Aristotle's wheel paradox

[anthony]

6 hours ago, Max said:

 

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

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

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.

1

Nope. The paradox is clear and simple. Described somewhere above in Aristotle's (?) words. This is the crux: a).The large circle travels the distance of its circumference. b).The small circle travels the identical distance--but more than its circumference. Since they are fixed they both roll once, naturally. So - Apparent contradiction.(My words). And there is no slippage.

THIS far, without another track mentioned, this raises a valid query - or - "paradox". ("Strange"- was A.'s observation). Except, it is what really happens, for a wheel within a wheel. (And no slippage). So there's no contradiction in reality. 

THEN, and stranger still, there is a demanded insistence on another track. By what he remarked, I have doubts this was Aristotle's doing. More likely an add-on by someone later to complicate the original paradox, someone who took a dotted line/path to be a 'track'.

[anthony]

20 hours ago, Darrell Hougen said:

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

 

Thanks Darrell. It won't be credited but I was first to mention the differing tangential velocities in a wheel as the probable explanation. I misnamed this "rotational" speed, since corrected it. 

 

[Jon Letendre]

28 minutes ago, anthony said:

Nope. The paradox is clear and simple. Described somewhere above in Aristotle's (?) words. This is the crux: a).The large circle travels the distance of its circumference. b).The small circle travels the identical distance--but more than its circumference. Since they are fixed they both roll once, naturally. So - Apparent contradiction.(My words). And there is no slippage.

THIS far, without another track mentioned, this raises a valid query and potential "paradox". ("Strange"- was A.'s observation). Except, it is what really happens, for a wheel within a wheel. (And no slippage). So there's no contradiction in reality. 

And THAT is called me winning the Egghead Challenge.

[Darrell Hougen]

56 minutes ago, anthony said:

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.

Hi Tony,

You are correct that the glass will roll off to one side. But, there is nothing special about a level surface. If the surface were tilted, what do you think would happen? Imagine that the surface is tilted so that the ends of the cup are straight up and down. Won't the glass continue to curve in the same direction as long as it doesn't slip?

You can create the same effect by just putting the small end of the cup on a book of the appropriate height. Try it. You don't need any fancy scientific equipment.

Darrell

[Darrell Hougen]

15 minutes ago, anthony said:

Thanks Darrell. It won't be credited but I was first to mention the differing tangential velocities in a wheel as the probable explanation. I misnamed this "rotational" speed, since corrected it. 

 

Hi Tony,

Yes, I remember that you mentioned that. But, if V > v, that means that one wheel is rolling faster than the other. If one wheel is rolling faster than the other, then they can't get to the end point at the same time. Or something else has to give.

Darrell

[Max]

9 hours 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. 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.

Impossible, as I've proved many times. Replace your wheels by gearwheels (also forming one solid wheel) and the surfaces by corresponding racks, then you'll have perfect grip between gearwheel and rack (ensure that the gears cannot leave the racks). Now you'll observe that these wheels cannot roll at all. That is reality! Just try it if you don't believe. Explain why it is impossible for the gearwheels to roll. Hint: it has something to do with the fact that the small wheel is unable to do something, thanks to the perfect grip of the gear system. 

[anthony]

49 minutes ago, Darrell Hougen said:

Hi Tony,

Yes, I remember that you mentioned that. But, if V > v, that means that one wheel is rolling faster than the other. If one wheel is rolling faster than the other, then they can't get to the end point at the same time. Or something else has to give.

Darrell

The "give" is their different circumferences. They both exactly reach the end point after the same duration - because - the slower has a smaller rotation, the faster a greater one. Longer/shorter circumferences equals them out. No?

[Max]

1 hour ago, anthony said:

Nope. The paradox is clear and simple. Described somewhere above in Aristotle's (?) words. This is the crux: a).The large circle travels the distance of its circumference. b).The small circle travels the identical distance--but more than its circumference. Since they are fixed they both roll once, naturally. So - Apparent contradiction.(My words). And there is no slippage.

Yes, there is slippage, that is what Aristoteles was missing.

 

Quote

THEN, and stranger still, there is a demanded insistence on another track. By what he remarked, I have doubts this was Aristotle's doing. More likely an add-on by someone later to complicate the original paradox, someone who took a dotted line/path to be a 'track'.

Ah, rewriting history... You don't understand what Aristotle wrote, and therefore "some other person" (not too bright presumably) must have added the second track.

[Max]

1 hour ago, anthony said:

Thanks Darrell. It won't be credited but I was first to mention the differing tangential velocities in a wheel as the probable explanation. I misnamed this "rotational" speed, since corrected it. 

You may have mentioned it, but your argument was completely wrong. Running boys in a circle has nothing to do with rolling circles. As I've shown, it is the combination of translation speed and tangential speed that explains that the smaller wheel is slipping.

[Darrell Hougen]

19 minutes ago, anthony said:

The "give" is their different circumferences. They both exactly reach the end point after the same duration - because - the slower has a smaller rotation, the faster a greater one. Longer/shorter circumferences equals them out. No?

Hi Tony,

The one with the longer circumference also has the greater tangential velocity. In fact, it has a greater tangential velocity because it has a longer circumference. The circumference is proportional to the radius. The greater the radius, the greater the circumference. The same thing is true of the tangential velocity. The tangential velocity is proportional to the radius. The greater the radius, the greater the tangential velocity. So, if R = 2r for example, then C = 2c and V = 2v where R, r = radius, C, c = circumference and V, v = tangential velocity of the big and small wheels respectively.

If that doesn't make sense to you, stand up with your arms outstretched and turn around. Your hands move both farther and faster than your elbows or shoulders. See what I mean?

Darrell

[anthony]

1 hour ago, Max said:

Yes, there is slippage, that is what Aristoteles was missing.

 

Ah, rewriting history... You don't understand what Aristotle wrote, and therefore "some other person" (not too bright presumably) must have added the second track.

Absolute nonsense. Without a second track (for the inner wheel) there cannot be internal slippage. "Slip" ... on what?  

I understand what Aristotle's conclusion was. It was totally about the relationship of the two circumferences(different) and their traversed distance(identical). Read it. Nowhere did he mention a track, not even in his preamble. That makes it doubtful a track was his addition.

[anthony]

35 minutes ago, Darrell Hougen said:

Hi Tony,

The one with the longer circumference also has the greater tangential velocity. In fact, it has a greater tangential velocity because it has a longer circumference. The circumference is proportional to the radius. The greater the radius, the greater the circumference. The same thing is true of the tangential velocity. The tangential velocity is proportional to the radius. The greater the radius, the greater the tangential velocity. So, if R = 2r for example, then C = 2c and V = 2v where R, r = radius, C, c = circumference and V, v = tangential velocity of the big and small wheels respectively.

If that doesn't make sense to you, stand up with your arms outstretched and turn around. Your hands move both farther and faster than your elbows or shoulders. See what I mean?

Darrell

My point also, made without your impeccable math. I gave one example of a children's simple merry go round, and the speed greater at the edge, reducing proportionately towards the middle. 

[anthony]

1 hour ago, Max said:

Impossible, as I've proved many times. Replace your wheels by gearwheels (also forming one solid wheel) and the surfaces by corresponding racks, then you'll have perfect grip between gearwheel and rack (ensure that the gears cannot leave the racks). Now you'll observe that these wheels cannot roll at all. That is reality! Just try it if you don't believe. Explain why it is impossible for the gearwheels to roll. Hint: it has something to do with the fact that the small wheel is unable to do something, thanks to the perfect grip of the gear system. 

You mean this gear train "cannot roll at all"? I've seen gears fixed solidly onto larger gears, and run on racks.

Double reduction gear[edit]

 

Double reduction gears

A double reduction gear comprises two pairs of gears, as single reductions, in series.^[3] In the diagram, the red and blue gears give the first stage of reduction and the orange and green gears give the second stage of reduction. The total reduction is the product of the first stage of reduction and the second stage of reduction.

It is essential to have two coupled gears, of different sizes, on the intermediatelayshaft. If three gears were used, the overall ratio would be simply that between the first and final gears, the intermediate gear would only act as anidler gear: it would reverse the direction of rotation, but not change the ratio

[Darrell Hougen]

1 hour ago, anthony said:

My point also, made without your impeccable math. I gave one example of a children's simple merry go round, and the speed greater at the edge, reducing proportionately towards the middle. 

Right. I remember that. So, okay, that means that if you have a large wheel turning at N rotations per minute and a small wheel turning at N rotations per minute, the large wheel will travel farther than the small wheel. Right?

Darrell

[Darrell Hougen]

 

2 hours ago, anthony said:

You mean this gear train "cannot roll at all"? I've seen gears fixed solidly onto larger gears, and run on racks.

Double reduction gear[edit]

 

Double reduction gears

A double reduction gear comprises two pairs of gears, as single reductions, in series.^[3] In the diagram, the red and blue gears give the first stage of reduction and the orange and green gears give the second stage of reduction. The total reduction is the product of the first stage of reduction and the second stage of reduction.

It is essential to have two coupled gears, of different sizes, on the intermediatelayshaft. If three gears were used, the overall ratio would be simply that between the first and final gears, the intermediate gear would only act as anidler gear: it would reverse the direction of rotation, but not change the ratio

I think Max was thinking of something more like this:

But with two concentric circular gears and two linear gears or "pinions."

Darrell

 

[Jon Letendre]

 

[Max]

5 hours ago, Darrell Hougen said:

 

I think Max was thinking of something more like this:

But with two concentric circular gears and two linear gears or "pinions."

Darrell

Exactly, you got it.

[Max]

6 hours ago, Jon Letendre said:

 

Good example, but I'm sure Tony doesn't understand what's happening here.

[merjet]

On 11/30/2018 at 8:02 PM, 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

LOL. A jeer from the peanut gallery. Your double standard is clear. You are upset that I vastly improved Wikipedia. But you have no complaints about an obnoxious ignoranus, lying, contradictory, reality-faking jackass named Jonathan. And apparently you were also duped by Jonathan and believe that gratuitously adding a second surface isn’t “beyond the pale.”

I bet you couldn’t find any errors in my solutions!

[merjet]

On 12/1/2018 at 2:29 PM, Jonathan said:

Yeah, and not only that, but they also then accuse us of adding those essential elements, claiming that our recognition of their inclusion in the original formulation, and of their importance to the alleged "paradox," is dishonest, a "crutch," a "scam," etc.

They say that we're showing doctored videos, tricky illusions, and con art. 

....

The dopey ancient geniuses accepted and agreed with that "should." That is the essence of the alleged "paradox" -- their belief that the smaller wheel "should" cover a distance greater than its circumference without slipping/skidding on the surface upon which it rolls.

You got that right.

....

That’s the habitual liar, psychologizer, and snooty reality-faker again. The author of Mechanica never said what Jonathan tries to cram in his mouth and mind. The author called the phenomena strange and remarkable, but “should” and “ought” do not appear in his description of it. He knew the smaller circle, when dependent on a larger circle, covered a distance greater than its circumference, and he did not posit two surfaces simultaneously.

[merjet]

On 12/1/2018 at 7:10 PM, Ellen Stuttle said:

Right, and thanks.  The "should" was that of the poser of the problem. 

....

Regarding Tony's continuing to call the track superfluous:  The track, like the "should," was put into the problem by the person who formulated it.

You got it wrong, propagandist trying to rewrite history.

[anthony]

13 hours ago, Darrell Hougen said:

Right. I remember that. So, okay, that means that if you have a large wheel turning at N rotations per minute and a small wheel turning at N rotations per minute, the large wheel will travel farther than the small wheel. Right?

Darrell

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 ).

[merjet]

On 12/1/2018 at 6:01 PM, Max said:

It is telling that the people who don't accept the slippage explanation of the paradox apparently feel compelled to remove essential elements from its original formulation. Why would that be so? 

It is telling that the people who claim slippage apparently feel compelled to gratuitously add elements to its original formulation. Why? With their crutch, they have a “solution.” Without their crutch, they are helpless.

 

[merjet]

On 12/2/2018 at 12:57 PM, Jonathan said:

I don't recall having seen this degree of reality-denying obstinacy outside of O-land, but I've seen it many times inside of O-land.

You are a perfect example. Beep, beep. Vrooom.

[merjet]

On 12/3/2018 at 5:13 PM, Darrell Hougen said:

Merlin edited the Wikipedia page so that it no longer contains an accurate description of Aristotle's paradox. The figure has also been edited and is no longer illustrative of the paradox.

Hogwash. The page didn’t have an accurate description before I edited it.