Circle revolutions rolling around another circle

Question

I just watched this video, and I'm a bit perplexed.

Problem:

The radius of Circle A is 1/3 the radius of Circle B.

Circle A rolls around Circle B one trip back to its starting point.

How many times will Circle A revolve in total?

The intuitive answer is 3, but the correct answer is 4. I understand the trick -- that the center of Circle A must travel a distance of $2\pi(r_B + r_A)$, not $2\pi r_B$ -- but I'm still confused on one item.

At the risk of sounding very un-mathematical, how do the (infinite set of) points on the circumference of each circle map to each other to accomplish this?

Consider Circle A rolling along a straight line the length of the circumference of Circle B. Then it will revolve 3 times. It's like the universe "knows" when to apply a different point mapping when you change the arrangement of matter.

Answer 1

Thank you everyone for your answers. They were all informative, but I wasn't able to intuitively understand them until I saw this visual:

https://www.geogebra.org/m/v3a437ux

The key for me was to see that there are two kinds of revolution happening:

1. Revolutions of Circle A with respect to Circle B, and
2. Revolutions of Circle A with respect to the (overhead) observer.

To the observer, the revolutions of the type (2) complete before revolutions of type (1). The revolutions of type (1) happen at the $r_B/r_A$ roots of unity, and the revolutions of type (2) happen at the $(r_B/r_A + 1)$ roots of unity. It was very helpful to see precisely where these revolutions occur, because it was mentally impossible to unify the two types of revolution without the visual. It's also useful to notice that at any given moment, with respect to the observer, the points of Circle A on the far side from Circle B are moving faster than the points of Circle A on the side that is touching Circle B.

David K's answer is great for understanding that the parametric mapping of time to point pairs is the same whether Circle A is rolling along a straight line or a circle, and that we are dealing with frames of reference. I simply didn't believe the mapping was the same until I saw the visual.

zoli and Hans Lundmark's answers are great for understanding that one extra revolution must occur at some point along the rolling path. The complete answer, of course, is that this last mysterious revolution does not happen all at once, but gradually along the entire roll.

The visual was discovered in the comments on this page, which is another discussion of the same problem:

https://plus.maths.org/content/circles-rolling-circles

Answer 2

As it is depicted on the figure below the smaller circle has to turn around $3$ times if it travels along a straight distance equaling the perimeter of the larger circle. In this case the center of the smaller circle takes the same distance.

Now imagine that the straight line of length $6\pi r_B$ takes a complete rotation around its center while the small circle rolls along. Also, suppose that the straight takes one full turn while the circle reaches the other end.

As a result the small circle takes a fourth revolution.

The number of rotations is not different if the path is circular. The circularity of the path was modeled in the case of the straight path by turning around the segment.

The video explains the same by claiming that the small circle has to travel a distance of $8\pi r_B$. This is the same as having to travel on the straight line of length $8\pi r_B$ which does not rotate about its center.

Answer 3

The mapping of points from one circle to the other is no different than when the small circle is rolling on a straight line; the extra revolution is purely due to bending.

It might be easier to see this in the case of gliding, when there is a single point $A$ on the small circle which is in contact with the line or the large circle throughout the process. When gliding on a line, the small circle doesn't revolve at all. But if it's supposed to glide along a large circle in such a way that the same point $A$ is in contact all the time, then it also has to revolve as at glides (and it will make one revolution as it glides one lap along the large circle).

Answer 4

I don't know what you mean by "the points map to each other". To "map" one set to another implies a function that maps each point in one set to a unique point in the other; to "map to each other" implies that the function is one-to-one. There is no one-to-one mapping between points: each point on the small circle meets three different points on the large circle. If you roll a circle of radius $2$ twice around the circle of radius $3$ then the correspondence between points that meet isn't even a function in one direction, let alone a one-to-one function. I think perhaps you mean there is a mapping from some parameter to the pairs of points along the two circles that are brought into contact; for example, if the parameter is time in seconds since the motion started, the mapping says which two points will be in contact at each time $t$.

I prefer a completely different approach than the one in the video. Seat yourself on a frame that is attached to the centers of the two circles. As the small circle rolls around the larger one, the motion of the center of the small circle rotates the frame (and you) around the center of the larger circle.

While sitting on the frame, what you see is the two centers of the circles remaining in the same place within your field of view, while the larger circle rotates around its center and the smaller circle also rotates without slipping against the larger circle. And of course you see the smaller circle rotate three times.

But someone who remained motionless relative to the larger circle, not sitting in your rotating frame, sees the smaller circle rotate four times: the three times that you observed, plus one rotation that you did not observe because you were doing a full rotation in the same direction yourself. If you step off the frame and "undo" the effect of its rotation upon you by turning yourself once in the opposite direction, you'll observe the fourth rotation of the small circle.

Answer 5

Nothing new, but a diffent way to look at the question:

First consider a cirlce with an inside hexagon (in general an n-figure). Then the circle will roll the distance of the circumference PULS an angle wihich is due to tilting at the edges. Thus, the additional angle (the $+1$ you look for) amounts in total to

$$6 \cdot \frac{2\pi}{6} = 2\pi, (n \cdot \frac{2\pi}{n} = 2\pi)$$

explaining the extra turn - no matter how many edges you take for your approximation.

Second consider an infinitisimal approach: For $x \in [0, R)$, $f(x) = \sqrt{R^2-x^2}$ is a circle with radius $1$. Later, we also need

$$f'(x) = \frac{-x}{\sqrt{R^2-x^2}}$$

A tangent vector $\vec{T}$ at $x$ is $$\vec{T}=\begin{pmatrix}1\\f'(x)\end{pmatrix}$$ and $\vec{T'}$ at $x+\Delta x$ is $$\vec{T}=\begin{pmatrix}1\\f'(x+\Delta x)\end{pmatrix} = \begin{pmatrix}1\\f'(x)+f''(x)\cdot \Delta x + \frac{f''(x)}{2}\Delta x^2\end{pmatrix}$$.

Now, the cosine of the angle $\Delta \alpha$ between the to vectors is

$$\cos(\Delta \alpha)=\frac{\vec{T}\cdot \vec{T'}}{\left|\vec{T}\right|\cdot \left|\vec{T'}\right|}$$

To first order this is

$$h(\Delta x) = \frac{1+\frac{f'f''\Delta x}{1+f'^2}+\frac{0.5f'f'''\Delta x^2}{1+f'^2}}{\left(1+\frac{f''^2\Delta x^2}{1+f'^2}+\frac{2f'f''\Delta x}{1+f'^2}+\frac{f'f'''\Delta x^2}{1+f'^2}\right)^{1/2}}$$

Use the following abbreviations for a Talor-series (which equals $\cos(x) = 1 - x^2/2$): $$N=\frac{1}{1+f'^2}, a = Nf'f'', b = 0.5Nf'f''', c = Nf''^2, d = c + 2b$$

Then

$$h(\Delta x) = \frac{1 + a\Delta x + b \Delta x^2}{\left(1 + 2a \Delta x + d \Delta x^2\right)^{1/2}}$$

$$h'(\Delta x) = \frac{(a^2-c)\Delta x + 3ab \Delta x^2 + bd \Delta x^3}{\left(1 + 2a \Delta x + d \Delta x^2\right)^{3/2}}$$ and

$$h''(\Delta x = 0) = \frac{(a^2-c)\Delta x + 3ab \Delta x^2 + bd \Delta x^3}{\left(1 + 2a \Delta x + d \Delta x^2\right)^3/2}$$

Finally we obtain

$$1 - 0.5\frac{f''^2}{\left(1+f'^2\right)^2} \Delta x^2 = 1 - \frac{\Delta \alpha^2}{2}$$

or

$$\int_{x_0}^{x_1} \frac{f''}{1+f'^2}dx = \arctan(f'(x_1)) - \arctan(f'(x_0))$$.

This again gives for a full circle $8\cdot\frac{pi}{4} = 2 \pi$. Furthmore we observe, that in the end, only the difference between the angles at $x_0$ and $x_1$ play a role.

Answer 6

Really enjoyed this, thanks so much. I thought I would just contribute something of a more "mechanical" view of this problem. I've used clockfaces just to keep things familiar.

In the top image, image $A$, the two clockfaces are aligned next to each other, with the "$3$" on the left touching at the "$9$" on the right black clockface.

In the middle image, image $B$, I've started the rotation of the right black clockface around the left white clockface, lining up the $4$ on the left with the $8$ on the right.

In the bottom image, image $C$, I've done exactly the same transformation, only this time I've rotated both clockfaces. (If you tilt your head to the right, image $B$ will look like image C does with your head level.)

So what this shows, mechanically, is when we appear to rotate one circle around the other, this is an illusion. In fact we are always rotating both circles at the same time. Because of that, the total distance that needs to be travelled is the sum of the circumferences of both circles. In this case, as the circles are the same size, it requires $2$ apparent rotations of one of the circles to get around the other.

If you want to introduce a bit more maths into this, you can look at the tangents at $4$ and $8$ as they differ from the vertical perpendicular.