Aristotle's wheel paradox

[Jonathan]

39 minutes ago, anthony said:

You don't seem to make the translocation from static wheels to rolling wheels. "Any point on the smaller wheel travels faster..."

Not in this universe. Smaller = slower. (in this context).

You can find no principle connecting athletic tracks, archery targets, wine bottles - and I am sure, orbiting planets - if you haven't conceptualized the common denominators of circle/wheel..

The tangential speed of two circles or wheels or planets, or sprinters, must always be greater on the outer circumference. IF - they stay in alignment.

Spinning or rolling, no diff. The outer rim has farther to travel in one revolution and the equivalent time. Get it?

 

 

You dodged and evaded the question.

Here it is again:

"Which point, E or A, is traveling faster? Which is covering more ground/space in the same amount of time?"

Answer the question instead of yapping and blabbering and lying.

J

[Jonathan]

2 minutes ago, Jon Letendre said:

The cable would snap, break.

The premise is that the cables are unbreakable. (They are stronger than any friction that would result from one of the wheels skidding/sliding.)

So, with that in mind, what must happen? Which wheel must skid/slide? Which must roll true/free?

J

[Jon Letendre]

My windshield droppings art is better than yours.

Yours look like they were made by deranged fingaloes, or flamingos or whatever they call themselves.

[Max]

2 hours ago, anthony said:

"...does not mean that the *concept* track is absent". (Logically, it doesn't mean that the concept is present, either).

But I showed you that the concept is present: Unrolling the large circle to the line ZI  means that ZI is the track over which the large circle rolls, and unrolling the smaller circle to the line HK means that HK is the track over which the small circle rolls

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But, good that you didn't pretend that "track" was explicitly mentioned.

As if that matters.

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All that we read here is of a "line" - i.e. a possible representation of a track, more like an imaginary "path".

Ever heard of technology? Of people who make machines, cars, airplanes, bridges, cranes, etc. etc.? Where everything depends on the fact that we can use mathematics to reliably calculate forces, distances, stresses, angles, speeds, torques etc.? Do you think they're worrying about the question whether a line on a technical drawing could be the representation of a track, or that it is more like an imaginary path? Miraculously, airplanes built based such drawing boards with imaginary paths can fly! That is reality!

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But you all need to have a tangible, physical "track" to fulfil the "slippage solution" - so, track it must be...You are "destroying" reality, not solving the paradox.

Nonsense. I've shown that a completely mathematical treatment also gives the slippage solution. As you'd expect from a correct solution, a physical realization confirms the fact that slippage occurs. Theory and practice are in agreement. You are the one who is destroying reality, by refusing to see what nearly everybody can see, and refusing to consider the mathematical treatment.

 

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"So those two tracks are an essential part of the original paradox".

NO. You are including your conclusion in validating your conclusion. Two tracks it must be, so that's the only thing that makes sense, which is illogical.

Two tracks it is. Why do you think Aristotle (or whoever that old Greek was) wrote about two lines, along which the circles unroll. Just to make a pretty picture? No, that was because those two lines are essential to the formulation of the paradox. If you don't understand that, you don't understand the paradox.

 

2 hours ago, anthony said:

Simply, again:

A wheel of circumference x rotates once, moving distance x. An fixed inner wheel of circumference y rotates once -- but moves also distance x.

How can it be!! For reasons I've repeated.

Slippage must be introduced!!

Aristotle: "The problem is then stated"-

"...and since the smaller does not leap over any point, it is strange [...] that the smaller traverses a path equal to the larger".

"Strange", and perhaps counter-intuitive, but that is indeed what happens.

And that what happens is called slipping. When you realize that, all strangeness disappears and the paradox is solved, as it was generated by the assumption that the smaller wheel could also roll without slipping - mathematically: the false assumption that the smaller circle could also trace out its circumference by rolling one revolution.

[Jon Letendre]

Pigeons cannot refuse to consider a new idea.

Pigeons cannot refuse to see.

[Jonathan]

It's kind of fun going back to the beginning of this thread and looking at the first videos that I had posted. At the time, I thought that I had slowed the motion down to the point that anyone should be able to see what's happening. Heh. And now, comparing them to the later videos, they look like fast-forwarded clips. That's how slow we've gotten, and the geniuses still don't grasp the in-motion relationships.

Astounding.

J

[Max]

6 hours ago, anthony said:

You don't seem to make the translocation from static wheels to rolling wheels. "Any point on the smaller wheel travels faster..."

Not in this universe. Smaller = slower. (in this context).

You can find no principle connecting athletic tracks, archery targets, wine bottles - and I am sure, orbiting planets - if you haven't conceptualized the common denominators of circle/wheel..

The tangential speed of two circles or wheels or planets, or sprinters, must always be greater on the outer circumference. IF - they stay in alignment.

Spinning or rolling, no diff. The outer rim has farther to travel in one revolution and the equivalent time. Get it?

That is true in the rest frame of the circle, the tangential speed of the outer circle is greater than the tangential speed of the smaller circle. But wait! We are considering the system in the rest frame of the track, where we see the wheel rolling to the right. In that frame you have to add the translation speed to the speed of the points on the circles. Due to the rotation, a point on the large circle continuously changes direction. In the lower half of the figure the horizontal component of the velocity vector of that point is directed to the left. So we have to subtract that horizontal component from the speed due to the translation to the right. In our rest frame, the point is moving slower than the center. In the 6 o'clock position the tangential velocity vector is exactly directed to the left. The speed in the rest frame (subtracting now the tangential speed from the translation speed) zero. At that one moment the point stands still. That is equivalent with the condition "rolling without slipping". *)Further rolling of the circle decreases the horizontal component of the velocity vector, so the speed in the rest frame increases again.

In the upper half of the figure the opposite happens. After passing the 9 o'clock position the speed becomes greater than the translation speed of the center.At the 12 o'clock position the velocity vector points to the right and now the tangential speed is added to the translation speed, the point has now a speed twice that of the center. Logical, because after one revolution every point on the circles must have  traveled the same distance to the right, so what they lose in the lower half, they must make up for in the upper half and vice versa.

Now look at the small circle. When the segment AB of the large circle lines up with CD of line 1, around the point of zero speed, you see that the corresponding segment EF of the small circle is swept to the right along a much larger segment GH of line 2. If the small circle would roll without slipping, like the large circle, it would in the same way line up with an segment GH that is just as small as EF. But as the tangential speed of the smaller circle is smaller than that of the large circle, the amount that is subtracted from the translation speed is smaller, and therefore it doesn't cancel the translation speed at that point (as in the case of the large circle), therefore instead of zero speed, there is a net translation to the right. That net translation we call slipping, and it is very well visible in this animation.

*) For cycloid lovers: this is the point where the cusp of the cycloid touches the line.

[Max]

Observe that it is just the fact that the tangential speed of the large circle is greater than that of the small circle,  is the cause that the lower part of the small circle moves faster to the right than the lower part of the large circle, as the horizontal component of the tangential vector has to be subtracted there from the translation speed, and subtracting a smaller value results in a larger speed than subtracting a greater value! 

[KorbenDallas]

5 hours ago, Jonathan said:

11 hours ago, Jonathan said:

And the word "unrolls" is important in that it means that the circles become the lines as they roll, as depicted in the Wolfram animated diagram that people have been posting.

its an idea that Ellen addressed a few pages back asking people to imagine strings wrapped around the wheels and unfurling when contacting the surfaces on which they roll.

Hmmm. Let's take that a step further. Referring to the animated diagram above, the two wheels are cable spool cylinders, rigidly affixed to one another. Red represents cables, which are unbreakable and do not stretch. The beginning end of each cable is affixed, at the starting point, to the surface on which its wheel will roll. Each cable is wrapped around its wheel exactly once, and the other end of each cable is then attached to its wheel aligned with the starting point. Thus, we have two different lengths of cable -- each is the circumference of the wheel around which it wrapped.

Now, we apply an overwhelming force to push the wheels forward and unfurl the cables as the wheels roll. Given that all items in the scenario are unbreakable, cannot be stretched or otherwise distorted in form, and that the wheels must move due to the overwhelming force, what will happen? What MUST happen, and why?

If marks were to be placed on both wheels, and their movements traced, what paths must they trace?

J

Here's a diagram of the above cable/spool concept:

What must happen when the wheels roll in the direction of the arrow?

J

Jonathan,

If everything in the scenario is unbreakable and can't be distorted in any way---which includes the spools themselves from crushing, the axle from bending, and the wheels becoming unaffixed---nothing can make them move.
 

[BaalChatzaf]

2 hours ago, KorbenDallas said:

Jonathan,

If everything in the scenario is unbreakable and can't be distorted in any way---which includes the spools themselves from crushing, the axle from bending, and the wheels becoming unaffixed---nothing can make them move.
 

The entire rig comes to a screeching halt when the inner wheel has rotated once. Which means the combined wheel does not make it to the other side. 

[william.scherk]

An exaltation of larks ...

26 minutes ago, BaalChatzaf said:

The entire rig comes to a screeching halt

The leash yoke squealed on the cable railway carriage as it jolted to a halt and hats, bags and old people went flying. I concur with Bob's pithy summation. The yank of the yoke on the rig. The end of the line. The one thing she is shorter than the other.

This thread unreeling could get Tony all the way up the hill, I figger.  The sound of larks instead of the squeal of flanged wheels resisting a curve. 

Doves and larks.

[KorbenDallas]

1 hour ago, BaalChatzaf said:

The entire rig comes to a screeching halt when the inner wheel has rotated once. Which means the combined wheel does not make it to the other side. 

Gotchya

Edited November 30, 2018 by KorbenDallas
brainfart

[merjet]

On 11/28/2018 at 6:16 PM, Max said:

That the word "track" isn't mentioned there, does of course not mean that the concept "track" is absent. Unrolling the large circle to the line ZI  means that ZI is the track over which the large circle rolls, and unrolling the smaller circle to the line HK means that HK is the track over which the small circle rolls. So those two tracks are an essential part of the original paradox. Taking those away is destroying the paradox, not solving it. Child and bathwater.

 

16 hours ago, Max said:

Two tracks it is. Why do you think Aristotle (or whoever that old Greek was) wrote about two lines, along which the circles unroll. Just to make a pretty picture? No, that was because those two lines are essential to the formulation of the paradox. If you don't understand that, you don't understand the paradox.

The paradox can be illustrated in two ways:

1. The larger circle is on a “track”/surface and the smaller circle is dependent upon the larger.

2. The smaller circle is on a “track”/surface and the larger circle is dependent upon the smaller.

Solve either 1 or 2, and solving the other is trivial.

Taking away both “tracks” destroys the paradox. Assuming both “tracks”/surfaces exist simultaneously muddles it. All of Jonathan’s videos and drawings show two “tracks”/surfaces simultaneously. That is the con in Jonathan’s con art. That’s the crutch, without which Jonathan and you are helpless.

[Jonathan]

2 hours ago, merjet said:

 

The paradox can be illustrated in two ways:

1. The larger circle is on a “track”/surface and the smaller circle is dependent upon the larger.

2. The smaller circle is on a “track”/surface and the larger circle is dependent upon the smaller.

Solve either 1 or 2, and solving the other is trivial.

Taking away both “tracks” destroys the paradox. Assuming both “tracks”/surfaces exist simultaneously muddles it. All of Jonathan’s videos and drawings show two “tracks”/surfaces simultaneously. That is the con in Jonathan’s con art. That’s the crutch, without which Jonathan and you are helpless.

False. The Aritotle's Wheel Paradox specifically identifies and requires two surfaces, one beneath each wheel. The entire point of the "paradox" is that the ancient geniuses believed that both wheels could roll freely at the same time without skidding/slipping.

[anthony]

20 hours ago, Max said:

But I showed you that the concept is present: Unrolling the large circle to the line ZI  means that ZI is the track over which the large circle rolls, and unrolling the smaller circle to the line HK means that HK is the track over which the small circle rolls

As if that matters.

Ever heard of technology? Of people who make machines, cars, airplanes, bridges, cranes, etc. etc.? Where everything depends on the fact that we can use mathematics to reliably calculate forces, distances, stresses, angles, speeds, torques etc.? Do you think they're worrying about the question whether a line on a technical drawing could be the representation of a track, or that it is more like an imaginary path? Miraculously, airplanes built based such drawing boards with imaginary paths can fly! That is reality!

Nonsense. I've shown that a completely mathematical treatment also gives the slippage solution. As you'd expect from a correct solution, a physical realization confirms the fact that slippage occurs. Theory and practice are in agreement. You are the one who is destroying reality, by refusing to see what nearly everybody can see, and refusing to consider the mathematical treatment.

 

Two tracks it is. Why do you think Aristotle (or whoever that old Greek was) wrote about two lines, along which the circles unroll. Just to make a pretty picture? No, that was because those two lines are essential to the formulation of the paradox. If you don't understand that, you don't understand the paradox.

 

And that what happens is called slipping. When you realize that, all strangeness disappears and the paradox is solved, as it was generated by the assumption that the smaller wheel could also roll without slipping - mathematically: the false assumption that the smaller circle could also trace out its circumference by rolling one revolution.

You are reading more than is there. I gather you see a 'design' (as required for "airplanes", etc.,) when there's no more than a description, by Aristotle. Maybe you don't know that Aristotle, or whoever, was making a non-literal explanation of his observation? But instead you want the  "unrolling lines" (a descriptive tool) to ~literally~ be or become tracks, for the purpose of inducing 'slippage'. 

Re-read his preamble and then (at what must be his conclusion): "The problem as stated".

In the latter as the former, he does not suggest any remedy - the "strange" "problem" stands as he states it until the end of his text. If he'd wanted, surely (to solve his own 'problem') he could have here converted his rolling lines into tracks. He didn''t. No tracks are suggested, no slippage is implied or suggested anywhere.

Inferred: He wasn't making a teaching moment, he observed and described a phenomenon of reality.

No escaping the unassailable fact one should accept - The inner wheel within an outer wheel traverses a distance further than its circumference when rotated once - and, logically - identical to the circumference of the larger.  That is the entire 'paradox' ("problem"). He as good as states it with his last remarks.

If the actions of one wheel or both the wheels ~could~ slip - or be mechanically contorted to induce slippage, we woudn't have either or both wheels behaving by their identity anymore. (I.e., a 'correction' causes a jammed wheel)

Aristotle made an identification - only. This discussion centers on the law of identity and causality (identity in action). An entity acts according to its nature. The entity is the outer wheel.

 

 

[merjet]

3 hours ago, Jonathan said:

False. The Aritotle's Wheel Paradox specifically identifies and requires two surfaces, one beneath each wheel. The entire point of the "paradox" is that the ancient geniuses believed that both wheels could roll freely at the same time without skidding/slipping.

Wrong. “If the smaller circle is rolled along its tangent without sliding or slipping, the paths traced out in a single revolution are, in the case of both circles, equal to the circumference of the smaller, whereas If the larger circle is rolled along its tangent without sliding or slipping, the paths traced out in a single revolution are, in the case of both circles, equal to the circumference of the larger. In this the author of Mechanica saw a difficulty and sought to find a solution.” (Drabkin, 'Aristotle's Wheel: Notes on the History of a Paradox', p. 162-3)

Even Mechanica says: " ....  When I move the smaller circle .... when I move the large circle ...."

 

[anthony]

23 hours ago, Jonathan said:

No, that does not logically follow. Not even close. You're still not getting it. You're never going to grasp it.

 

So, you're saying that we don't see what we see? We just believe that we see it? And your proof of this is the fact that you don't see it? If you don't see or understand something, then, therefore, no one else does either? No one can see or understand anything that Tony can't, because Tony is the universal limit of human cognition?

 

You've gotten it backwards. As usual. The opposite is true. Any point on the smaller wheel travels faster than its corresponding point on the larger wheel.

Look again at the animated diagram. Slow down, try to focus, and pay attention.

Do you notice anything in addition to the circles and lines?

Can you see the yellow and orange segments? How about the letters identifying point on the circles and lines? Can you see them?

See them now? Okay, now watch point E in comparison to point A (actually, first spit out your gum -- we don't want you multitasking while trying to do this). Okay. Which point, E or A, is traveling faster? Which is covering more ground/space in the same amount of time?

See? It's pretty easy if you look and keep your attention on it.

This is where your little theory goes to hell. Your theory is about wheels rotating on a point, like a ferris wheel. That doesn't work here because these wheels are ROLLING!!!! Different stuff happens when something is rolling versus when it is only rotating on a stationary point. See?

No, of course you don't see.

J

You take reality from animations. Experiments online. Any and all may have a bias to what the maker wishes.

Explain why there is, apparently, 'slippage' in this depiction of the inner wheel's motion. Or unequal contact.

Is there grease on the track? Are the wheels not supported equally? One track slightly too low for the different diameters? More friction on the lower surface?

Just as easily, the greater wheel can be "made to appear" to 'slip' instead.

[Jon Letendre]

[Max]

 

23 minutes ago, anthony said:

You take reality from animations. Experiments online. Any and all can have a bias to what the maker wishes.

Explain why there is, apparently, 'slippage' in this depiction of the inner wheel's motion. Or unequal contact.

I've explained that in detail in this post (click on the arrow): 

 

 

Didn't you read that? It answers all your questions.

 

23 minutes ago, anthony said:

Is there grease on the track? Are the wheels not supported equally? One track slightly too low for the different diameters? More friction on the lower surface?

Just as easily, the greater wheel can be "made to appear" to 'slip' instead.

And you'll also see why these are silly questions.

[Jonathan]

3 hours ago, merjet said:

Wrong. “If the smaller circle is rolled along its tangent without sliding or slipping, the paths traced out in a single revolution are, in the case of both circles, equal to the circumference of the smaller, whereas If the larger circle is rolled along its tangent without sliding or slipping, the paths traced out in a single revolution are, in the case of both circles, equal to the circumference of the larger. In this the author of Mechanica saw a difficulty and sought to find a solution.” (Drabkin, 'Aristotle's Wheel: Notes on the History of a Paradox', p. 162-3)

Even Mechanica says: " ....  When I move the smaller circle .... when I move the large circle ...."

 

Neither the Drabkin quote above nor the original Mechanica says anything about ignoring or eliminating either line while any rolling takes place. Nothing in the above quotes state, or even suggest, anything about disregarding either of the lines at any time.

It's just something that you've made up.

J

[Jonathan]

2 hours ago, anthony said:

You take reality from animations. Experiments online. Any and all may have a bias to what the maker wishes.

Show the bias. Empty accusations aren't enough. Objectively demonstrate any errors or falsehoods in the diagrams and sequences that I've posted.

 

2 hours ago, anthony said:

Explain why there is, apparently, 'slippage' in this depiction of the inner wheel's motion. Or unequal contact.

It has already been explained to you in the simplest terms possible. You are not cognitively capable of grasping it.

J

[Jon Letendre]

Does the pigeon hear the words spoken to it?

Yes, it hears every single one of them.

[Jonathan]

16 hours ago, BaalChatzaf said:

The entire rig comes to a screeching halt when the inner wheel has rotated once. Which means the combined wheel does not make it to the other side. 

Indeed! But, you haven't answered any of the questions.

Describe what must happen during the rolling, not after it. Which of the wheels must roll without slipping, and which must roll with slipping, and why? Do you see the blue and green dots on the wheels? One is on the small wheel, and the other is on the large wheel. What path will each take, and why? One will create a proper cycloid. What type of cycloid will the other create, curtate or prolate? Why?

J

[Jonathan]

7 hours ago, merjet said:

 

The paradox can be illustrated in two ways:

1. The larger circle is on a “track”/surface and the smaller circle is dependent upon the larger.

2. The smaller circle is on a “track”/surface and the larger circle is dependent upon the smaller.

Solve either 1 or 2, and solving the other is trivial.

Taking away both “tracks” destroys the paradox. Assuming both “tracks”/surfaces exist simultaneously muddles it. All of Jonathan’s videos and drawings show two “tracks”/surfaces simultaneously. That is the con in Jonathan’s con art. That’s the crutch, without which Jonathan and you are helpless.

As people have asked you several times, and which you've evaded answering, what is paradoxical about a wheel rolling over a surface while an inner wheel attached to it is not rolling on its own surface?

J

[merjet]

1 hour ago, Jonathan said:

As people have asked you several times, and which you've evaded answering, what is paradoxical about a wheel rolling over a surface while an inner wheel attached to it is not rolling on its own surface?

I've explained it many times. Wikipedia and Drabkin explain it. However, I'm not surprised at your mental inability to get it.