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Circles rolling on circles |

Circles rolling on circles |

2023-05-29 03:56:01

Coin rolling

What number of revolutions will the smaller coin make when rolling across the larger one?

Think about a circle with radius 1 cm rolling utterly alongside the circumference of a circle with radius 4 cm. What number of rotations did the smaller circle make?

The circumference of a circle with radius $r$ is $2pi r$, so the circumference of a circle with radius $4r$ can be $8pi r$. Since

$8pi r div 2pi r = 4,$

I figured the reply have to be 4 revolutions. So think about my shock once I noticed that the reply was given to be 5!

I learn the reason of why this was certainly the proper reply, and though the reasoning appeared sound, it took a while earlier than I may really persuade myself that my answer was in error. It’s an fascinating downside, so I offered it to a lot of individuals, most of whom instantly answered “4” as I did, and, like me, have been tough to persuade in any other case; solely a only a few may intuitively see “5” as the proper reply.

Right here’s a greater manner to think about this downside: quite than rolling alongside the bigger circle, begin by imagining the smaller circle as rolling on a line the identical size because the circumference of the bigger circle. On this case it’s simple to think about the road as being $8pi r$ items lengthy, and so the smaller circle clearly will need to have made $8pi r div 2pi r = 4$ rotations. Subsequent, consider the circle as sliding alongside the road, with out rolling, in order that the purpose on the coin at which it touches the road stays the identical. Now contemplate the distinction between sliding alongside a straight line and doing the identical alongside the circumference of a circle; in case you slide items $8pi r$ alongside a straight line, you arrive at your vacation spot simply as you began, with out having ever modified orientation. However in case you do the identical alongside the circumference of a circle you’ll have made a full rotation on the time you come back to your place to begin. When rolling alongside the identical circumference, due to this fact, you’ll have made the 4 rolling rotations plus the one sliding revolution, for a complete of 5!

Put in another way: when the small circle rolls alongside the circumference of the bigger circle, two sorts of motion concurrently happen, revolution and rotation. The 4 actions one initially considers are the 4 revolutions, maybe as a result of these are readily seen. Rotation, alternatively, is harder to know.

Coin rolling

It’s tough to make progress on this type of downside simply by fascinated about it, so it is very important confirm the state of affairs by experimentation. For instance, you must strive modelling this downside utilizing two cash; if the issue adopted the predictions of most individuals, then when utilizing two same-sized cash the shifting one would rotate $2pi r div 2pi r = 1$ occasions, however as you will note it does so twice. For instance, you may predict that rolling from the highest of the fastened coin to its backside would outcome within the rolling coin ending upside-down, however in truth it should have unexpectedly carried out a whole rotation by this level. I extremely advocate making an attempt this your self.

See Also

Should you discover it tough to know how issues work on a circle, you may also need to think about what would occur on a sq.. When a circle rolling alongside the outer periphery of a sq. encounters the primary nook, it should rotate an “further” 90° to proceed alongside the following aspect. It will occur once more at every nook, and since 90° x 4 = 360°, this accounts for an extra full revolution.

Every of the above explanations describes the circle’s motion as a
decomposition into rotation and revolution, however in actuality no such decomposition is going down. Simply as a human’s coronary heart and lungs work concurrently, rotation and revolution happen collectively. Separating revolution from rotation is useful for understanding, however doing so doesn’t present a elementary answer. Some say that the make-up of our brains doesn’t permit for multitasking, however studying to concurrently comprehend such phenomena can be of nice worth.

An identical downside appeared in Aha! Gotcha: Paradoxes to Puzzle and Delight by Martin Gardner and likewise in Scientific American in 1868. Should you can consider an alternate proof or rationalization for this downside, please put up a remark or email us!

Concerning the creator


Yutaka Nishiyama is a professor at Osaka College of Economics, Japan. After finding out arithmetic on the College of Kyoto he went on to work for IBM Japan for 14 years. He’s within the arithmetic that happens in each day life, and has written ten books concerning the topic. The newest one is The Mysterious Quantity 6174: Considered one of 30 mathematical subjects in each day life, printed by Gendai Sugakusha in July 2013 (ISBN978-4-7687-6174-8). You may go to his web site here.

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