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### One Circle Revolving Around Another

```
Date: 05/26/99 at 09:16:45
From: Mike Aumueller
Subject: Circle circumference/revolution question

If one circle has a circumference of 7.8 cm and a second circle has a
circumference of 2.6 cm, how many revolutions will the smaller circle
make while rotating around the perimeter of the larger circle?

I have tried for hours to create a formula that will answer this
question, but I have had no success. Help.

I have tried the following:

C1/C2 = Revolution1/Revolution2 where I have measured the revolution
of circle 2 around circle 1. But I would like to figure out the answer
without actually measuring.

Thanks.
```

```
Date: 05/26/99 at 12:18:47
From: Doctor Rick
Subject: Re: Circle circumference/revolution question

Hi, Mike.

This is pretty hard to picture, because it involves a sort of
"motion-picture geometry." I hope I can help a little.

Let the center of the big circle (radius R1) be A and let the center
of the small circle (radius R2) be B. Label one point on the
circumference of the small circle as point C.

Start with AB at angle 0 (say, the x-axis), and with point C at the
point of tangency of the two circles. As AB rotates, the small circle
rolls along the big circle. When the center of the small circle has
moved to B', sweeping out the angle alpha = BAB', the point of
tangency has moved along a certain arc of the big circle. Because the
circle rolls without slipping, point C will have moved to point C'
sweeping out an equal arc of the small circle.

Draw this out and think about it. There is enough information here for
you to figure out how the angle beta = ABC' is related to the angle
alpha. But then you must be careful, because angle beta is NOT what

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 05/26/99 at 13:22:30
From: Mike Aumueller
Subject: RE: Circle circumferece/revolution question

Thank you very much for your assistance and have a great day.

```

```
Date: 06/22/99 at 15:00:59
From: Mike Aumueller
Subject: Going in circles

Dear Sir, I need to know the answer and proof to the question I have
presented. A mechanical engineering professor states that the answer
is 4 revolutions; however many other people insist that the answer is
3.

Thanks.
```

```
Date: 06/22/99 at 21:38:34
From: Doctor Rick
Subject: Re: Going in circles

Hi again, Mike.

I gave you the outline of the proof last time. The key to the
difference between the answers of 3 revolutions and 4 revolutions is
in my last statement: "But then you must be careful, because angle

Let's draw a picture.

*******
**       **
*           *
*             *          C'
*      B      *       *******
*      |      *     **   |   **
*     |     *     *     |     *
**   |   **     *      |      *
*******       *      B'     *
*       S|C      *  *     /|      *
*           |          **  /  |     *
*             |            *    |   **
*               |          / D*******
*                |        /      * S'
*                 |      /         *
*                  |    /            *
*                   |  /               *
*                   |/                 *
*                   A                  *
*                                      *
*                                      *
*                                    *
*                                  *
*                                *
*                              *
*                          *
*                      *
*                *
******

Let the center of the big circle (radius R1) be A and let the center
of the small circle (radius R2) be B. Label one point on the
circumference of the small circle as point C.

Start with AB at angle 0 (say, the y-axis), and with point C at the
point of tangency of the two circles. As AB rotates, the small circle
rolls along the big circle without slipping. This means that, if the
circle has rolled so that the point of tangency is at D, then the arc
length CD on the big circle is equal to the arc length DC' on the
small circle. Since the radius of the small circle is 1/3 the radius
of the large circle,

angle DB'C' = arc DC' / R2 (radians)
= arc CD / (R1/3)
= 3 arc CD / R1

But we are not interested in angle DC'B'. The angle of rotation of the
small circle is the angle S'B'C', where B'S' is parallel to BS, so
that we are reckoning the rotation relative to a fixed direction. Now

angle S'B'C' = angle S'B'D + angle DB'C'

Thus when the small circle has gone 360 degrees, that is, angle CAD =
1 revolution, angle S'B'C' will have swept out 4 revolutions.

See if this satisfies both factions.

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 06/23/99 at 09:40:27
From: Mike Aumueller
Subject: RE: Going in circles

Thank you very much for your help. The only other question I have is:
Why is it that when you physically simulate the rotation of the two
circles the result is 3 rotations, not 4?
```

```
Date: 06/23/99 at 10:07:36
From: Doctor Rick
Subject: RE: Going in circles

Hi, Mike.

Are you absolutely SURE you are measuring the rotations relative to a
fixed direction (like north) and not relative to the line from the
center of the big circle to the small circle?

And are you holding the big circle fixed and revolving the small
circle around it, rather than turning the big circle like a gear and
keeping the small circle in one place?

If you answer yes to both questions, then you should count 4 rotations
per revolution. You should find that, when the small circle has gone
from north to east of the big circle, the point that was touching the
big circle at first has come back around to the bottom (south) of the
small circle. That's one full rotation in 1/4 revolution.

I remember in fifth grade being asked by my teacher what was the
period of the moon's rotation. I said one month, and my teacher said
no, it does not rotate at all. I didn't talk back to my teacher, but
the fact is, I was thinking of the sidereal period - counting
rotations relative to the stars - while he was thinking of the
geocentric period, I guess you'd call it, counting rotations relative
to a line from the earth to the moon. It's the same problem of being
off by one rotation per revolution because the reference direction is
itself rotating. It can be easy to miss!

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry

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