Wikipedia:Reference desk/Archives/Mathematics/2012 November 8

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November 8

Perfect circles

This is a purely theoretical question, I understand perfect circles do not exist and probably never shall, but could someone answer this for me?

If perfect circles existed, as in Aristotle's wheel paradox, then a small circles circumference could be made to trace a ridiculous length by being attached to a larger circle. My question is this, would this make it possible to create infinite energy?

Different sized cogs turn at different rates due to the way their teeth are arranged. This is due to how many teeth fit on cogs relative to their size and the size of the cogs around them, right?

But if perfect circles existed, surely systems could be developed where cogs were replaced with completely circular pieces of metal or wood (as a circle could be described as a cog with an infinite amount of infinitesimal teeth), right? And if this was so different sized circular pieces, unlike cogs, could turn at the same rates regardless of their size. Going by set theory both would have an amount of points in their circumference and the teeth could be put into 1-1 correspondence.

Now, my question is this, given that all cogs/circles could turn at the same rate regardless, could a cog-based device be created that gave infinite energy? If so how? Equations would be greatly welcomed!

Note: If you disagree with the idea that the points on the cogs/circles would be able to be put into one-to-one correspondence, please just humor me and assume that all circles could turn at the same rate regardless to address the question!

(I have also posted this in the science desk as I was unsure if it was a math or physics question) — Preceding unsigned comment added by 109.157.115.249 (talk) 19:40, 8 November 2012 (UTC)

How much wood could a woodchuck chuck if a woodchuck could chuck wood? — Preceding unsigned comment added by 96.36.119.54 (talk) 14:56, 9 November 2012 (UTC)

A woodchuck would chuck, as much wood as a woodchuck could chuck, if a woodchuck could chuck wood. StuRat (talk) 18:50, 9 November 2012 (UTC)

Unwisely attempting a sensible answer to what appears as a strange question.

If you have gearwheels of differing sizes, they don't all rotate at the same rate. The teeth move past each other at the rate of one tooth of gear A for one tooth of gear B (unless weird stuff is happening). Thus if a gear of circumference 100 with 100 teeth engages with a smaller gear of circumference 50 with only 50 (The teeth have the same size), while the big gear rotates once, the small gear will rotate twice.

If you do this with perfect circles and assume good friction and no slippage then for every rotation of a circle of circumference 100, a meshed snmaller circle of circumference 50 will go round twice. One of circumference 10 would go round 10 times. One of circumference 1, a hundred times. One of circumference 0.001 would go round 100,000 times.

There's no magic, no infinite energy involved. Just ratios. -- SGBailey (talk) 19:42, 9 November 2012 (UTC)

I see three aspects of your question, one related to logic, one related to geometry, and one related to physics.

1. Logically, you assume that no perfect circle exists, and your question is this: "If all circles could turn at the same rate, could a device be created that gave infinite energy?" Your premise (that all circles could turn at the same rate) is true (because when no perfect circle exists then "all circles" means "no circles", and any statement concerning all circles is then true). Whether it is possible to create infinite energy has nothing to do with circles.
2. Geometrically, a circle could be described as a cog with an infinite number of infinitesimal teeth. A cog can turn around because the changed position of the teeth can be observed, but there is no way to observe if a circle turns around. What is true for a cog with a finite number of teeth may not be true for the limiting case.
3. Physically, the motion of an electron around an atomic nucleus is not really a motion, even if the state has nonzero angular momentum, because the direction from the nucleus to the electron is undefined. This is a modern version of Aristotle's wheel paradox. It is not a source of infinite energy, however.

Bo Jacoby (talk) 21:39, 9 November 2012 (UTC).

We can already make different sized cogs turn at the same rate, simply by putting them on a common shaft or by connecting their shafts with a belt. This does not result in infinite energy.--80.109.106.49 (talk) 22:20, 9 November 2012 (UTC)