merjet

Aristotle's wheel paradox

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8 hours ago, merjet said:

You got it wrong, propagandist trying to rewrite history.

No, I didn't.  And copycatery - which you resort to a lot - is such a confession of weakness.  Plus, you're lousy at it.

Ellen

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9 hours ago, anthony said:

Yes, Darrell - if you mean the two wheels are separate, rolling independently.

Back to the wheel in a wheel:

The entire paradox is premised on the pesky small wheel which pops up at the end, having (we see) rolled only once--and having traveled at an identical forward speed (transitional velocity) to its big brother (self-evidently) and ending up in its exact original location within the large wheel.

How did it get there?

Why has it laterally traveled further than its own circumference, in a single revolution? "Surely" - some will believe - "It has to have skidded/etc./etc. to have moved so far in its one (smaller) revolution, in the same time?".

1. Such "slippage" contradicts the identity of the wheel. And one's experience in reality. 2. The explanation (how and why) is clear when one accepts (as one induces from experience -and- formally learns) that any inner circle/wheel/point within a wheel, is turning slower than any other circles, (etc.) outside of its circumference - up to and including the main wheel.  Therefore, it is able to rotate once, slower, (in the same period the big wheel rotates once, a little quicker) -- while moving a distance a few or several times its length of circumference.  A distance determined by the large wheel's circumference.

To look at this in reverse, if the (erroneous) assumption is made by casual observation, that the small wheel 'turning-speed' and the big wheel 'turning-speed' are identical, then the paradox remains a paradox. Although one knows, self-evidently, that the wheels always 'work', in reality, one can't explain this phenomenon.

Relative *tangential velocity* is the full explanation for the paradox.

(I suspect more than ever, the second 'track' was added in later. Not just to complicate, but more to attempt to justify "slippage" where there is none ).

Hi Tony,

No offense, but it does seem as though you're having trouble visualizing what is happening, so I am attempting to take you through a logical progression of steps in order to prove the point. Please forget about what your eyes are telling you and focus on the inescapable logic of the derivation.

1. You agreed that if a big wheel and a small wheel rolled independently N times, then the big wheel would roll farther than the small wheel. Correct?

2. Now, if the big wheel and small wheel were rigidly connected to opposite ends of an axle, then the big and small wheels would have to turn the same number of times. Correct? So, if the big wheel turned N times then the small wheel would also have to turn N times. Right?

You said earlier that if a person were to roll a tapered glass or party cup on the table or floor, it would veer off to one side. Correct? The reason the cup would veer off to the side is that the large end rolls farther than the small end.

3. Similarly, if two wheels that were the same sizes as the ends of the cup were rigidly connected by an axle the same length as the height of the cup and both were in contact with a flat surface so that they rolled without slipping, they would veer off to the side. That follows logically from facts (1) and (2), namely (1) that the big wheel will travel farther than the small wheel and that (2) that the wheels are rigidly connected. Correct?

4. Now, if the small wheel is placed on a support so that the axle connecting the small wheel to the big wheel is parallel to the ground, the same thing would happen as in statement 3, namely the wheel assembly would veer off to the side. That follows from the fact that placing the small wheel on a support doesn't alter the fact that it won't travel as far as the big wheel nor does it change the fact that the wheels are rigidly connected. Therefore, since (1) and (2) are still valid, the pair of wheels must veer off to the side. Correct?

Aristotle's paradox implies that a pair of wheels joined by an axle won't veer off to the side. It implies that they will both roll the same distance without slipping. See how that contradicts the logic of the proof? That's why it is a paradox. It asserts a contradiction. In reality, either the small wheel must slip or the big wheel must slip or the wheel assembly must veer off to the side.

Darrell

 

 

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On December 4, 2018 at 7:56 AM, Max said:

 

I can't remember, but you stated your position again so explicitly, that I was wondering why it should be so important. As you'll have seen, there are many different definitions of a paradox, and also many different kinds of paradoxes. No problem for me, I don't believe so much in the "one and only" correct definition à la Rand (e.g. her definition of altruism). My viewpoint is, that so many of those classic "paradoxes" are known as "paradoxes", that I see no reason not to use that term for that kind of "paradoxes", genuine or not.

For myself I use the definition: an apparent contradiction in an argument caused by a more or less hidden error in the argument or in the premises. In general it isn't difficult to move the error from a false premise to an error in the argument, and an error in the argument can always be thought of as the result of an implicit false premise, there is no sharp distinction between the two options. Changing the formulation a bit can change the formal expression of a paradox, without really changing its essence. Therefore I think my definition isn't that much different from yours, only less restricting, while I also admit false premises. But as I said, I find definitions not that important (the only correct one!) as long as you state them clearly.

Max, I'm not of the "one and only" correct definition viewpoint either.

It's just in the case of this Aristotle's Wheel thing, the false assumption seems so glaringly obvious to me, I wonder at people's being thrown by it even in Greek times, many centuries before the development of calculus.  I wouldn't have thought that anyone today would have any trouble with it, prior to the present enormously long discussion.

By contrast, some of the "paradoxes" you listed are truly difficult, especially, I'd say, the twin paradox.  So I think that at minimum, there's significant variation in degree among the various problems called "paradoxes," even if you want to call them all of the same kind.

Ellen

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1 hour ago, Darrell Hougen said:

Hi Merlin,

I agree that Jonathan is sometimes an obnoxious jackass, but the fact of the matter is that his analysis of Aristotle's paradox is correct. He has also been very patient at times, going out of his way to produce illustrative videos. We all owe him a debt of gratitude for that. And, so far as I know, he hasn't taken this dispute outside of OL. You really should put the Wikipedia page back the way it was or let us do it.

Since you laid down the gauntlet, I'll take a look at your solutions later, when I get the chance.

Cheers,

Darrell

Hi Darrell.

I don’t dispute any of the above. I just wanted to add something to tie it all up, something more than just your “He has also been ...”

My observation has always been that he is patient until he gets obnoxious shit. Then he returns fire. Not just sometimes he’s this way, sometimes that, no. I’ve never seen him go at someone who didn’t earn it. Others may not know the previous interactions and they may think they are seeing Jonathan rip into people unearned and from nowhere, but they are just uninformed about what came before what they are seeing.

Everyone deals with obnoxious shit differently and that’s ok. Some ignore it resolutely and politely carry on with the discussion, even though the other side is not doing so. Some worship civility and beg of others that they would see its eternal, intrinsic importance. Some refuse to continue, saying why and expressing disappointment. Jonathan returns fire. They are all reasonable, defensible responses to obnoxious shit.

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5 hours ago, anthony said:

Hi Darrell, believe me, I put in some time playing around with books and bottles and canisters!

What I assume we are doing here is re-creating a wheel+wheel, experimentally,  by means of other objects. I understand that a cone also has a greater and smaller diameter. However, it is unsuitable for experiment in that it has an inherent bias. You try to force it to roll it straight - but the only way is by inducing slippage. The built-in "bias" biases the effects. So to say.

So the cylinder is the closest to an "extruded" wheel, which rolls straight and true. One cylinder will roll on a surface after it's pushed, in a straight line. Connect another cylinder, of lesser diameter, and repeat - the same outcome. (No slip)

Now place the small cylinder on a 'ledge' of sorts so that both cylinders are supported on surfaces.  "All things being equal" - all the factors I've mentioned have to be precisely right -  there is no reason whatsoever why the cylinder combination(e.g. a wine bottle) will not roll as it did without a second platform--straight and slip-less. The 'rule of tangential velocity' equally applies here, to both 'wheels'.

All it is is an accurate reproduction of what we all know a rolling bottle does when without a 'track'. Bottle plus neck rolls - one rotation - with no slippage. Add a track, and albeit some friction/drag which has to be equalised on both wheels, one can reproduce the same scenario. 

By that standard of reality, IF one finds slippage, IF the bottle rolls skew, we know the setup of the experiment is imprecise.

 

Hi Tony,

The "bias" is that one end is bigger than the other. That's the whole point. If two wheels that matched the sizes of the ends of the cup were rigidly connected to each other by means of an axle the same length as the height of the cup, the assembly would have the same "bias" as the cup. It wouldn't roll straight unless it was forced to roll straight by inducing slippage. But, Aristotle's paradox says that it will roll straight. That's the problem.

You say that if the bottle rolls skew, the setup of the experiment is imprecise, but it is the fact that of the bottle rolling skew that proves the point. It isn't a bug. It's a feature. The fact that the bottle rolls skew proves that Aristotle's paradox is a paradox --- an impossible contradiction.

Darrell

 

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>>> = T 

>> = M 

> = T

>>> Does it occur to anyone that the teeth of the two gears are unequally spaced

>> Are they?

> Evidently, or else they would both mesh. 

Interesting. Most people see immediately that those two gears are equally spaced, it is really obvious. Now you see that the small wheel somehow doesn't mesh well with the chain, and as you can't imagine another cause than a different spacing, you suppose that this gear is differently spaced, your brain overruling the obvious fact that both gears are equally spaced. So you see how unreliable human senses can be for judging what is "reality".

When Jon shows you unambiguously that those gears are indeed equally spaced, you reply:

Not getting this, still. The "same tooth-spacing" on two different circumferences, is non-identical. 'The curves of these two gears are dissimilar, being smaller and larger 'wheels'.

Do you think that using gear wheels of different size might cause such a problem? Then I'd suggest you study some bicycle gearing. Well, I even have a suspicion that the gears in this video have something to do with bicycles.

And of course, one or other can engage with 'no slip'. Not both at once.

Aha! That's what we've told you already a few thousand times. If both wheels cannot engage with "no slip", that means that at least one of them must slip!

The example with gears is chosen as it forces rolling without slipping (which is supposed in the original formulation of the paradox). If one gear must slip, it cannot move, unless it can escape from the chain, that is what you see happening in the video, and what you erroneously attributed to a different spacing.

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5 minutes ago, Jon Letendre said:

Hi Darrell.

I don’t dispute any of the above. I just wanted to add something to tie it all up, something more than just your “He has also been ...”

My observation has always been that he is patient until he gets obnoxious shit. Then he returns fire. Not just sometimes he’s this way, sometimes that, no. I’ve never seen him go at someone who didn’t earn it. Others may not know the previous interactions and they may think they are seeing Jonathan rip into people unearned and from nowhere, but they are just uninformed about what came before what they are seeing.

Everyone deals with obnoxious shit differently and that’s ok. Some ignore it resolutely and politely carry on with the discussion, even though the other side is not doing so. Some worship civility and beg of others that they would see its eternal, intrinsic importance. Some refuse to continue, saying why and expressing disappointment. Jonathan returns fire. They are all reasonable, defensible responses to obnoxious shit.

Hi Jon,

Perhaps I was too hasty since I haven't been reading all the posts on OL, but I just noticed that some people are quick to engage in name calling. I'm not an absolutist when it comes to being polite. If someone is being flagrantly rude and disparaging, I'll sometimes get down in the gutter and engage in a little tit-for-tat. However, I generally dislike being impolite just because someone else doesn't seem to understand something, frustration notwithstanding.

Cheers,

Darrell

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2 hours ago, Darrell Hougen said:

Hi Tony,

No offense, but it does seem as though you're having trouble visualizing what is happening, so I am attempting to take you through a logical progression of steps in order to prove the point. Please forget about what your eyes are telling you and focus on the inescapable logic of the derivation.

1. You agreed that if a big wheel and a small wheel rolled independently N times, then the big wheel would roll farther than the small wheel. Correct?

2. Now, if the big wheel and small wheel were rigidly connected to opposite ends of an axle, then the big and small wheels would have to turn the same number of times. Correct? So, if the big wheel turned N times then the small wheel would also have to turn N times. Right?

You said earlier that if a person were to roll a tapered glass or party cup on the table or floor, it would veer off to one side. Correct? The reason the cup would veer off to the side is that the large end rolls farther than the small end.

3. Similarly, if two wheels that were the same sizes as the ends of the cup were rigidly connected by an axle the same length as the height of the cup and both were in contact with a flat surface so that they rolled without slipping, they would veer off to the side. That follows logically from facts (1) and (2), namely (1) that the big wheel will travel farther than the small wheel and that (2) that the wheels are rigidly connected. Correct?

4. Now, if the small wheel is placed on a support so that the axle connecting the small wheel to the big wheel is parallel to the ground, the same thing would happen as in statement 3, namely the wheel assembly would veer off to the side. That follows from the fact that placing the small wheel on a support doesn't alter the fact that it won't travel as far as the big wheel nor does it change the fact that the wheels are rigidly connected. Therefore, since (1) and (2) are still valid, the pair of wheels must veer off to the side. Correct?

Aristotle's paradox implies that a pair of wheels joined by an axle won't veer off to the side. It implies that they will both roll the same distance without slipping. See how that contradicts the logic of the proof? That's why it is a paradox. It asserts a contradiction. In reality, either the small wheel must slip or the big wheel must slip or the wheel assembly must veer off to the side.

Darrell

 

 

I pick up at point 4. When the axle has been raised onto a rail (or track), to exactly compensate for the radii disparity, you have eliminated the bias. If indeed the conjoined wheels would rotate at the same tangential velocity, the assembly would veer off. But do they? No, and you have also confirmed this recently. Now they are fixed together as a single unit, the small wheel rotates one rev to the big wheel's one revolution. The small one must turn slower, proportionately, so completing its rotation at the same point, same distance, and no veering- off.

Darrell, the "logic" begins with that circle diagram, which one can confirm or reject by a comparison to facts: Seeing and knowing wheels in motion. In all wheels there cannot -possibly - be slippage. Otherwise, a wheel can't function.

Observe a 'circle' inside any wheel - i.e. an imaginary inner wheel - Watch it closely, and it does and must outrun the length of its circumference (for one turn of the main wheel). No one can refute this fact.

It *has to*, by definition, it is a smaller i.e., a lesser circumference circle.  If it were extremely tiny in relation to the big wheel, anyone could see this effect instantly, I surmize. Lateral distance covered would now be many times this little circumference.

And visibly, it doesn't slip. Now, if that 'circle' were "extruded" to be a second, inner wheel - and placed level on a track, with all things being equal - what can change?

So Aristotle's diagram is true to reality - uncontradictory. (in his terms, one might call the large wheel the "prime mover"- the cause - of which the small wheel's motion is an effect and which has no influence on proceedings). 

All that's missing is an explanation of the phenomenon, and most certainly in my view, it is in that disparity between the tangential velocity of the wheels. You've not taken that into account. 

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2 hours ago, Max said:

>>> = T 

>> = M 

> = T

>>> Does it occur to anyone that the teeth of the two gears are unequally spaced

>> Are they?

> Evidently, or else they would both mesh. 

Interesting. Most people see immediately that those two gears are equally spaced, it is really obvious. Now you see that the small wheel somehow doesn't mesh well with the chain, and as you can't imagine another cause than a different spacing, you suppose that this gear is differently spaced, your brain overruling the obvious fact that both gears are equally spaced. So you see how unreliable human senses can be for judging what is "reality".

When Jon shows you unambiguously that those gears are indeed equally spaced, you reply:

Not getting this, still. The "same tooth-spacing" on two different circumferences, is non-identical. 'The curves of these two gears are dissimilar, being smaller and larger 'wheels'.

Do you think that using gear wheels of different size might cause such a problem? Then I'd suggest you study some bicycle gearing. Well, I even have a suspicion that the gears in this video have something to do with bicycles.

And of course, one or other can engage with 'no slip'. Not both at once.

Aha! That's what we've told you already a few thousand times. If both wheels cannot engage with "no slip", that means that at least one of them must slip!

The example with gears is chosen as it forces rolling without slipping (which is supposed in the original formulation of the paradox). If one gear must slip, it cannot move, unless it can escape from the chain, that is what you see happening in the video, and what you erroneously attributed to a different spacing.

Hmm. I suppose a gear of one-meter diameter - and everything the same with teeth-size, frequency (etc.), as a gear 10cm in diameter are both going to properly mesh with the exact same cogs or pinion?

Tell me another one. 

You too, are not getting the disparate outer curvatures of the two gear-wheels. This affects the meshing, and even moreso when you try to engage both gears ~ simultaneously ~ into two identical pinions/chains.

Seriously, this is Engineering 101.

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2 hours ago, Ellen Stuttle said:

Slippage / Spin-out

I typed in Xeno and got Xenophanes. And Zeno and got:

Scientology, Seizures, and Science by Edward Hudgins. January 13, 2009 -- Jett Travolta, the sixteen-year-old son of actors John Travolta and Kelly Preston, died recently of what the autopsy found to be a seizure. The boy had a history of seizures and unconfirmed reports suggest that his parents acted responsibly to ensure he was on medication to mitigate his condition . . . . end quote

Excellent letter. You should look it up. Peter

And this. The Problem of Induction (1953, 1974) by Carl Popper. For a brief formulation of the problem of induction we can turn to Born, who writes: '. . . no observation or experiment, however extended, can give more than a finite number of repetitions'; therefore, 'the statement of a law - B depends on A - always transcends experience. Yet this kind of statement is made everywhere and all the time, and sometimes from scanty material. ' 1

. . . . The question of the sources of our knowledge, like so many authoritarian questions, is a genetic one. It asks for the origin of our knowledge, in the belief that knowledge may legitimize itself by its pedigree. The nobility of the racially pure knowledge, the untainted knowledge, the knowledge which derives from the highest authority, if possible from God: these are the (often unconscious) metaphysical ideas behind the question. My modified question, 'How can we hope to detect error?' may be said to derive from the view that such pure, untainted and certain sources do not exist, and that questions of origin or of purity should not be confounded with questions of validity, or of truth. This view may be said to be as old as Xenophanes. Xenophanes knew that our knowledge is guesswork, opinion - doxa rather than episteme - as shown by his verses [quoted on p. 31 above]. Yet the traditional question of the authoritative sources of knowledge is repeated even today - and very often by positivists and by other philosophers who believe themselves to be in revolt against authority.

The proper answer to my question 'How can we hope to detect and eliminate error?' is, I believe, 'By criticizing the theories or guesses of others and - if we can train ourselves to do so - by criticizing our own theories or guesses.' (The latter point is highly desirable, but not indispensable; for if we fail to criticize our own theories, there may be others to do it for us.) This answer sums up a position which I propose to call 'critical rationalism'. It is a view, an attitude, and a tradition, which we owe to the Greeks. It is very different from the 'rationalism' or 'intellectualism' of Descartes and his school, and very different even from the epistemology of Kant. Yet in the field of ethics, of moral knowledge, it was approached by Kant with his principle of autonomy- This principle expresses his realization that we must not accept the command of an authority, however exalted, as the basis of ethics. For whenever we are faced with a command by an authority, it is for us to judge, critically, whether it is moral or immoral to obey. The authority may have power to enforce its commands, and we may be powerless to resist. But if we have the physical power of choice, then the ultimate responsibility remains with us. It is our own critical decision whether to obey a command; whether to submit to an authority . . . . end quote

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2 hours ago, Darrell Hougen said:

Hi Tony,

The "bias" is that one end is bigger than the other. That's the whole point. If two wheels that matched the sizes of the ends of the cup were rigidly connected to each other by means of an axle the same length as the height of the cup, the assembly would have the same "bias" as the cup. It wouldn't roll straight unless it was forced to roll straight by inducing slippage. But, Aristotle's paradox says that it will roll straight. That's the problem.

You say that if the bottle rolls skew, the setup of the experiment is imprecise, but it is the fact that of the bottle rolling skew that proves the point. It isn't a bug. It's a feature. The fact that the bottle rolls skew proves that Aristotle's paradox is a paradox --- an impossible contradiction.

Darrell

 

Darrell, This is why I said a conical shape is unsuitable as a model for the wheels. Being biased, it isn't "neutral". ;)

The cylindrical shape answers the purpose much better.

As I said, a cylinder rolling on the floor hasn't a bias. Two connected cylinders - like a bottle (an extended wheel in a wheel) -  don't have a bias. Why should anything change when you place the assembly on two tracks, AND, compensating for their diameter differences? Rolling skew would be the bug. Rolling straight without slip, is the "feature".

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6 hours ago, Jon Letendre said:

Evidently not. The wheels have the same tooth spacing and the chains are also dimensionally the same.

This video proves it...

 

 

It "proves" this. The 2 gear-wheels are observably turning at different speeds while staying in synch (exactly as do the wheels in the paradox) so they cannot mesh simultaneously on the 2 chains. 

Remember? Different circumferences - unequal tangential speeds.

"Proves" my explanation of the 'paradox'.

Thanks.

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U R Ell ...

47 minutes ago, Peter said:

For a brief formulation of the problem of induction we can turn to Born, who writes: '. . . 

http://dieoff.org/page126.htm

37 minutes ago, anthony said:

Rolling straight without slip, is the "feature".

Can you explain "flange squeal" on curves?

 

Edited by william.scherk
Canada Line railcars have flange squeal on tight curves. Expo Line railcars do not. Why not?

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1 hour ago, anthony said:

I pick up at point 4. When the axle has been raised onto a rail (or track), to exactly compensate for the radii disparity, you have eliminated the bias. If indeed the conjoined wheels would rotate at the same tangential velocity, the assembly would veer off. But do they? No, and you have also confirmed this recently. Now they are fixed together as a single unit, the small wheel rotates one rev to the big wheel's one revolution. The small one must turn slower, proportionately, so completing its rotation at the same point, same distance, and no veering- off.

Darrell, the "logic" begins with that circle diagram, which one can confirm or reject by a comparison to facts: Seeing and knowing wheels in motion. In all wheels there cannot -possibly - be slippage. Otherwise, a wheel can't function.

Observe a 'circle' inside any wheel - i.e. an imaginary inner wheel - Watch it closely, and it does and must outrun the length of its circumference (for one turn of the main wheel). No one can refute this fact.

It *has to*, by definition, it is a smaller i.e., a lesser circumference circle.  If it were extremely tiny in relation to the big wheel, anyone could see this effect instantly, I surmize. Lateral distance covered would now be many times this little circumference.

And visibly, it doesn't slip. Now, if that 'circle' were "extruded" to be a second, inner wheel - and placed level on a track - what can change?

So Aristotle's diagram is true to reality, uncontradictory. (in his terms, one might call the large wheel the "prime mover"- the cause - of which the small wheel's motion is an effect and which has no influence on proceedings). 

All that's missing is an explanation of the phenomenon, and most certainly in my view, it is in that disparity between the tangential velocity of the wheels. You've not taken that into account. 

Hi Tony,

The "bias" is not caused by the fact that the cone or cup or wheel assembly is leaning to one side. It is caused by the fact that one wheel is larger than the other. Therefore, leveling the assembly has no effect on the "bias."

Tangential velocity is the velocity of a point on the circumference of a wheel relative to the center point of the wheel. The assembly veers off because the tangential velocities of the two wheels are different. If the tangential velocities were the same, the assembly would not veer off even if one end were higher than the other.

Darrell

 

 

 

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4 hours ago, Ellen Stuttle said:

Slippage / Spin-out

 

[...]

-

 

The difference between the distance around the larger circle's circumference and the lateral distance traversed is what's being called "spinning-out."

Ellen

 

I can see the gist (of what I think is in error), and I did a way back. Ellen, there is a glaringly false premise everyone has accepted, which is that the two wheels' circumferences turn at the same speed. While a few here have referred in passing to tangential velocity, I am not sure they realise its ramifications. 

For rolling wheels or planets in orbit, or runners round a track, the principle is the same. The outer ring has to move further and -therefore - faster - than the inner ring, in order to maintain synchronisation.

Two fixed wheels revolving one inside the other,  do not revolve at the identical tangential speed. Logically, they cannot. Because of this there isn't any "spinning out" or slip. The small wheel turns slower in its single revolution than the outer one. It 'arrives' at the same time and place as the outer one. (While of course traversing a greater distance than its circumference) . What puzzled Aristotle, very possibly, was finding an explanation for what he observed in reality, and varying tangential speed in a circle is what I think explains it.

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2 hours ago, Ellen Stuttle said:

It's just in the case of this Aristotle's Wheel thing, the false assumption seems so glaringly obvious to me, I wonder at people's being thrown by it even in Greek times, many centuries before the development of calculus.  I wouldn't have thought that anyone today would have any trouble with it, prior to the present enormously long discussion.

Amazing, isn't it?

2 hours ago, Ellen Stuttle said:

By contrast, some of the "paradoxes" you listed are truly difficult, especially, I'd say, the twin paradox.  So I think that at minimum, there's significant variation in degree among the various problems called "paradoxes," even if you want to call them all of the same kind.Ellen

Oh, sure there is a large variation in difficulty among those paradoxes. Not surprisingly those ancient ones are easy to us, while the more recent ones can be difficult. 

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Children of a Lesser God:

 

Trying to share the experience of music with a deaf person: Easier than trying to share the experience of visuospatial/mechanical reasoning competence with someone who lacks it.

J

 

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21 minutes ago, anthony said:

I can see the gist (of what I think is in error), and I did a way back. Ellen, there is a glaringly false premise everyone has accepted, which is that the two wheels' circumferences turn at the same speed. While a few here have referred in passing to tangential velocity, I am not sure they realise its consequences. 

For rolling wheels or planets in orbit, or runners round a track, the principle is the same. The outer ring has to move further and -therefore - faster - than the inner ring, in order to maintain synchronisation.

Two fixed wheels revolving one inside the other,  do not revolve at the identical tangential speed. Logically, they cannot. Because of this there isn't any "spinning out" or slip. The small wheel turns slower in its single revolution than the outer one. It 'arrives' at the same time and place as the outer one. (While of course traversing a greater distance than its circumference) . What puzzled Aristotle, very possibly, was finding an explanation for what he observed in reality, and varying tangential speed in a circle is what I think explains it.

Well, I tried.

According to you, Tony, if the outer wheel revolves at one revolution per year, it goes farther in one revolution than if it revolves at one revolution per second.  Oy.

Ellen

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1 hour ago, Darrell Hougen said:

Hi Tony,

The "bias" is not caused by the fact that the cone or cup or wheel assembly is leaning to one side. It is caused by the fact that one wheel is larger than the other. Therefore, leveling the assembly has no effect on the "bias."

Tangential velocity is the velocity of a point on the circumference of a wheel relative to the center point of the wheel. The assembly veers off because the tangential velocities of the two wheels are different. If the tangential velocities were the same, the assembly would not veer off even if one end were higher than the other.

Darrell

 

 

 

I feel we are talking past each other, Darrell. I haven't that I recall, talked about leveling (maybe, once, loosely) and certainly not "leaning".

To get back on point,  I believe this is all about a second 'track' for the inner wheel? Is that right? This is best depicted by the cylinders I mention. The cylinder assembly is indeed not leaning to one side--a sectional view represents two wheels, one "larger than the other" accurate to the wheel diagram.

Briefly, I maintain that placing the bottle/combined cylinders onto two tracks (one beneath the neck, the other beneath the bottle, and compensating for their radii difference) is not going to, in essence, change anything about their combined motion. The "neck" or smaller wheel, turns slower (as we know)  in the same time as the main bottle turns slightly faster, therefore without slippage, and the two finish at the same endpoint after a rotation by each.

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1 hour ago, anthony said:

Hmm. I suppose a gear of one-meter diameter - and everything the same with teeth-size, frequency (etc.), as a gear 10cm in diameter are both going to properly mesh with the exact same cogs or pinion?

Don't evade the question: can the two gear wheels in the video properly mesh with the chain? 

1 hour ago, anthony said:

You too, are not getting the disparate outer curvatures of the two gear-wheels. This affects the meshing, and even moreso when you try to engage both gears ~ simultaneously ~ into two identical pinions/chains.

Separately they mesh perfectly. Simultaneously they cannot. That is not some discovery of yours, I've argued that countless times in this thread, just read all my posts! The reason why? Because if the large wheel rolls without slipping, i.e. meshes perfectly, the small wheel must slip to cover the same distance. With gears this is not possible, unless the wheel moves out of the chain so that it no longer meshes, to make that slipping movement possible. That is the truth that you are continuously evading, pretending that you yourself have solved the paradox. Not!

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39 minutes ago, Ellen Stuttle said:

Well, I tried.

According to you, Tony, if the outer wheel revolves at one revolution per year, it goes farther in one revolution than if it revolves at one revolution per second.  Oy.

Ellen

Jesus, try harder! ;) Does nobody get relative velocity in a circle? If two moons are in orbit around your planet (!) and visibly stay permanently close together, but you know that one is much more distant - which one is moving faster?

If a point on the outer rim of a wheel moves at 15m per sec when revolving, does a point anywhere inside the wheel a. move faster b. move slower c. the same speed?

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On 11/25/2018 at 7:53 PM, Jonathan said:

Merlin has made this same stupid comment.

So, Tony,  your "idea" here, your "argument," is that since you don't see tracks or ledges out on the road while you drive your car, the idea of them is ridiculous?

 

Doh! I found an example right away on the Wikipedia page! Thank God that I got to it before Merlin dishonestly erased it:

A modern approximation of such an experiment is often performed by car drivers who park too close to a curb. The car's outer tire rolls without slipping on the road surface while the inner hubcap both rolls and slips across the curb; the slipping is evidenced by a screeching noise.

Oh noes, Tony! Now what?

J

 

 

This has to take the prize for the most specious explanation of the paradox in Wikipedia. I hope it did get erased. Bad parking and - "the slipping is evidenced by a screeching noise", all to demonstrate that it CAN happen that slippage between hub and tire occurs. Another million times it does not. What logical fallacy to call this?

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10 hours ago, anthony said:

This has to take the prize for the most specious explanation of the paradox in Wikipedia. I hope it did get erased. Bad parking and - "the slipping is evidenced by a screeching noise", all to demonstrate that it CAN happen that slippage between hub and tire occurs. Another million times it does not. What logical fallacy to call this?

The Indertminacy Fallacy?

Hubs and tires only sometimes behave according to their physical nature? They disobey the laws of physics as we know them? When tested under the same conditions, they do not deliver repeatable results? Science and reality do not apply to them? And we know this because we trust Tony when he says that he has rigorously tested such things. He is a master of precision and scientific control. There is no one who is more aware and observant. Nothing gets past him. He's on top of the shit. So, therefore, we totally know without any doubt.

Wheels and hubs have volition, and sometimes choose to slip, and sometimes not.

Anyway, Tony, you've once again, as predicted, misrepresented what you read. You wrote, "all to demonstrate that it CAN happen that slippage between hub and tire occurs."

FALSE!!!! We've told you this several times now. You keep going back to it. No one has or is claiming that there is slippage between the hub and tire. Again: No one is claiming that there is slippage between the hub and tire. That is not where the slippage occurs. The point of the comment that you quoted from Wikipedia is that there is slippage between the hub and the curb that it contacts.

J

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13 hours ago, Max said:

Don't evade the question: can the two gear wheels in the video properly mesh with the chain? 

Separately they mesh perfectly. Simultaneously they cannot. That is not some discovery of yours, I've argued that countless times in this thread, just read all my posts! The reason why? Because if the large wheel rolls without slipping, i.e. meshes perfectly, the small wheel must slip to cover the same distance. With gears this is not possible, unless the wheel moves out of the chain so that it no longer meshes, to make that slipping movement possible. That is the truth that you are continuously evading, pretending that you yourself have solved the paradox. Not!

"The small wheel *must* slip to cover the same distance".

The only viable explanation has been staring you in the face, but you prefer the "slippage" theory that has no factual base? Incredible.

Problem: Gear A will engage a chain; gear B will engage a chain. But A and B will not at the same time.

You yourself have made an allusion to the formulae of tangential velocities in a wheel, but apparently it's a theory you have not tried to apply. The theory is perceivable in action and can be logically validated . It has formulations in mathematics and geometry.  One can show it experimentally. And yet you ignore these in favor of an explanation that contradicts the identity of wheel motion..

Q. Why do A and B not mesh simultaneously?** 

A. Each is turning at a different velocity.

Have another look below. 

(** over and above the reason of different curvatures on each gear) 

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20 hours ago, Jon Letendre said:

This time the large wheel rolls without slip

 

Here it is. A crystal clear demonstration of differences in 'turning' speed, or 'rotational' speed or technically, "tangential velocity".

A lower rate of turn for the small wheel allows it, by necessity, to travel the same distance as the outer one, in the same time, with the same forward speed and complete exactly one revolution, too - and yet, travel past its own circumference length. 

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