Aristotle's wheel paradox

[Peter]

Beavis?

What Butthead?

Snort. Ba'al said naked.

[Jonathan]

56 minutes ago, BaalChatzaf said:

34 pages!!  This is what happens when philosophers discuss a straightforward  problem in mechanics!   We are sufficiently advanced scientifically and historically to safely ignore anything the Aristotle had to say on matter and motion.  He totally  failed in addressing these matters.  Listen to Aristotle on descriptive biology, politics, rhetorics and  literary style.   The only part of science that Aristotle did well on was descriptive biology based on naked eye observations (the Greeks never developed lenses).  No less a naturalist than Charles Darwin gives Aristotle high marks in this endeavor.

Live Long and Prosper  \\//

You've heard the metaphor "thick skull," no? Well, the unit of measure of "thickness of skulls" -- stubbornness, obstinacy, pigheadedness -- is number of OL pages. We're in the process of measuring Merlin's obstinacy, and, so far, it has crossed over into 35 OL pages thick. That's pretty damned thick!

J

[Peter]

I've heard that blind Merlin could hear a smile or a frown from across the room. Now, stop wrinkling your nose so I can think.

[Brant Gaede]

4 hours ago, Jonathan said:

But, what if we change the conditions of the "paradox" so that certain arbitrarily selected parts of it are ignored or treated as being "metaphorical"? Let's say that the upper line, the one that contacts the base of the smaller wheel, doesn't exist, and that the large wheel sort of exists, but a section of it that I don't like is only "metaphorical." Let's also stipulate that there might be an invisible wheel behind the two visible wheels which is rolling freely on a ledge which isn't there but which I want to believe exists.

Okay, so the invisible wheel rolls freely on the non-existent ledge that I want to believe exists, but this wheel does not create a proper cycloid, but a curtate one. Instead, the large wheel, which under these conditions in reality would create a prolate cycloid, in this case creates a proper cycloid, because I want it to.

Considering all of the above, what's your solution to the "paradox"?

J

The Gordian Knot solution.

--Brant

[Jon Letendre]

8 hours ago, anthony said:

I covered that, and you missed it. I said:  "One higher than the other to finely adjust for the differing diameters".

I did ask you to roll a bottle along a surface, say the floor, and observe. It runs straight - right? 

Meaning, logically, the smaller diameter of the neck has ~zilch~ to do with the rotation of the entire bottle.

Now, take that to the two tracks. If and when it is perfectly aligned and balanced on its levels on two tracks, the bottle must turn as it would on a flat surface. Yes? If not, why not? (And clearly, frictional resistance, air resistance, gravity does not fall within the bounds of the "Paradox") . Anyhow, it is not clear - and superfluous - that that line is a second "track", or simply the path of travel. But in controlled laboratory conditions, two tracks will work equally well. 

Really, I don't know why nobody gets this. I think context has constantly been dropped by focusing too closely on the inner (fixed, attached) wheel.

As I keep repeating, the inner wheel/circle must move (laterally) the identical distance the outer one does!! (Or else, we can never trust a wheel again - they will break apart).

20 meters or 0.5 meter; a partial revolution - or a dozen revolutions - no difference. Fact remains: Each, single inner point and circle within the main wheel body will traverse precisely 20m or 0.5m - or whatever - as well. The inner line of travel in the paradox diagram will be the same for every internal circle, starting and ending at the same place. 

The clever placement and length of the inner line wrt. to the inner circle's circumference creates the illusion of a paradox. But *all* lines connecting all inner points from their beginning of motion to their rest, must always be that same length! (For any given circle and forward motion).I.e. the same length as the big wheel's total travel. 

The inner wheel is not an independent entity!

Its circumference has no bearing on the larger context.

Its motion and travel is extraneous to the main wheel - and must be considered only within the greater context of the larger entity.

I think this confusion between logic and fact, or, putting the theoretical in conflict with the evidence of our senses, seems to point to the old (false) dichotomy, Analytic vs Synthetic. That's well overturned by objectivity and Objectivism. Smart fellow, Aristotle, wasn't he?

No, sorry, Tony, but you are not visualizing it correctly.

"One higher than the other to finely adjust for the differing diameters" doesn’t adjust anything that matters to the bottle rolling straight versus veering off.

Your bottle will not roll straight because any amount* of rotation advances the base (with more circumference) farther than the small neck (with less circumference.)

* For example, roll the bottle a half of a rotation. The base will advance on its track about three inches while the neck will advance only about one inch.

If you want the bottle to roll straight, then the base of the bottle can stay in contact with its track without slip while the neck skids, slips. Or, the neck can stay in contact with its track without slip while the base over-spins or spins-out.

[Jon Letendre]

Anyone can confirm the above at home:

Lie a bottle down on a smooth table. Place a book under the neck to fill the gap between it and the table.

Press down on the neck while rolling and while keeping the bottle going straight and observe that the base of the bottle wants to over-spin, relative to the table.

Now press down on the base of the bottle while rolling and observe that the neck is skidding, relative to the book/table.

[Jon Letendre]

Let us examine the video animation Billyboy offered. I took a still picture by screen-saving while it was playing. I marked it up a bit to point some things out...

By the orientation of the purple “start line” we can see that the wheel has only completed a little more than a quarter of a rotation.

The centers of both the wheel and the “small wheel” have traversed the road an equal amount, equal to the yellow line segment.

But notice that the video is a scam as neither wheel nor “small wheel” is executing true rolling. The large wheel is over-spinning or “spinning-out,” evidenced in the black arc’s length exceeding the yellow line segment length. The black arc is the length of wheel that has made contact with the road so far, and it is longer than the yellow line segment, which proves over-spin.

Meanwhile, the green arc is shorter than the yellow line segment. This proves that the “small wheel” is slipping, skidding along its “road.” It is rotating, but not fast enough to true roll on its road.

[Jon Letendre]

On 9/16/2017 at 9:23 PM, Jon Letendre said:

The video Merlin offered is honest in that it depicts the main wheel performing a true roll on the road (which is a couple millimeters below the black deck.) True rolling is evidenced by the equal lengths of the pink arc and pink line.

The slip or skid of the “small wheel” relative to its “road,” which many of us have been referring to from the beginning, is evidenced by the blue arc’s length being less than the blue line’s length.

The blue wheel has traversed a length of blue road equal to the length of the blue line segment, and yet all the while it has made contact with the road with only a short length of wheel (the blue arc,) which is insufficient for all that traversing, thus we can see it has been skidding, slipping on its “road.”

[Jon Letendre]

There is yet another way to detect the over-spin of the wheel in Billyboy’s video and also in the picture used in the Wiki article.

The picture used in the Wiki article https://en.wikipedia.org/wiki/Aristotle's_wheel_paradox is almost a correct depiction of traversed road length given the wheel diameter. On my screen at this moment, the blue wheel has 1.5 inch diameter, so the blue road should be 1.5 * 3.14 = 5 inches, yet it is only 3.5 inches. In other words, the two wheels below, depicting start and end positions, should be spread even farther apart. Another way of saying the same thing is that if you rolled the blue wheel below, it would not have completed its one rotation yet when it got to where it is shown on the right. It would have farther to go to complete one rotation.

Here is the correct depiction of ratio of wheel diameter to road length traversed in one rotation...

Now observe the highly compressed road length from Billyboy’s video... 

[Jon Letendre]

I spin wheels every day and do my own wrenching on all my bikes.

At the moment I have twelve tuned and ready bicycles and another approximately twelve not in operating order, projects.

Also four plated and insured motorcycles and a scooter.

Thats seventeen bikes I currently ride and keep in working order.

Here I am on my 4-foot diameter high-wheel...

 

[Jon Letendre]

Here I am on one of this year’s additions: my Nicky Hayden Edition 2004 HONDA RC51...

[merjet]

On 11/13/2018 at 7:23 PM, Max said:

Well, at least the article gives the correct solution

I get it. You believe there is only one correct solution -- merely because you like it. There are many proofs of the Pythagorean Theorem. Is only one of them correct merely because you like it best? 

On 11/13/2018 at 7:23 PM, Max said:

The part with the cycloids doesn't explain the paradox.

Maybe to you. The cycloids solution is correct. What part of it do you not understand?

On 11/13/2018 at 7:23 PM, Max said:

That there must be an error is trivial, 2*pi*r < 2*pi*R for r < R, that impossibility is what makes it a paradox (an apparent contradiction) but that still doesn't tell us what exactly the error in the presentation of the paradox is. That is namely the supposition that both wheels can roll without slipping. 

You still have a big hang-up about two wheels. Get over it. 

I disagree. The false supposition is that the smaller circle's circumference is relevant to its horizontal movement, when in fact it is entirely irrelevant. You and others make the same false supposition. A smaller circle's radius is also not relevant, but its center is.

This is easily shown. Add two more inner concentric circles to the usual two depicting Aristotle's wheel paradox, such as to represent the inner and outer edges of the white ring on a white wall tire. When all four are rolled together, all three inner concentric circles move the identical horizontal distance, the circumference of the largest circle.  Their circumferences, all different, are irrelevant, but their centers, all the same, are relevant. 

On 11/13/2018 at 7:23 PM, Max said:

But cycloids are in fact just an unnecessary distraction for explaining the paradox.

'Slipping' is in fact just an unnecessary distraction compared to the second, translation solution on the Wikipedia page. The translation solution is simpler and far more elegant than 'slippage'. The center of the smaller circle matters; its circumference doesn't.

On August 14 I challenged you to quantify the three terms on the right of this equation: 

2*pi*R =  Rotation + Translation + Slippage 

You haven't answered yet. Is answering it too difficult for you?

[merjet]

A large part of the 35 pages consists of Jonathan's nonsense, infantile behavior, and obsessed ad hominem. Moreover, he is an exemplar of stubborn, obstinate, pigheaded, closed-minded, and snooty, when he believes he has the only correct solution -- merely copied from somebody else -- to the paradox. For me to not totally agree with him offends his fragile ego. That I could have another, better, and original solution -- perish the thought, two such solutions -- offends his fragile ego.

Poor Jonathan. Like Wile E. Coyote, his tactics backfire. Beep-beep, vroooom. Maybe he is beyond reform. He is what he is. A is A. 

[Brant Gaede]

4 hours ago, merjet said:

A large part of the 35 pages consists of Jonathan's nonsense, infantile behavior, and obsessed ad hominem. Moreover, he is an exemplar of stubborn, obstinate, pigheaded, and snooty, when he believes he has the only correct solution -- merely copied from somebody else -- to the paradox. For me to not totally agree with him offends his fragile ego. That I could have another, better, and original solution -- perish the thought, two such solutions -- offends his fragile ego.

Poor Jonathan. Like Wile E. Coyote, his tactics backfire. Beep-beep, vroooom. Maybe he is beyond reform. He is what he is. A is A. 

None of this is true--except A is A.

--Brant 

[Max]

8 hours ago, merjet said:

(>> = me, > = merjet, black = me )

>> Well, at least the article gives the correct solution

> I get it. You believe there is only one correct solution -- merely because you like it. There are many proofs of the Pythagorean Theorem. Is only one of them correct merely because you like it best?

So you admit that it is a solution. However, it is the same solution that has been given and defended by Jon, Jonathan, Ellen, Baal and me. Yet you’ve many times stated that our solution was wrong, that seems to me to be a contradiction. Further it is a simple solution that goes to the heart of the paradox.

>> The part with the cycloids doesn't explain the paradox.

> Maybe to you. The cycloids solution is correct. What part of it do you not understand?

There is nothing to understand. Your own summary states: “Summarizing, the smaller circle moves horizontally 2πR because any point on the smaller circle travels a shorter, more direct path than any point on the larger circle.” Well, duh. That the smaller circle moves 2πR when the large circle rolls without slipping one revolution is trivial, you don’t need any cycloids to prove that. Moreover, in your “second solution” you say the same thing without any cycloids, implying that these are just unnecessary embellishments.

But the fact that the smaller circle also moves over a distance of 2πR is just the first part of the solution. The paradox is generated by the supposition that the small circle also rolls without slipping, implying that after one revolution it would move over a distance 2πr < 2πR: contradiction. Conclusion: the small circle cannot roll without slipping, it must slip to make up the difference 2π(R – r). QED.

> This is easily shown. Add two more inner concentric circles to the usual two depicting Aristotle's wheel paradox, such as to represent the inner and outer edges of the white ring on a white wall tire. When all four are rolled together, all three inner concentric circles move the identical horizontal distance, the circumference of the largest circle. Their circumferences, all different, are irrelevant, but their centers, all the same, are relevant.

The same as above: it is trivial that all those circles move the same horizontal distance, but that is just the first part of the paradoxical statement, the second part being the supposition that those smaller circles also can roll without slipping. It is only by combining those two parts that the paradox arises. Therefore the notion of slippage is essential for understanding and solving this paradox.

> 'Slipping' is in fact just an unnecessary distraction compared to the second, translation solution on the Wikipedia page. The translation solution is simpler and far more elegant than 'slippage'. The center of the smaller circle matters; its circumference doesn't.

Wrong. As I’ve shown above, the “translation solution” isn’t a solution, it’s just stating one half of the paradox problem. The circumference of the smaller circle is essential to the paradox and its solution. If the radius of the smaller circle equals the radius of the large circle, the paradox disappears.

> On August 14 I challenged you to quantify the three terms on the right of this equation:

> 2*pi*R = Rotation + Translation + Slippage

> You haven't answered yet. Is answering it too difficult for you?

You can’t expect me to “answer”, as you didn’t ask me anything in that post.

 

 

 

 

[Max]

13 hours ago, Jon Letendre said:

There is yet another way to detect the over-spin of the wheel in Billyboy’s video and also in the picture used in the Wiki article.

The picture used in the Wiki article https://en.wikipedia.org/wiki/Aristotle's_wheel_paradox is almost a correct depiction of traversed road length given the wheel diameter. On my screen at this moment, the blue wheel has 1.5 inch diameter, so the blue road should be 1.5 * 3.14 = 5 inches, yet it is only 3.5 inches. In other words, the two wheels below, depicting start and end positions, should be spread even farther apart. Another way of saying the same thing is that if you rolled the blue wheel below, it would not have completed its one rotation yet when it got to where it is shown on the right. It would have farther to go to complete one rotation.

Here is the correct depiction of ratio of wheel diameter to road length traversed in one rotation...

Now observe the highly compressed road length from Billyboy’s video... 

Good points to check such diagrams. In fact any distance between the wheels after one revolution in such diagrams is physically possible, but if one of the wheels rolls (rotates without slipping), only two possibilities remain, corresponding to the circumference of respectively the small and the large wheel.

[Jon Letendre]

Here I correct the Wiki picture.

The drawing used in the Wiki article depicts an arrangement where neither wheel is performing true rolling without slip or skid.

As displayed on the Wiki page, the drawing depicts start and end positions that have the blue wheel spinning-out over the road and the red wheel skidding somewhat along its “road.”

In order to depict the correct distance between start and end positions where the blue wheel makes one rotation without any slip, the positions must be widened to 3.14 times the wheel diameter, like this...

 

In order to depict the correct distance between start and end positions where the small, red wheel makes one rotation without any slip over its “road,” the positions must be shortened to 3.14 times the red ”wheel” diameter, like this...

[anthony]

On 11/15/2018 at 10:58 PM, Jon Letendre said:

Anyone can confirm the above at home:

Lie a bottle down on a smooth table. Place a book under the neck to fill the gap between it and the table.

Press down on the neck while rolling and while keeping the bottle going straight and observe that the base of the bottle wants to over-spin, relative to the table.

Now press down on the base of the bottle while rolling and observe that the neck is skidding, relative to the book/table.

But yet, I assume, you will confirm that without any book ('upper track') - all on its own - the bottle will roll evenly and straight? How do you account for that, Jon?

This is all self-evident.

Evidence: An inner wheel-circle turns with the outer wheel which contains it -- we can see it does. A circle is a circle, and any circle within it, rotates accordingly and travels the identical distance. Period.

It is clear that the 'paradox' does not involve going into friction, drag or weight or anything experimental, it's an exercise in abstract cognition - perception, identification and integration of a real thing. You guys are looking for an engineering solution or a mathematical exercise, and losing the plot. A  is A. Trust your senses.

[Jon Letendre]

So that’s our error, Tony? That we are looking for a mechanical understanding of a question of mechanics?

Did you roll the bottle as I described, Tony?

To review, you thought that the bottle can roll without slip at the wide base and at the narrow neck and go straight:

“Bob: Nothing changes if there is to be a 'track' the inner wheel is on, or if the line is an imaginary 'path'. You get the same result with two surfaces, or one. Set up or picture an experiment with any bottle (best, because it has a protruding "inner circle", the neck and cap seen from the side). Construct two parallel tracks for the bottle to rest on. One higher than the other to finely adjust for the differing diameters. When the bottle is evenly balanced and leveled on both tracks, roll it along them and observe that both bottle and neck will roll evenly, with no skipping, slipping or jamming. (I'm sure if it's not balanced well, it will veer off course).“

Did the bottle experiment cure your misunderstanding of the matter?

[anthony]

23 minutes ago, Jon Letendre said:

Did you roll the bottle as I described, Tony?

Did you find that I corrected your misvisualization of the matter?

Heh, got ahead of you.  I experimented with some round articles (and books, etc.) months ago. Nothing invalidated what I had previsualized. (Also. "Press down" on any part of the bottle and you can make anything happen...). I far rather relate to all I've seen of wheels in action.

I maintain, still, this is an abstract mental exercise of how one perceives real, concrete things and their motions. It precedes math and scientific experiment, which applied prematurely, are misleading distractions.

[Jon Letendre]

1 minute ago, anthony said:

Heh, got ahead of you.  I experimented with some round articles (and books, etc.) months ago. Nothing invalidated what I had previsualized. (Also. "Press down" on any part of the bottle and you can make anything happen...).

I maintain, still, this is an abstract mental exercise of how one perceives real, concrete things and their motions. It precedes math and scientific experiment, which applied prematurely, are misleading distractions.

So you have not conducted the bottle experiment?

[Jon Letendre]

3 minutes ago, anthony said:

Heh, got ahead of you.  I experimented with some round articles (and books, etc.) months ago. Nothing invalidated what I had previsualized. (Also. "Press down" on any part of the bottle and you can make anything happen...).

I maintain, still, this is an abstract mental exercise of how one perceives real, concrete things and their motions. It precedes math and scientific experiment, which applied prematurely, are misleading distractions.

No, it doesn’t “make anything happen,” but it ensures roll without slip.

[Max]

If a randroid resorts to his magic mantra "A is A", you know that he's run out of arguments.

[anthony]

2 minutes ago, Jon Letendre said:

So you have not conducted the bottle experiment?

Just said so.

[anthony]

LOL.