Aristotle's wheel paradox

[Ellen Stuttle]

On December 9, 2018 at 3:00 PM, anthony said:

The Paradox was set up like a 'trick question' to lead you astray with slippage. The 'paradox' is dispensed with when you can see the different roll-speeds.

 

On December 9, 2018 at 6:36 PM, Jonathan said:

Tony knows the intentions of the author of the "paradox." He doesn't even know for certain who the author is (nor does anyone else), but he's certain that he knows his intentions. When the author mentioned lines and wheels unrolling on them, he didn't mean it. Only some of the entities and conditions should be treated as being real: Only so many as Tony can keep in his mind at once and feel that he is grasping their movement.

J

I think we can know that the intentions of the author of the "paradox" were NOT as Tony surmises.  The author was puzzled by the problem, what with (incorrectly) expecting the small wheel to roll only the distance of its own circumference.  And although "slippage" leads Tony astray because he fails to understand its meaning, it doesn't lead astray those who understand what it means.

Ellen

[Jonathan]

On 12/10/2018 at 9:19 PM, william.scherk said:

 

The very first rail carriages had no bogies -- no independent bogies  that could rotate on  a horizontal  axis. Instead of paired fixed-axle flanged wheels free to turn left or rightways, the first railway vehicle wheels were stiffly attached to the the carriage as in a simple four-wheeled drive.

 

 

I, too, love trains, Billy. I love everything mechanical. But is this the place for it? Do you really want to tempt discussing matters of differential systems with Merlin and/or Tony? As I said earlier:

Quote

Imagine confronting them with something a bit more complex, like, say, comparing a standard bevel gear differential system with a worm gear differential system (Torsen), and explaining the whats, whys and hows of it all. If A happens to B, then C will cause D to rotate faster than E.

We'd be standing here thousands of times more tardfounded than we are right now.

J

[Jonathan]

52 minutes ago, anthony said:

To "perform the experiment properly" would require fine adjustments. Height, weight distribution, and friction/drag - equalized. I've mentioned that near-perfect "balance" is the prerequisite. The elevated track needs precise compensation for the different diameters, and fine measurements  using specialized instruments to observe the contact/or slippage of the neck. I have got close to achieving balance with rough tests.

Ba haha hahahahaha!!!!

 

53 minutes ago, anthony said:

When "slippage"? - when firm contact? When sliding and when rolling? How does one observe the distinction, in practice? This cannot be validated in an average home experiment I think.

Well, it certainly can't be done in a less than average home experiment. Ahem.

 

57 minutes ago, anthony said:

As read above from Max: "The difference in tangential velocities explains why slipping *does* occur". Untrue. A causal reversal. Tangential velocity is a characteristic of the wheel, slippage is an abnormality/and or intervention.

Anyhow, for now I have been entertaining the idea of a track without slippage, while not accepting a 'track' with slippage. As close an approximation to the 2-D Aristotle diagram, as possible, can be seen in that video demonstration above. All ~three~ circles conform to each other and to the lines, the represented 'tracks'.

Keep entertaining ideas, Tony! Build on the foundation that you've established! Go, Tony, go!

J

[anthony]

28 minutes ago, Ellen Stuttle said:

 

I think we can know that the intentions of the author of the "paradox" were NOT as Tony surmises.  The author was puzzled by the problem, what with (incorrectly) expecting the small wheel to roll only the distance of its own circumference.  And although "slippage" leads Tony astray because he fails to understand its meaning, it doesn't lead astray those who understand what it means.

Ellen

The answer is in the question. There is nothing abnormal about a small wheel out-rollng its circumference. This is a characteristic.

"Strange" to Aristotle at first, we read. This is a "paradox" -- only an ~apparent~ contradiction, which will turn out to be non-contradictory with simple observation and logic.

Now, tell me where is the 'slippage" and why the track?

The power of suggestion. 

[Jonathan]

15 minutes ago, Ellen Stuttle said:

 

I think we can know that the intentions of the author of the "paradox" were NOT as Tony surmises.  The author was puzzled by the problem, what with (incorrectly) expecting the small wheel to roll only the distance of its own circumference.  And although "slippage" leads Tony astray because he fails to understand its meaning, it doesn't lead astray those who understand what it means.

Ellen

Exactly.

J

[Jonathan]

1 minute ago, anthony said:

The answer is in the question. There is nothing abnormal about a small wheel out-rollng its circumference.

"Strange" to Aristotle at first, we read. This is a "paradox", only an ~apparent~ contradiction, which will turn out to be non-contradictory with simple observation and logic.

Now, tell me where is the 'slippage" and why the track?

The power of suggestibility. 

We've answered those questions.

You are not cognitively capable of grasping reality in this case. It's too complex for your mind to handle.

J

[anthony]

An anti-metaphysician can't handle the concept "characteristic". 

"Proof" by numbers and demonstration or he's lost... 

[anthony]

"Nothing can be A and non-A at the same time and in the same respect". 

Did anybody stop to consider that the guy who came up with this - could not possibly accept "rolling" *plus* "slippage"--

as a valid answer to the simple paradox which he observed and now bears his name?

[Jonathan]

8 minutes ago, anthony said:

"Nothing can be A and non-A at the same time and in the same respect". 

Indeed! A person who is visuospatially/mechanically incompetent cannot be visuospatially/mechanically competent.

J

[Jonathan]

29 minutes ago, anthony said:

An anti-metaphysician can't handle the concept "characteristic". 

"Proof" by numbers and demonstration or he's lost... 

 

10 minutes ago, anthony said:

"Nothing can be A and non-A at the same time and in the same respect". 

Did anybody stop to consider that the guy who came up with this - could not possibly accept "rolling" *plus* "slippage"--

as a valid answer to the simple paradox which he observed and now bears his name?

Yeah, run to mother.

Quoting Rand will give you comfort and solace. It'll make you feel that you're right, and also heroic!

Mama's gonna keep baby cozy and warm.

J

[Jonathan]

Tony and Merlin,

Can you solve this one:

These are the premises:

Three gears are lying flat on a plane as depicted above. The axles on all three gears are fixed in position, therefore rotation of the gears is possible but translation is not.

Gear A rotates clockwise, which causes gear B to rotate counterclockwise.

Do not object to the premises. Accept them. This is a thought experiment. In thought experiments, you must accept the premises as true.

Now, in which direction does C rotate, clockwise or counterclockwise?

Have at it! Bonehead-egghead the hell out of it!

J

[Jules Troy]

Erm..none of the above?  Unless of course A or B gears teeth get shorn off or their axel breaks in which case winner takes all..

[anthony]

If I may sum up, what I gather from the slip-ist p.o.v.  The problem one is posed is observing the small wheel making a single rotation - and - exceeding its own circumference in distance traveled.

An explanation/justification: The inner wheel is - or has to be - retarded in its rotation  - while, simultaneously, accelerated in its distance traveled.

Enter the necessary devices of "slippage" - and a track to slip on. When the inner wheel skids, it goes further - right?

But, it must also turn exactly one revolution (in conjunction with the large wheel to which it's fixed).  

Therefore, the slipist justification goes, the small wheel must skid - AND - roll : Rolling, just sufficient to rotate once, skidding just enough to travel the full distance. (By what means this is supposedly accomplished, I can't figure. Nor, how it's envisaged, by a "visuo-spatial' imagination. A few practical experiments might ~seem~ to bear out sporadic slip-and-roll, while fully explained by inconsistencies of grip and weight).

So here's when an initial, *apparent contradiction* (a paradox) turns into an outright contradiction in terms. Rolling accompanied by slipping. Slip, but with roll. A and non-A, "at the same time and in the same respect". 

The first 'problem' is there isn't a problem. Anything within the wheel will travel laterally the distance the wheel does, and if it is another wheel or circle, will travel further than its own circumference (and the large wheel's distance). The second 'problem' is eliminated when realizing that the rotation of the inner wheel already IS "retarded". I.e., The effect of tangential velocity, increasing from zero m/s at the center (of a circle) to its outside perimeter where it is at maximum velocity. Therefore, the inner wheel rotates at a proportionately lesser speed in accordance with where it is positioned relative to the outer wheel.

 

[anthony]

12 hours ago, Jules Troy said:

Erm..none of the above?  Unless of course A or B gears teeth get shorn off or their axel breaks in which case winner takes all..

You noticed how Mr. Tricky gave only two options.

[Jonathan]

1 hour ago, anthony said:

If I may sum up, what I gather from the slip-ist p.o.v.  The problem one is posed is observing the small wheel making a single rotation - and - exceeding its own circumference in distance traveled.

An explanation/justification: the inner wheel is - or has to be - retarded in its rotation  - while, simultaneously, accelerated in its distance traveled.

Enter the necessary devices of slippage and a track to slip on. When the inner wheel skids, it goes further - right?

But, it must also turn exactly one revolution (in conjunction with the large wheel to which it's fixed).  

Therefore, the justification goes, the small wheel must skid - AND - roll : Rolling, just sufficient to rotate once, skidding just enough to travel the full distance. (By what means this is supposedly accomplished, I can't figure. Nor, how it's envisaged, by "visuo-spatial' imagination. A few practical experiments might ~seem~ to bear out sporadic slip-and-roll, fully explained inconsistencies of grip and weight).

So here's when an initial, *apparent contradiction* (a paradox) turns into an outright contradiction in terms. Rolling accompanied by slipping. Slip, but with roll. A and non-A, "at the same time and in the same respect". 

The first 'problem' is there isn't a problem. Anything within the wheel will travel the distance the wheel does, and if it is another wheel or circle, will travel further than its own circumference (and the large wheel's distance). The second 'problem' is eliminated when realizing that the rotation of the inner wheel already IS "retarded". I.e., The effect of tangential velocity, increasing from zero m/s at the center (of a circle) to its outside perimeter where it is at maximum velocity. Therefore, the inner wheel rotates at a proportionately lesser speed in accordance with where it is positioned relative to the outer wheel.

 

You're never going to understand it.

But thanks for participating. It's been very instructive observing the routes that your and Merlin's minds have taken to protect your beliefs.

J

[Jonathan]

1 hour ago, anthony said:

You noticed how Mr. Tricky gave only two options.

Jules, I think he's calling me Mr. Tricky, not you.

Tony believes that an accusation alone is enough. I'm accused of creating diagrams and animations that have some sort of trick in them, despite all aspects of them being precisely, impeccably accurate, and despite Tony's failure to present any evidence of trickery.

Anyway, Tony, can't you solve the gear problem? Quit griping and solve it.

J

[Jonathan]

2 hours ago, anthony said:

You noticed how Mr. Tricky gave only two options.

Tony, if you use the scientifically proper amount of weight, balance, friction, etc., and you accept the problem as a thought experiment, then the two options that I gave are more than enough.

Start by figuring out rotational speeds and tangential speeds!

J

[anthony]

Witness this guy's dishonesty and smug superiority. He knows he allowed only two false options, but everyone is too stupid to see through it.. And let's see him give his understanding of the paradox. Not without help from others, he won't. 

[Max]

2 hours ago, anthony said:

If I may sum up, what I gather from the slip-ist p.o.v.  The problem one is posed is observing the small wheel making a single rotation - and - exceeding its own circumference in distance traveled.

An explanation/justification: the inner wheel is - or has to be - retarded in its rotation  - while, simultaneously, accelerated in its distance traveled.

Enter the necessary devices of slippage and a track. When the inner wheel skids, it goes further - right?

But, it must also turn exactly one revolution (in conjunction with the large wheel to which it's fixed).  

Therefore, the justification goes, the small wheel must skid - AND - roll (in precise turns, I suppose): Rolling, just sufficient to rotate once, skidding just enough to travel the full distance. (By what means this is supposedly accomplished, I can't figure. But some practical experiments might ~seem~ to bear out sporadic slip-and-roll, although explained by varied, inconsistent degrees of grip and weight).

So here's when an initial, *apparent contradiction* (a paradox) turns into an outright contradiction. Rolling accompanied by slipping. Slip, but with roll. A and non-A, "at the same time and in the same respect". 

 

There is no contradiction. It is perfectly possible to add independent motions like rolling and slipping to a combined motion, that is a characteristic of vector quantities. If you throw a ball to someone, its motion is a combination of a forward motion (in the x-direction) and an upward motion (in the z-direction). (In that case those vectors change continuously due to effects from gravity and air resistance). Rolling itself is a combination of rotation and translation, such that the contact point with the support has speed zero. Add an extra translation movement and you get a combination of rolling and slipping.

Those "precise" and "just enough" factors are not some miracles but are forced by the fact that the wheel is a solid body: rotation of the outer wheel forces an equal rotation of the smaller wheel, while the slipless translation of the outer wheel forces the equal translation of the smaller wheel. The amount of slipping is simply the difference between the forced translation and the translation due to the forced rotation of the smaller wheel, which can be easily calculated. No miracle, no contradiction. 

 

[Brant Gaede]

2 hours ago, anthony said:

You noticed how Mr. Tricky gave only two options.

You were given the simplest possible problem. But the correct answer would wipe out all your previous going to the moon verbiage. That's what you aren't going to deal with.

--Brant

[anthony]

1 minute ago, Brant Gaede said:

You were given the simplest possible problem. But the correct answer would wipe out all your previous going to the moon verbiage. That's what you aren't going to deal with.

--Brant

The correct answer would sanction this guy's deceitfulness. And it is obvious to any primary schooler. Where this links - in the least - with my moon verbiage, you'll have to tell me.

[Brant Gaede]

3 minutes ago, anthony said:

The correct answer would sanction this guy's deceitfulness. And it is obvious to any primary schooler. Where this links - in the least - with my moon verbiage, you'll have to tell me.

Ah, it's a moral issue. That's a standard Ayn Rand default. Something to do with the sanction of the victim--NOT! But she then usually left it at that for she had spent 13 years on a novel that didn't leave it at that. Your position is you're not going to be bitch slapped by reality. One couldn't do that to her for she refused discussion. You pretend to discuss, she didn't. That was her public face. When she didn't see it as a moral issue she could be quite patient with the seeker of the truth. So I have been told and read and seen enough personally to believe it.

--Brant

[anthony]

13 minutes ago, Brant Gaede said:

Ah, it's a moral issue. That's a standard Ayn Rand default. Something to do with the sanction of the victim--NOT! But she then usually left it at that for she had spent 13 years on a novel that didn't leave it at that. Your position is you're not going to be bitch slapped by reality. One couldn't do that to her for she refused discussion. You pretend to discuss, she didn't. That was her public face. When she didn't see it as a moral issue she could be quite patient with the seeker of the truth. So I have been told and read and seen enough personally to believe it.

--Brant

I have had in this thread, many occasions to make a "moral" complaint. After many insults and other put-downs. Did you notice, I did not? The first time I have, and this is a problem for you? C'mon. For once I hit back? Maybe you have got too accustomed to the bad manners of a few and to my previous restraint.

[Brant Gaede]

3 minutes ago, anthony said:

I have had in this thread, many occasions to make a "moral" complaint. After many insults and other put-downs. Did you notice, I did not? The first time I have, and this is a problem for you? C'mon. For once I hit back? Maybe you have got too accustomed to the bad manners of a few and to my previous restraint.

Jonathan is rough but he always goes back to the subject at hand. I don't like him for the rough, but he's not into being liked. That's between you and him.

--Brant  

[Jonathan]

57 minutes ago, anthony said:

He knows he allowed only two false options...

Can't solve it, eh? Just as I thought.

If you ever work up the courage to attempt to solve it, remember that you're not allowed to induce anything into the scenario, like friction, or slippage, or resistance, or anything like that. That's according to your own rules that we learned from your handling of the Aristotle's Wheel Paradox. So, don't try to sneak in any "necessary devices" or insert any of that kind of trickery.

With that in mind, in which direction does gear C turn, clockwise or counterclockwise?

J