Aristotle's wheel paradox

[anthony]

3 hours ago, Max said:

 

I've explained that in detail in this post (click on the arrow): 

 

 

Didn't you read that? It answers all your questions.

 

And you'll also see why these are silly questions.

Right. I see it. Good effort. One helluva investment for so little return. 

(I do not think Aristotle was looking for solutions to the phenomenon, it puzzled him, that's all).

[Jon Letendre]

Does a pigeon see what you show him?

Of course! The light enters the iris and signals are sent to the brain.

It sees everything you show him.

[BaalChatzaf]

1 hour ago, Jonathan said:

Indeed! But, you haven't answered any of the questions.

Describe what must happen during the rolling, not after it. Which of the wheels must roll without slipping, and which must roll with slipping, and why? Do you see the blue and green dots on the wheels? One is on the small wheel, and the other is on the large wheel. What path will each take, and why? One will create a proper cycloid. What type of cycloid will the other create, curtate or prolate? Why?

J

If the outer wheel does not slip, the inner wheel does.

 

[anthony]

You induce slippage to either of two fixed wheels, and something breaks, or both will slip.

But there's the math and animations to prove otherwise.

A long way from reality.

[Jon Letendre]

5 minutes ago, anthony said:

You induce slippage to either of two fixed wheels, and something breaks.

But there's the math and animations to prove otherwise.

A long way from reality.

Tony’s current mental state:

When you roll the archery target, the red, blue and black circles don’t shred apart from one another.

Therefore, the math and animations, which imply they do or could or should or would, must be false.

[anthony]

 

No, the way to make them "shred apart" is make every circle a wheel and to place every wheel on its own track (fixed - where, how..? no matter).  Induce selected slippage, roll the target down a slope, stand back - et voila, destruction.

Duh.

 

[Jonathan]

1 hour ago, merjet said:

I've explained it many times.

Where? When?

Cite the post in which you identified what is paradoxical about a wheel rolling over a surface while an inner wheel attached to it is not rolling on its own surface.

[Jonathan]

43 minutes ago, BaalChatzaf said:

If the outer wheel does not slip, the inner wheel does.

 

False.

[Max]

1 hour ago, anthony said:

Right. I see it. Good effort. One helluva investment for so little return. 

(I do not think Aristotle was looking for solutions to the phenomenon, it puzzled him, that's all).

We don't know for sure that Aristotle wrote that text at all, perhaps it was Archytas of Tarentum, as has been suggested. But whoever it was, we can't know whether he was looking for solutions or not. I think he was (it would be rather unnatural for such a person not to try to solve the puzzle), but that he couldn't find the solution. After all, after him people like Galileo, Mersenne, Fermat and Boyle also tried to solve the puzzle. 

[Jonathan]

If we were to duct tape a pigeon to a wheel...

 

...and then roll the wheel, eggheaded geniuses would be confused, and they'd ponder how it is possible that, after one rotation of the wheel, the pigeon moves the distance of the wheel's circumference rather than the distance of the pigeon's perimeter. It's a danged paradox! It don't make no sense! It's Hawrgwarsh!

Everyone knows that anything that you stick to a wheel should not move the distance that the wheel rolls, but should move the distance of the thing's own perimeter or circumference, right?

J

[Max]

1 hour ago, BaalChatzaf said:

If the outer wheel does not slip, the inner wheel does.

It's the other way around. The shortest cable determines the movement, as it can't be lengthened, but the large cable can be loosened. The small wheel rolls without slipping, generating a proper cycloid. The large wheel is slipping backwards, loosening its cable, generating part of a prolate cycloid.

[Jonathan]

10 minutes ago, Max said:

It's the other way around. The shortest cable determines the movement, as it can't be lengthened, but the large cable can be loosened. The small wheel rolls without slipping, generating a proper cycloid. The large wheel is slipping backwards, loosening its cable, generating part of a prolate cycloid.

Correct. The shorter cable on the small wheel limits the entire rig to moving with the small wheel's roll, while the large wheel over-spins/spins-out, and generates slack it its cable.

J

[merjet]

1 hour ago, Jonathan said:

...and then roll the wheel, eggheaded geniuses would be confused, and they'd ponder how it is possible that, after one rotation of the wheel, the pigeon moves the distance of the wheel's circumference rather than the distance of the pigeon's perimeter. It's a danged paradox! It don't make no sense! It's Hawrgwarsh!

Everyone knows that anything that you stick to a wheel should not move the distance that the wheel rolls, but should move the distance of the thing's own perimeter or circumference, right?

Bubba thought it, but you corrected him -- link.

[Jonathan]

1 hour ago, merjet said:

Bubba thought it, but you corrected him -- link.

And you think it. You think that there is a paradox whenever something is attached to a wheel, and that it rolls the distance that the wheel rolls. You expect it to roll the distance of the perimeter or circumference of the object that is attached to the wheel.

J

[Jonathan]

4 hours ago, merjet said:

I've explained it many times. Wikipedia and Drabkin explain it. However, I'm not surprised at your mental inability to get it.

I responded:

Where? When?

Cite the post in which you identified what is paradoxical about a wheel rolling over a surface while an inner wheel attached to it is not rolling on its own surface.

Still no answer.

J

[Jon Letendre]

3 hours ago, anthony said:

 

No, the way to make them "shred apart" is make every circle a wheel and to place every wheel on its own track (fixed - where, how..? no matter).  Induce selected slippage, roll the target down a slope, stand back - et voila, destruction.

Duh.

 

Let’s leave the archery target whole and round.

We roll it down the hill, and it stays whole, right? Because it’s a single solid thing.

But Jonathan thinks there is slippage, so he thinks the target would have to tear or otherwise come apart.

But it doesn’t come apart.

So Jonathan must be wrong. 

Do I have it right?

[Darrell Hougen]

On 11/10/2018 at 3:58 AM, merjet said:

The Wikipedia page about Aristotle's wheel paradox (link) has been vastly improved!

By me. I included an image that shows (a) the circles before and after rolling one revolution and (b) paths of motion for three points. The image thus helps to show why the smaller circle moves 2*pi*R -- the path of motion of every point on the smaller circle is shorter and more direct than the path of motion of any point on the larger circle.

Unbelievable! I don't know why I'm reading this thread, but I cannot believe you actually edited the Wikipedia page to support your argument. People can say whatever they want on here, but taking this fight outside of OL is way beyond the pale. No one outside of OL asked to be part of this dispute.

--- Darrell

[anthony]

1. You bring in a 'track' to carry the inner wheel - "a track" presupposes friction, drag and a braking effect.

2. But- you introduce "slippage" , intermittent or constant - or..?

so 2. cancels out 1. 

so, why the track and what for, slippage?

1. Will affect the rolling of the large wheel (a braking effect) and therefore of the combined wheel assembly.

2. 'Constant slippage' nullifies the need of a superfluous and cumbersome 'track', as the large wheel will roll freely, as normal (without imposed drag on the inner wheel).  'Intermittent slippage' will cause sporadic rotation of the whole assembly, a "jerkiness".

Everything then is a contradiction in terms. You can't change the relation of outer to inner wheel, they're fixed.

And how any of this attempts an answer to the 'paradox', Alice in Wonderland might guess.

[anthony]

12 hours ago, Jon Letendre said:

Let’s leave the archery target whole and round.

We roll it down the hill, and it stays whole, right? Because it’s a single solid thing.

But Jonathan thinks there is slippage, so he thinks the target would have to tear or otherwise come apart.

But it doesn’t come apart.

So Jonathan must be wrong. 

Do I have it right?

And I was just mentioning Alice!

Now here come Tweedledum and Tweedledee, (which are you - dee or dum?)  Little difference.

I admit you have the patter down well, rehearsed backstage, no doubt.

[BaalChatzaf]

18 hours ago, Jonathan said:

False.

No  true.  The outer wheel move further per revolution because its radius is larger. That means the inner wheel is partly dragged, partly rolled. Because one revolution of the inner wheel corresponds to to shorter length than one revolution of the outer wheel. Since the wheels are rigidly affixed to a common axel  both turn together rigidly.  Pay attention to the mechanics, the motion and the geometry of the rig..

[BaalChatzaf]

17 hours ago, Jonathan said:

Correct. The shorter cable on the small wheel limits the entire rig to moving with the small wheel's roll, while the large wheel over-spins/spins-out, and generates slack it its cable.

J

Even with the cable the inner wheel is dragged of the outerwheel does not slip.

 

[Brant Gaede]

15 hours ago, Darrell Hougen said:

Unbelievable! I don't know why I'm reading this thread, but I cannot believe you actually edited the Wikipedia page to support your argument. People can say whatever they want on here, but taking this fight outside of OL is way beyond the pale. No one outside of OL asked to be part of this dispute.

--- Darrell

Crass is believable.

--Brant

almost, sort of, maybe

arrogance--crass arrogance: Bingo!

[Ellen Stuttle]

3 hours ago, anthony said:

1. You bring in a 'track' to carry the inner wheel - "a track" presupposes friction, drag and a braking effect.

2. But- you introduce "slippage" , intermittent or constant - or..?

so 2. cancels out 1. 

so, why the track and what for, slippage?

1. Will affect the rolling of the large wheel (a braking effect) and therefore of the combined wheel assembly.

2. 'Constant slippage' nullifies the need of a superfluous and cumbersome 'track', as the large wheel will roll freely, as normal (without imposed drag on the inner wheel).  'Intermittent slippage' will cause sporadic rotation of the whole assembly, a "jerkiness".

Everything then is a contradiction in terms. You can't change the relation of outer to inner wheel, they're fixed.

And how any of this attempts an answer to the 'paradox', Alice in Wonderland might guess.

 

Tony,

The track is part of the original problem.

Here's how the problem goes:  Suppose you have a wheel which rolls on a surface.  After one revolution, the wheel will have moved laterally a distance equal to its circumference.

Now suppose that there's an inner circle or wheel, like the hub of a chariot wheel.  Now imagine that the inner circle or wheel is rolling on a surface.  After one revolution it should have traveled laterally on that surface a distance equal to its circumference.

But it doesn't do that.  Instead, it travels the same distance as the outer wheel.

Why?

What is your answer to why the inner circle or wheel, in one revolution of the outer wheel, revolves the same distance as the outer wheel instead of only revolving the distance of its circumference?

Ellen

[Ellen Stuttle]

15 hours ago, Darrell Hougen said:

Unbelievable! I don't know why I'm reading this thread, but I cannot believe you actually edited the Wikipedia page to support your argument. People can say whatever they want on here, but taking this fight outside of OL is way beyond the pale. No one outside of OL asked to be part of this dispute.

--- Darrell

Agreed about Merlin's Wiki editing to support his argument here being "beyond the pale," although I don't find his doing it "unbelievable," since he characteristically gets awfully ego-invested in being right.

Ellen

[anthony]

1 hour ago, Ellen Stuttle said:

 

Tony,

The track is part of the original problem.

Here's how the problem goes:  Suppose you have a wheel which rolls on a surface.  After one revolution, the wheel will have moved laterally a distance equal to its circumference.

Now suppose that there's an inner circle or wheel, like the hub of a chariot wheel.  Now imagine that the inner circle or wheel is rolling on a surface.  After one revolution it should have traveled laterally on that surface a distance equal to its circumference.

But it doesn't do that.  Instead, it travels the same distance as the outer wheel.

Why?

What is your answer to why the inner circle or wheel, in one revolution of the outer wheel, revolves the same distance as the outer wheel instead of only revolving the distance of its circumference?

Ellen

Yes, Ellen, almost precisely how I stated the paradox. I understand. But, to ask, what does the insertion of a second 'track' have to do with anything? Relative to the large wheel and basic or main surface, the inner wheel will - must- do the same as you describe: in short, over-run its own circumference. I've maintained the line, or path (which I think it is) or 'track" - if you like - is a distraction to the main event. So dispense with a superfluous 'track', get the same result.

"Should have traveled laterally on that surface..." Your "should" says it all. I think that's a assumption one might make at first, by associating the valid *circumference=distance* (that applies to the large wheel, only) with the minor wheel. However, why "should" it, its travel is determined only by the outer wheel's movement? My "answer" [to the paradox] is in the fact that the outer wheel is a self-contained entity, and the cause of the effect on the small wheel, which has no reverse relation to the main wheel (unless it - the small one- is fiddled with, like I pointed out).