Aristotle's Wheel "Paradox"

[dream_weaver]

9 minutes ago, StrictlyLogical said:

The key I think is to remember the purpose of the exercise is not to impress others or improve other people's thinking (when not asked to) but to clarify one's own.

 

I think in this case the problem is actually ill-posed, and uses a kind of fundamental misdirection.

The wording in the initial setup draws attention to the difference in circumference. As if identifying it were not enough, eliciting for a paradox or problem casts the initial shadow here.

Quote

The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem.

Later redressing of the original setup tends to add other elements to take into consideration as it goes from antiquity to modern times.

[whYNOT]

On 5/31/2021 at 1:34 PM, StrictlyLogical said:

That's "affirming", not refuting. 

Something jams when things cannot slip, and otherwise would: it's just a matter of friction at the contact points, gears/cogs are equivalent to very high friction which does not allow slippage, whereas normal rolling surfaces (wood, rubber, metal etc). when there is an inconsistency of speed between touching surfaces, would tend to skid or slip.

True for a "track" which is single rigid object (with a sort of stair step) moving at a single speed. 

But you could use a double wheel type thing, for example, to slide or feed work pieces, packages, or whatever, underneath it, shallower ones at one wheel and thicker ones at the other. Those different objects would be conveyed, at different speeds.

Yes, but the mechanical applications are not an integral part of the paradox. Certainly, there are endless possible uses.

Anyway, the ~solution~ to the puzzle wasn't needed or asked for; the implied intent of the diagram by whoever posed it surely was simply to examine the "paradox", (an ~apparent~ contradiction of dissimilar circumferences but traveling the same distance in one rev), and at most, explain it - since it is what exactly happens with wheels and hubs (circles and radii).

If one set up such an experiment (wheels on twin tracks), one would need to equalize the weight distribution and the friction of both wheels on both surfaces. Some extra force would be required to roll the assembly at a steady speed, because the smaller, slower turning wheel would act as a drag on the larger one. Then "slippage" ~would~ be artificially induced, in one wheel or both.

Edited June 1, 2021 by whYNOT