# the circumference of a circle

• Jun 19th 2012, 05:08 AM
kingman
the circumference of a circle
I need help in the below problem
thanks

A circle of diameter 2cm rolls along the circumference of a circle of diameter 12cm, without slipping until it returns to its starting position. Given that the smaller circle has turned x degrees about its centre. Find the value of x
• Jun 19th 2012, 06:09 AM
Reckoner
Re: the circumference of a circle
Edit: My post is incorrect. Please refer to Soroban's explanation. I apologize for the error.

The smaller circle has a circumference of $2\pi$ cm and the larger circle has a circumference of $12\pi$ cm. To roll all the way around the larger circle, the smaller circle would have to make $\frac{12\pi}{2\pi} = 6$ full rotations about its center. 6 complete rotations equals how many degrees?
• Jun 19th 2012, 06:46 AM
kingman
Re: the circumference of a circle
Thanks but why sometimes we have to add one more round when the small circle has completed one full round about the
centre of bigger circle
• Jun 19th 2012, 07:08 AM
richard1234
Re: the circumference of a circle
@kingman no...the small circle starts and ends at its original position; it makes six full revolutions (because the ratio of the circumferences of the larger, smaller circles is 6:1).
• Jun 19th 2012, 12:50 PM
Soroban
Re: the circumference of a circle
Hello, kingman!

This is a classic trick question . . .

Quote:

A circle of diameter 2cm rolls along the circumference of a circle of diameter 12cm,
without slipping until it returns to its starting position.
Given that the smaller circle has turned x degrees about its centre, find the value of x

You are correct about that "extra revolution".

Consider a circle with radius $R.$
Roll it along a line.

Code:

              * * *                          * * *           *          *                  *          *         *              *              *              *       *                *            *                *       *                  *          *                  *       *        *        *          *        *        *       *        |        *          *        |        *                 |                              |       *        |R      *            *        |R      *         *      |      *              *      |      *           *    ↓    *                  *    ↓    *     ----------*-*-*---------------------------*-*-*----------                 : - - - - - -  2πR  - - - - - - :
At the start, the "initial radius" (IR) is pointing down (at the line).
After one revolution, the circle has moved $2\pi R$ units
. . and the IR is again pointing at the line.

Now revolve the circle around a circle with twice its radius.

Code:

              * *             *    *             *  *  *                 |R             *  |  *               * * *           *          *         *              *       *                *       *                  *       *        * - - - - *       *              2R  *       *                *         *              *           *          *               * * *               * *             *  |  *                 |R             *  *  *             *    *               * *
The circle starts with its IR pointing at the circle.
In one revolution, it moves $2\pi R$ units around the large circle
. . and its IR is again pointing at the circle.

But that radius is pointing upward.
. . How did that happen?

While the small circle rolled around half of the large circle,
. . the IR made ${\color{blue}1\tfrac{1}{2}}$ revolutions.

So in making one "orbit", the smaller circle makes three revolutions.
• Jun 19th 2012, 02:28 PM
richard1234
Re: the circumference of a circle
Huh, very interesting. I guess it's because the "path" is circular, not a straight line, so there's an extra revolution. My apologies.

Turns out the College Board screwed up on this type of problem as well, in 1982:
Brain teaser: rolling one quarter around another. Rotation vs. revolution.
• Jun 19th 2012, 02:42 PM
kingman
Re: the circumference of a circle
Thanks but can we conclude that in general for any small circle of radius "r" revolving round a bigger circle of radius '' R " one has to add one more revolution after the smaller circle has made one Orbit.
• Jun 19th 2012, 06:01 PM
Soroban
Re: the circumference of a circle
Hello, kingman!

Quote:

Thanks, but can we conclude that in general for any small circle of radius "r"
revolving round a bigger circle of radius ''R", one has to add one more revolution
after the smaller circle has made one Orbit?

Yes, that is true.
• Jun 21st 2012, 05:53 AM
kingman
Re: the circumference of a circle
Sorry Can you please explain from your diagram how you get 1/1/2 revolutions by just noticing that radius is pointing upward.

I wonder whether it is true to say the half revolution is due the small circle rotating about its own axis and the remaining half revolution is due to the half revolution after the small circle has revolved about the centre of the big circle. In concludion one is due to roatation about its ( small circle )own sxis and another is due the small circle orbiting about the centre of the big circle.
• Jun 21st 2012, 06:08 AM
Reckoner
Re: the circumference of a circle
Quote:

Originally Posted by kingman
I wonder whether it is true to say the half revolution is due the small circle rotating about its own axis and the remaining half revolution is due to the half revolution after the small circle has revolved about the centre of the big circle. In concludion one is due to roatation about its ( small circle )own sxis and another is due the small circle orbiting about the centre of the big circle.

Yes, I think that interpretation makes sense.

If the circle was rolling along a flat surface for the same distance, there would not be an extra revolution. The extra rotation comes from the fact that the surface on which the circle is rolling is itself curved.