectively). Circle B rolls -without slippage- along the circumference of Circle A until it reaches its starting point. At the end of the trip, how many times has Circle B revolved on its center?

To make the question precise, we look at the radius at the initial point of tangency (in green, at the extreme of the radius from the center -in yellow- of Circle B). In the rolling process, the center of Circle B travels along the dashed circumference, and the behavior of that radius can be observed by sliding the control. The question is then, how many complete turns does this radius give in the process?

Discussion

There appears to be a paradox here, which we will try to resolve. For a given ratio of the radius of Circle A to that of Circle B, one might conclude that the point of tangency revolves a number of times equal to that ratio along its trip. Yet, as one may check in the dynamic figure below, the yellow radius makes one extra turn in this process.

Out[]=

Fig. 1. The green circle rolls, without slipping, around the white

circumference. Different sizes of the radii are considered. The

revolutions of the rolling circle during the process may be

observed by watching the behavior of the yellow radius.

radius of the

rolling circle

1

2

3

4

1 ordinary view

2 constructive view

1

2

roll the

green circle

back to start

The rolling process of the green circle implies that its center (in yellow) rolls together with it, and it gives two whole revolutions as it travels along the dashed circumference in yellow. We confirm this by looking at the motion of the yellow radius joining the center with the green point. As soon as the rolling process begins, the green point moves away from the white circumference and, together with the yellow radius, shows clearly the turning of the circle about its center (in yellow)

I am using a vector approach for drawing the dynamic image. The rolling process is described by the sum of two vectors: a), the position of the center of the rolling circle (called redVector in the code), and b) the position of the whiteVector, which controls the turning speed of the rolling circle around its center. Both vectors are given in parametric form and depend on the angle ϕ. These elements may be seen using the constructive view control.

Using the terms borrowed from Astronomy, the green circle is subject to two motions: 1) the rotation about its center, and 2) the translation around the white circle. Now, what is the meaning of non-slippage? Looking at the image in Fig.1 above (with the thumbnail control set to 1), when the yellow point has travelled, say 1/4 of its circular path (that is, in its translation motion), the same must be true of the rotation motion, which is identified by the yellow radius of the green circle and is now in a horizontal position -for the first time. If the translation proceeds to 1/2 of the circular path, then, again, the yellow radius reaches a horizontal position. And we realize that at this moment it has given a complete revolution (the extremes of the radius, green and yellow, are in the same relative position as in the beginning). The translation process goes on, and at the end, we see that two complete revolutions of the green circle have been produced.

Now consider the image in Fig. 2 below. Here the rotation motion has been inhibited: the green circle does not rotate about its center; the yellow radius with its green point at the point of tangency of the two circles, only changes its position due to the translation motion. This green point never leaves the wihite circumference. Yet, at the end of the process the yellow radius will have given one complete turn. The three points, yellow, green and red, remained aligned all the time.

Out[]=

Fig. 2. The green circle does not roll around the white

circumference. The two circles have the same radius. The

green circle does not rotate: the green point remains on

the white circumference all the time.

slide the

green circle

back to start

So, under null rotation, the translation motion has, by itself , produced one revolution of the green circle. Going back to Fig. 1, we conclude that this explains the apparent paradox mentioned at the beginning of this notebook. In general, the process of rolling a circle, without slipping, along the circumference of another circle will generate r + 1 complete turns, where r is the ratio of the fixed circle radius to the rolling circle radius, for r = 1, 2,...n (see Fig. 1 above). Informally, we could say that r revolutions have been generated by the rotation motion, and an extra one which is due to the translation.

In order to ensure that there is no slippage, the rotation of the green circle must be faster than the translation of its center, so that when the translation has advanced by an angle ϕ, the rotation must have advanced to 2 ϕ. In general, in the case of circles of different size, and r is the ratio of the radii of the fixed and the rolling circles, a translation by an angle ϕ will need an angle of (1 + r) ϕ for the rotation. These conditions are easily identified in the code.

In the case of equal radii, the yellow circumference has a length of 2π, since its radius is twice the radius of the green circle ( equal to 1). The next figure describes the motion of the yellow circle along a line of length 2π, together with the original two-circles setup, and the non-slippage is evident since both the yellow and the orange circles run at the same speed.

Out[]=

Fig. 3. The green circle rolls, without slipping, around the white circumference.

The two radii are equal to 0.5. The rolling process implies that the center

of the green circle describes a corresponding circular path of length 2π,

thus making the green circle to revolve twice.

roll the

green circle

back to start

Comments

This problem has aroused much interest, at least among a number of Mathematica practitioners. There are, of course, good reasons for that, since there seems to be no satisfactory explanation to the paradox I refer to above. In recent times, there was a curious event which sparked this interest, to wit, a multiple-choice question in the 1982 SAT (Scholastic Aptitude Test, in the U.S., an examination for students entering college). The question was essentially the problem we have been studying in this notebook, and none of the 5 options offered was the correct one: there was a grave mistake in the exam sheet. The reader may watch the video here .
A number of discussions may be found in https://math.stackexchange.com/questions/1351058/circle-revolutions-rolling-around-another-circle.

To conclude, a plausible explanation, then, is that the rotation motion contributes whatever number of revolutions result from the the ratio beetween the two circles, and the translation motion contributes one additional turn.

Cite this as: Tomas Garza, "How many times does a circle revolve as it rolls along the circumference of another circle?" from the Notebook Archive (2024), https://notebookarchive.org/2024-04-5ki5bol