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Aristotle's wheel paradox

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25 minutes ago, anthony said:

The entire wheel rolls forwards, on two tracks, where there was before just one surface. Assuming the weight on each track is carefully and evenly distributed, the outcome will be what the diagram denotes: the large wheel (tyre) rolls its circumference; the inner wheel (rim) rolls one revolution--but far past its circumference. It does not 'slip', it doesn't need to.

Really? Rolling one revolution, far past its circumference: that is by definition slipping, it is definitely not rolling without slipping, because then it couldn't get past its own circumference.

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30 minutes ago, anthony said:

Darrell, yeah. I later accepted for the sake of argument and to prove a point, that there is a track.

Yet, I maintain that in practice a wheel will rotate the same as it always does, as we experience it to act, and as it does in theory (a circle diagram). You can observe a car wheel turn normally - and, let us say, you delineate the metal wheel rim to be the 'inner wheel'. Now extend that rim outwards. Now place a track for the extended rim to roll on. 

Now, you roll the car wheel for a revolution. What is possibly going to occur which did not occur when it was simply a car wheel and tyre on the road? No difference - surely? The entire wheel rolls forwards, on two tracks, where there was before just one surface. Assuming the weight on each track is carefully and evenly distributed, the outcome will be what the diagram denotes: the large wheel (tyre) rolls its circumference; the inner wheel (rim) rolls one revolution--but far past its circumference. It does not 'slip', it doesn't need to. The wheel assembly acts now exactly as it did on a single surface. Nothing essential has changed, inner and outer wheels keep integrity. 'Slippage' would contradict and destroy that. For this reason, the 'track' was put in as a red herring, imo.

Hi Tony,

The inner wheel is fixed relative to the outer wheel. That is true. However, either the inner wheel or outer wheel must slip relative to its track.

At one point you wrote that both the inner wheel and outer wheel rotate at the same angular velocity. That is true. However, if the wheels rotate at the same angular velocities, then their tangential velocities must be different.

The tangential velocity is the linear velocity of a point on the outside of the wheel. If V and v are the linear velocities of points on the big and small wheels and omega is their shared angular velocity, then V = R * omega and v = r * omega. So, if R > r then V > v.

If the vehicle to which the wheel is attached is moving with velocity = V, then the outer wheel will maintain rolling contact with the road while the inner wheel will skid while rolling on its track.

I like Jon's suggestion of experimenting with a bottle. However, I would like to suggest performing a slightly different experiment. After finding a book that is the right height to support the neck of the bottle, press down on both the body and neck of the bottle simultaneously while trying to roll it. In other words, apply enough force to make sure that both the body and neck of the bottle remain in rolling contact with their respective surfaces. See what happens.

Darrell

 

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52 minutes ago, Max said:

Really? Rolling one revolution, far past its circumference: that is by definition slipping, it is definitely not rolling without slipping, because then it couldn't get past its own circumference.

That is quite disingenuous. Of course - the small wheel rolls its own circumference. It, too, revolves - once. BUT, the ~distance travelled~ is greater than its own circumference, and you know what I meant, in my brief way of stating that.  So, this fails: "that is by definition slipping"... 

No, it is by definition - "rolling".

Repeat: it does not "get past its own circumference". It ~traverses a distance~ greater than its own circumference. Geez. Poor attempt. 

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1 minute ago, anthony said:

That is quite disingenuous. Of course - the small wheel rolls its own circumference. It, too, revolves - once. BUT, the ~distance travelled~ is greater than its own circumference, and you know what I meant, in my brief way of stating that.  So, you fail: "that is by definition slipping"... 

No, it is by definition "rolling".

Wrong, it is rolling and slipping.

1 minute ago, anthony said:

Repeat: it does not "get past its own circumference". It ~traverses a distance~ greater than its own circumference. Geez. Poor attempt. 

And that is the definition of slipping. If it wouldn't slip, it could not traverse a distance greater than its own circumference. Show me how the wheel could traverse a distance greater than its own circumference without slipping, and without resorting to pure magic or meaningless objectivist buzzwords.

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43 minutes ago, Max said:

Wrong, it is rolling and slipping.

And that is the definition of slipping. If it wouldn't slip, it could not traverse a distance greater than its own circumference. Show me how the wheel could traverse a distance greater than its own circumference without slipping, and without resorting to pure magic or meaningless objectivist buzzwords.

First, go see a wheel and tyre in motion. Observe and establish that an 'internal' wheel does, indeed, "traverse a distance greater than its own circumference". Always.

I.e. it traverses the distance of the outer tyre's circumference, which exceeds its own. Without slipping. 

There's this "objectivist buzzword" called "reality" - what it is (and does). HOW it happens is something further. I suggested the differing tangential speeds as the cause.

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22 minutes ago, anthony said:

First, go see a wheel and tyre in motion. Observe and establish that an 'internal' wheel does, indeed, "traverse a distance greater than its own circumference". Always.

I.e. it traverses the distance of the outer tyre's circumference, which exceeds its own. Without slipping. 

There's this "objectivist buzzword" called "reality" - what it is (and does). HOW it happens is something further. I suggested the differing tangential speeds as the cause.

Hi Tony,

After reading MSK's post from Nov. 22nd --- I'll catch up eventually --- I realized that there are two ways to resolve the paradox. Perhaps the second way is easier for you.

Let R, W, and V be the radius, angular velocity and tangential velocity of the big wheel. Then V = RW.

Define r, w, and v similarly for the small wheel so that v = rw.

Then, if R > r either V > v or w > W. Either the tangential velocity of the big wheel is larger or the angular velocity of the small wheel is larger. So, another way of resolving the paradox is to say that the wheels are actually separate wheels that turn at different rates. If that is easier for you to visualize, that works too.

Darrell

 

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1 hour ago, Darrell Hougen said:

After reading MSK's post from Nov. 22nd --- I'll catch up eventually --- I realized that there are two ways to resolve the paradox.

Darrell,

That's just cruel.

My post is way back on Page 43 of posts. We are now on Page 55.

Who's gonna read it due to your mentioning it?

:) 

So here is the link to my post of Nov. 22nd.

Ah...

That's better...

:) 

(btw - I'm glad you liked it. The world was swimming through my brain at the time... I would like to restrict that to the past tense, too, but alas... :) )

Michael

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10 hours ago, anthony said:

First, go see a wheel and tyre in motion. Observe and establish that an 'internal' wheel does, indeed, "traverse a distance greater than its own circumference". Always.

Why should I? Did I deny anywhere that the internal wheel traverses a distance greater than its own circumference? That has always been part of my argument.

10 hours ago, anthony said:

I.e. it traverses the distance of the outer tyre's circumference, which exceeds its own. Without slipping. 

It can only traverse the distance of the outer tyre's circumference if it slips. Otherwise it is impossible.

10 hours ago, anthony said:

There's this "objectivist buzzword" called "reality" - what it is (and does). HOW it happens is something further. I suggested the differing tangential speeds as the cause.

No vague suggestions, show the calculations, just as I've done. I've also shown that it is just the differing tangential speeds that explain the slipping. Reality is not what you think it is, but what it is.

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10 hours ago, anthony said:

First, go see a wheel and tyre in motion. Observe and establish that an 'internal' wheel does, indeed, "traverse a distance greater than its own circumference". Always.

I.e. it traverses the distance of the outer tyre's circumference, which exceeds its own. Without slipping. 

Without slipping on what?

 

10 hours ago, anthony said:

There's this "objectivist buzzword" called "reality" - what it is (and does).

Oh, wait, did you say "reality"? Well, if you're Officially invoking reality, then I guess that's it, it's final, you win.

 

10 hours ago, anthony said:

HOW it happens is something further. I suggested the differing tangential speeds as the cause.

Thanks for being our teacher!

♫Those schoolgirl days of telling tales and biting nails are gone
But in my mind I know they will still live on and on
But how do you thank someone who has taken you from crayons to perfume?
It isn't easy, but I'll try...♫

J

 

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On December 2, 2018 at 9:48 AM, Max said:

 

So according to you, Zeno's paradoxes, the twin-paradox, the barn-pole paradox, the bug-rivet paradox, the Gibbs paradox, Olbers' paradox are not genuine paradoxes, although these are well-known as such? 

 

Some yes, some no.  Long subject, which I don't have time for.

The definition of "paradox" I used in the post you're referring to - "an apparent contradiction between two true premises" - is the definition which I thought Jon gave in a post somewhere up the thread, but I couldn't find where on searching.  Maybe Jon was quoting a dictionary source.  The Search function (very irritatingly to me) does not pick up material enclosed in a quote box.  Or maybe I misremembered the definition Jon gave.  Either way, I think it's a good definition, and that it doesn't cover the "Aristotle's Wheel" problem.

I'm curious:  Have you or Jon objected to Jonathan's saying that the problem isn't a genuine paradox?  (The post of his I was agreeing with isn't the first time he's said that.)

Ellen

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9 hours ago, Darrell Hougen said:

Let R, W, and V be the radius, angular velocity and tangential velocity of the big wheel. Then V = RW.

Define r, w, and v similarly for the small wheel so that v = rw.

Then, if R > r either V > v or w > W. Either the tangential velocity of the big wheel is larger or the angular velocity of the small wheel is larger. So, another way of resolving the paradox is to say that the wheels are actually separate wheels that turn at different rates. If that is easier for you to visualize, that works too.

But that doesn't solve Aristotle's paradox, it's avoiding it by changing the conditions. That there is no paradox when two concentric wheels can rotate independently from each other, Aristotle no doubt could have also figured out, but that wouldn't have helped him solving his paradox. The two wheels forming one rigid body can easily be realized in a physical system, no contradiction there. The contradiction emerges when you suppose that both wheels can roll without slipping/both circles can trace out their circumference at the same time. That is the essence of the paradox.

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20 hours ago, anthony said:

Exactly. You're bearing out my position, no paradox. "Should' is a general assumption, I should have made clear, and is the only premise of the paradox (i.e., the action of one wheel compared/contrasted to the action of the other).

Like this "should", why can't one also assume a second track "was put into the problem by the person who formulated it" ?Again, the 'track' serves no purpose but to lead astray with more "shoulds" or woulds.

This track is superfluous since it either a). inhibits/skews the rotation of the main wheel, and the 'paradox' is rendered null and void or b). it does nothing - has no effect (the small wheel continues to traverse a distance beyond its circumference). It can't be had both ways.

Tony, no, I am not "bearing out [your] position."  You claim that the small wheel does a true roll - a roll without slippage.  It does not.  If a wheel "traverse[s ] a distance beyond its circumference" in one revolution, it is not doing a true roll.  Call what happens the small wheel's being partly dragged while revolving if the word "slips" causes you such difficulty.

Ellen

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29 minutes ago, Ellen Stuttle said:

Some yes, some no.  Long subject, which I don't have time for.

The definition of "paradox" I used in the post you're referring to - "an apparent contradiction between two true premises" - is the definition which I thought Jon gave in a post somewhere up the thread, but I couldn't find where on searching.  Maybe Jon was quoting a dictionary source.  The Search function (very irritatingly to me) does not pick up material enclosed in a quote box.  Or maybe I misremembered the definition Jon gave.  Either way, I think it's a good definition, and that it doesn't cover the "Aristotle's Wheel" problem.

I'm curious:  Have you or Jon objected to Jonathan's saying that the problem isn't a genuine paradox?  (The post of his I was agreeing with isn't the first time he's said that.)

Ellen

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.

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14 hours ago, Darrell Hougen said:

Hi Tony,

The inner wheel is fixed relative to the outer wheel. That is true. However, either the inner wheel or outer wheel must slip relative to its track.

 

I like Jon's suggestion of experimenting with a bottle. However, I would like to suggest performing a slightly different experiment. After finding a book that is the right height to support the neck of the bottle, press down on both the body and neck of the bottle simultaneously while trying to roll it. In other words, apply enough force to make sure that both the body and neck of the bottle remain in rolling contact with their respective surfaces. See what happens.

Darrell

 

 

The bottle (and other things) was what I was testing months ago, and because of minor inconsistencies - the elongated neck (a lo-ong 'inner wheel') whose leverage exaggerates the slightest level and pressure discrepancies, and the slippery nature of glass - were inconclusive.

I think you will agree that the purpose of experiments is to bring all factors to neutral, i.e. reproduce what we see and know in nature (i.e., here, of a bottle rolling alone, unaided, straight and true) - and only then, to add or subtract variables and note whatever changes take place. Home experiments are not rigorous enough to draw any conclusions from. This requires a lab experiment, controlled conditions, utilizing laser measurements and constant downward force, etc. etc.

The closest I got to rolling evenly on twin tracks, was using a squat round object like a jar or canister having a short lid and almost no neck, the "inner wheel". I could give it a push and it rolled quite evenly - close to how it does normally - and on two surfaces. But the height and weight adjustments - and especially the grip on both surfaces - are essential to get as 'perfect' as possible.  

Go back to the auto wheel - having identical "grip" of the large AND small wheels is most critical. When that grip differential is just slightly out, you introduce a bias, and then slippage occurs in one wheel. If it means also fitting a rubber tread to the small (extended inner rim) wheel, and using an identical (road) surface for it to run on-- grip has to be equal for both wheels, and critical also, they are rolled on two precisely compensated levels. If the wheel combination is given a push and it turns smoothly, the inner track has made no difference to the outcome - one we know and accept from observation of all wheels, which is that an inner wheel/circle will travel laterally a distance in excess of its circumference - without slip - when the outer rolls once. The added track is then a superfluity.

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It seems paradoxes come either out of deficiencies in reasoning or knowledge, quantum physics not withstanding. Supporting the existence of a paradox is wrestling with reality.

Taken as a whole this thread supports the existence of paradoxes including the Wikipedia spillover. It should have died on the vine over a year ago. Mia culpa, Mia maxima culpa.

Now there's Objectivism for you! 

--Brant

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1 hour ago, Ellen Stuttle said:

Tony, no, I am not "bearing out [your] position."  You claim that the small wheel does a true roll - a roll without slippage.  It does not.  If a wheel "traverse[s ] a distance beyond its circumference" in one revolution, it is not doing a true roll.  Call what happens the small wheel's being partly dragged while revolving if the word "slips" causes you such difficulty.

Ellen

Do we all agree this far? I will be as unambiguous as I can: A wheel inside a wheel travels a distance in excess of its own (calculated or visible) circumference when the large wheel turns its own circumference. (And they both rotate, once, together doing a "true roll" and no slippage. 

A circle in a circle, ditto.

This is a fact about wheels and circles. Immediately, at this stage, the 'paradox' may be dispensed with. It is accurate to reality, non-contradictory.

The large circumference is the *only* determinant of lateral distance moved. (By definition, and very odd any other way...)

----

Then a 'second track' gets brought in, and the fun starts. This track - apparently - makes the wheel in a wheel behave differently, according to some.

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1 hour ago, anthony said:

Go back to the auto wheel - having identical "grip" of the large AND small wheels is most critical. When that grip differential is just slightly out, you introduce a bias, and then slippage occurs in one wheel. If it means also fitting a rubber tread to the small (extended inner rim) wheel, and using an identical (road) surface for it to run on-- grip has to be equal for both wheels, and critical also, they are rolled on two precisely compensated levels.

Heh.

He still thinks that controlling the rig and limiting it so that one of the wheels rolls without slipping is an act of "introducing bias," or of "inducing slippage" that would not exist in the system otherwise. Heh. To Tony, the use of scientific control taints and invalidates any experiment. Hahahah!

 

1 hour ago, anthony said:

If the wheel combination is given a push and it turns smoothly, the inner track has made no difference to the outcome - one we know and accept from observation of all wheels...

"We"? Who is "we"? You?

All of us are not knowing and accepting what you are based on your lack of observation. We don't share your limitations. I and others observe significantly more than you do.

J

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2 hours ago, anthony said:

Do we all agree this far? I will be as unambiguous as I can: A wheel inside a wheel travels a distance in excess of its own (calculated or visible) circumference when the large wheel turns its own circumference. (And they both rotate, once, together doing a "true roll" and no slippage. 

A circle in a circle, ditto.

This is a fact about wheels and circles. Immediately, at this stage, the 'paradox' may be dispensed with.

But you haven't identified any paradox yet. What was thought to be paradoxical in the first place? Specifically what do you think that you're dispensing with by saying the above? You get so easily lost in your own circular reasoning. You're telling us that you've answered the question. We respond that you haven't, and that you don't even know what the question was, and you don't answer.

 

2 hours ago, anthony said:

Then a 'second track' gets brought in, and the fun starts. This track - apparently - makes the wheel in a wheel behave differently, according to some.

The second track was not "brought in," but was an original element of the "paradox." It is essential to the "paradox." Without it, there is no apparent contradiction or logically unacceptable conclusion to bother Aristotle (or whomeverthefuck).

How many times do we have to remind you? No matter how many times, you keep reverting to your previous mistaken premise. Write the shit down, Tony. Make a checklist since you can't hold all of the information in your mind at one time. Stop going back to false arguments that have been soundly refuted already.

J

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1 hour ago, anthony said:

Do we all agree this far? I will be as unambiguous as I can: A wheel inside a wheel travels a distance in excess of its own circumference when the large wheel turns its own circumference. (And they both rotate, once, together doing a "true roll" and no slippage. 

False, the crux of the paradox is that both wheels cannot do a "true roll" without slippage. 

1 hour ago, anthony said:

A circle in a circle, ditto.

This is a fact about wheels and circles. Immediately, at this stage, the 'paradox' can be dispensed with. It is accurate to reality, non-contradictory.

Reality is that the smaller wheel is slipping, reality is not what you're imagining.

1 hour ago, anthony said:

The large circumference is the *only* determinant of lateral distance. (Very odd, any other way...)

----

Then a 'second track' gets brought in, and the fun starts. This track - apparently - makes the wheel in a wheel behave differently, according to some.

Aristotle brought that second track in, your suggestion that that is some newfangled invention of ours is disingenuous, we just keep to the original formulation! Further, nobody claims that that wheel behaves differently when this track is "brought in", it only is a reference that makes clear that the smaller wheel is not rolling out its circumference, but makes another movement that we call slipping.

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3 hours ago, Ellen Stuttle said:

Some yes, some no.  Long subject, which I don't have time for.

The definition of "paradox" I used in the post you're referring to - "an apparent contradiction between two true premises" - is the definition which I thought Jon gave in a post somewhere up the thread, but I couldn't find where on searching.  Maybe Jon was quoting a dictionary source.  The Search function (very irritatingly to me) does not pick up material enclosed in a quote box.  Or maybe I misremembered the definition Jon gave.  Either way, I think it's a good definition, and that it doesn't cover the "Aristotle's Wheel" problem.

I'm curious:  Have you or Jon objected to Jonathan's saying that the problem isn't a genuine paradox?  (The post of his I was agreeing with isn't the first time he's said that.)

Ellen

Yeah, I don't know that I have the time or interest either...well, at least not for much of anything beyond the following.

Is it an issue of semantics? Perhaps.

I guess what it comes down to, to me, is paradoxes being in the eyes of the beholders, and there aren't enough beholders believing in the false premise of the Aristotle's Wheel Paradox. Most people seem to be able to immediately identify the false premise. To all but a few dopey eggheads, the reality of movement of compound wheels on surfaces is easy to track/visualize/understand. The dopes' inabilities don't define things for the rest of us. It is only a paradox to them, due to their personal, abnormal cognitive inabilities in the realm of visuospatial/mechanical reasoning, and they are a tiny fraction of the population.

J

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8 hours ago, Max said:

But that doesn't solve Aristotle's paradox, it's avoiding it by changing the conditions. That there is no paradox when two concentric wheels can rotate independently from each other, Aristotle no doubt could have also figured out, but that wouldn't have helped him solving his paradox. The two wheels forming one rigid body can easily be realized in a physical system, no contradiction there. The contradiction emerges when you suppose that both wheels can roll without slipping/both circles can trace out their circumference at the same time. That is the essence of the paradox.

Hi Max,

Maybe "resolve" (my word) or "solve" isn't the right term. Of course, Aristotle's paradox cannot be "resolved" if all of the conditions are enforced. It is impossible for two wheels that are rigidly attached to each other to turn without slipping on two different tracks if the radii are different. I was simply trying to point out that mathematically, there must be slippage somewhere in the system. Formally, it is impossible to have:

V = RW

v = rw

V = v

W = w 

and

R > r

where V, v = tangential velocities of the big and small wheels respectively, W, w = their respective angular velocities, and R, r = their respective radii.

That is what Aristotle's paradox demands.

Darrell

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7 hours ago, anthony said:

The bottle (and other things) was what I was testing months ago, and because of minor inconsistencies - the elongated neck (a lo-ong 'inner wheel') whose leverage exaggerates the slightest level and pressure discrepancies, and the slippery nature of glass - were inconclusive.

I think you will agree that the purpose of experiments is to bring all factors to neutral, i.e. reproduce what we see and know in nature (i.e., here, of a bottle rolling alone, unaided, straight and true) - and only then, to add or subtract variables and note whatever changes take place. Home experiments are not rigorous enough to draw any conclusions from. This requires a lab experiment, controlled conditions, utilizing laser measurements and constant downward force, etc. etc.

The closest I got to rolling evenly on twin tracks, was using a squat round object like a jar or canister having a short lid and almost no neck, the "inner wheel". I could give it a push and it rolled quite evenly - close to how it does normally - and on two surfaces. But the height and weight adjustments - and especially the grip on both surfaces - are essential to get as 'perfect' as possible.  

Go back to the auto wheel - having identical "grip" of the large AND small wheels is most critical. When that grip differential is just slightly out, you introduce a bias, and then slippage occurs in one wheel. If it means also fitting a rubber tread to the small (extended inner rim) wheel, and using an identical (road) surface for it to run on-- grip has to be equal for both wheels, and critical also, they are rolled on two precisely compensated levels. If the wheel combination is given a push and it turns smoothly, the inner track has made no difference to the outcome - one we know and accept from observation of all wheels, which is that an inner wheel/circle will travel laterally a distance in excess of its circumference - without slip - when the outer rolls once. The added track is then a superfluity.

Hi Tony,

Actually, I don't think you need laboratory conditions. The effect of having wheels of different sizes is very pronounced. In fact, you could perform an experiment with an ordinary drinking glass. Find a glass that is tapered so that the two ends have different diameters and roll it on the table or floor and watch what happens. A simple Dixie cup or party cup should work just fine. You don't have to roll it fast.

Darrell

 

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12 minutes ago, Darrell Hougen said:

Hi Tony,

Actually, I don't think you need laboratory conditions. The effect of having wheels of different sizes is very pronounced. In fact, you could perform an experiment with an ordinary drinking glass. Find a glass that is tapered so that the two ends have different diameters and roll it on the table or floor and watch what happens. A simple Dixie cup or party cup should work just fine. You don't have to roll it fast.

Darrell

 

Heh.

Darrell, have you not been paying attention?

Prepared to be disappointed when Tony takes your suggestion, rolls the glass or cup, and then reports back that he observed something other than what happened in reality.

You're asking him to do something which he does not have the cognitive ability to do.

J

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35 minutes ago, Darrell Hougen said:

Hi Max,

Maybe "resolve" (my word) or "solve" isn't the right term. Of course, Aristotle's paradox cannot be "resolved" if all of the conditions are enforced. It is impossible for two wheels that are rigidly attached to each other to turn without slipping on two different tracks if the radii are different. I was simply trying to point out that mathematically, there must be slippage somewhere in the system. Formally, it is impossible to have:

V = RW

v = rw

V = v

W = w 

and

R > r

where V, v = tangential velocities of the big and small wheels respectively, W, w = their respective angular velocities, and R, r = their respective radii.

That is what Aristotle's paradox demands.

Darrell

Or, in terms of the original statement of the paradox, it is impossible to have:

X2 - X1 = R * (T2 - T1)

x2 - x1 = r * (t1 - t1)

X2 - X1 = x2 - x1

T2 - T1 = t2 - t1

 and 

R > r

where X2 - X1 and x2 - x1 are the distances traveled by the big and small wheels, respectively, T2 - T1 and t2 - t1 are the angles  (theta) that both wheels rotate (e.g. 2pi radians) and R and r are the radii.

That is a mathematical statement of Aristotle's paradox.

Darrell

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1 hour ago, Darrell Hougen said:

Hi Tony,

Actually, I don't think you need laboratory conditions. The effect of having wheels of different sizes is very pronounced. In fact, you could perform an experiment with an ordinary drinking glass. Find a glass that is tapered so that the two ends have different diameters and roll it on the table or floor and watch what happens. A simple Dixie cup or party cup should work just fine. You don't have to roll it fast.

Darrell

 

Darrell, The tapered glass is conical, which will, sure, roll in a curve off to one side on a flat surface. Until you raise the smaller diameter end to the same level (as the larger) on a second track.

To reproduce a wheel inside another wheel in motion, a simpler demonstration is 2 cylindrical shapes connected. I.e. a wine bottle, and many others. And again, to compensate for the 2 different diameters, the 2 tracks need to be adjusted precisely to support the bottle, one slightly above the other. Or a skew to one side recurs.

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