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Aristotle's wheel paradox

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3 hours ago, BaalChatzaf said:

Even with the cable the inner wheel is dragged of the outerwheel does not slip.

 

The inner wheel can't be dragged. Its cable prevents its being dragged. The only possible outcome is that it rolls without slipping, which drives the larger wheel to spin-out and throw out slack of its own cable.

J

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2 hours ago, BaalChatzaf said:

No  true.  The outer wheel move further per revolution because its radius is larger. That means the inner wheel is partly dragged, partly rolled. Because one revolution of the inner wheel corresponds to to shorter length than one revolution of the outer wheel. Since the wheels are rigidly affixed to a common axel  both turn together rigidly.  Pay attention to the mechanics, the motion and the geometry of the rig..

Heh.

As I said in my post just previous to this one, the small wheel cannot be dragged along its surface. You would be right were it not for the cables. My having added them to the experiment changes things. The cable wrapped around the small wheel prevents it from being dragged across its surface. It can only roll truly (roll without slipping/skidding). Its rolling truly must therefore drive the larger wheel to rotate faster than the distance that it covers, which means that it spins-out, which it is not prevented from doing by its cable.

J

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2 hours ago, anthony said:

"Should have traveled laterally on that surface..." Your "should" says it all.

It's not Ellen's "should," but Aristotle's (or whoever came up with the "paradox" if not Aristotle). 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.

Theirs was a mistaken premise, an error/contradiction rather than a paradox. Their "should" was stupid, it contradicted simple geometry and reality, and they couldn't figure it out, and largely due to their having gone all eggheady over it, and making it way more complex than it had to be.

J

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6 hours ago, BaalChatzaf said:

The outer wheel move further per revolution because its radius is larger. That means the inner wheel is partly dragged, partly rolled.

Bob,

It can go the other way, too. If the inner wheel is the one not dragging, the outer wheel will skip and partly roll.

There is nothing on the schematic that says the outer wheel is the only one that can roll evenly. This is implied because the mind has a bias toward giving bigger things the benefit of the doubt. We presume that the length of track is the same as the circumference of the larger wheel and the smaller wheel partly rolls and partly drags (in the case where the schematic represents two different tracks), but the schematic can just as easily represent that the length of track is the same as the circumference of the smaller wheel and the larger wheel partly rolls and partly skips.

This is the whole basis of how cog wheels work.

I'm pretty sure that's why Jonathan said, "False." You guys are doing a cable thing right now, but what I said above is true. And I've seen Jonathan and Jon demonstrate this.

Michael

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Just to be clear, remember that we're now talking about the new thought experiment that I proposed, the one which includes cables attached to the wheels and the surfaces on which they sit, and that the cables are attached so that they unfurl when the wheels are pushed and rolled to the right. The force used is irresistible/overwhelming, so that it easily overcomes any traction that either wheel might have with its surface.

Heres the original info and diagram once again:

   On 11/29/2018 at 6:15 AM,  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.

AristotlesWheel.gif

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:

45381043674_93c81ac422_b.jpg

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

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6 minutes ago, Michael Stuart Kelly said:

Bob,

It can go the other way, too. If the inner wheel is the one not dragging, the outer wheel will skip and partly roll.

There is nothing on the schematic that says the outer wheel is the only one that can roll evenly. This is implied because the mind has a bias toward giving bigger things the benefit of the doubt. We presume that the length of track is the same as the circumference of the larger wheel and the smaller wheel partly rolls and partly drags (in the case where the schematic represents two different tracks), but the schematic can just as easily represent that the length of track is the same as the circumference of the smaller wheel and the larger wheel partly rolls and partly skips.

I'm pretty sure that's why Jonathan said, "False."

Michael

That is half the story. The other half is the fact that in this case the large wheel cannot "roll without slipping". That would namely imply that the smaller wheel would be dragged along, slipping to keep up with the large wheel (as we've already shown in about 10000 posts). But the small wheel is held back by its shorter cable, so that's the only wheel that can roll without slipping. That again causes the large wheel to slip: it is held back, rotates more than its "rolling distance", causing its own cable to become slack.

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5 minutes ago, Max said:

That is half the story. The other half is the fact that in this case the large wheel cannot "roll without slipping". That would namely imply that the smaller wheel would be dragged along, slipping to keep up with the large wheel (as we've already shown in about 10000 posts). But the small wheel is held back by its shorter cable, so that's the only wheel that can roll without slipping. That again causes the large wheel to slip: it is held back, rotates more than its "rolling distance", causing its own cable to become slack.

Max,

That's another way. What's more, the chains move, not the wheels. That's correct. The chains move and the wheels move. That's correct. One chain moves and both wheels do. That's correct. Or how about a treadmill belt instead of a chain? :) Or geared cogwheels? Or road and wheel? They are all correct.

To me a schematic is a schematic is a schematic.

You can build all kinds of different things from that one schematic--all incompatible with each other and all correct according to the schematic. Or the reverse. That one schematic can be a correct representation of all kinds of different incompatible things.

All you need is enough ambiguity in the drawing to give the appearance of a paradox and off you go.

Oddly enough, I see this entire dissuasion more in terms of human behavior than math or geometry, or even schematic drawing. Have you noticed that most everyone wants to be the rule-giver?

That's a powerful motivation.

Welcome to the human race.

:) 

Michael

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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? 

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9 minutes ago, 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? 

So it will be simpler and they can then tackle it.

It is too complex in the original form for them to get their arms around it,

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6 minutes ago, Jon Letendre said:

So it will be simpler and they can then tackle it.

...and explain it away, telling us that they've "solved" the paradox.

 

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4 hours ago, Jonathan said:

It's not Ellen's "should," but Aristotle's (or whoever came up with the "paradox" if not Aristotle). 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.

Theirs was a mistaken premise, an error/contradiction rather than a paradox. Their "should" was stupid, it contradicted simple geometry and reality, and they couldn't figure it out, and largely due to their having gone all eggheady over it, and making it way more complex than it had to be.

J

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

I thought after I'd posted that Tony might mistakenly take the "should" as mine, even though I'd said "Here's how the problem goes" (i,e., here's how the person posing the problem was thinking), but I didn't have time to edit by adding a parenthetical stating that the "should" was the assumption of the problem's formulator.

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.

Quite agreed - and I've said before - that the problem isn't a genuine paradox - defined as "an apparent contradiction between two true premises" - but instead an actual contradiction between a false assumption and reality.

Ellen

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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? 

 

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18 hours ago, Max said:

That is half the story. The other half is the fact that in this case the large wheel cannot "roll without slipping". That would namely imply that the smaller wheel would be dragged along, slipping to keep up with the large wheel (as we've already shown in about 10000 posts). But the small wheel is held back by its shorter cable, so that's the only wheel that can roll without slipping. That again causes the large wheel to slip: it is held back, rotates more than its "rolling distance", causing its own cable to become slack.

the inner wheel will slip until its cable is taught at which point the entire rig stops.  The outer wheel will have done only part of one turn. 

 

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18 hours ago, Max said:

That is half the story. The other half is the fact that in this case the large wheel cannot "roll without slipping". That would namely imply that the smaller wheel would be dragged along, slipping to keep up with the large wheel (as we've already shown in about 10000 posts). But the small wheel is held back by its shorter cable, so that's the only wheel that can roll without slipping. That again causes the large wheel to slip: it is held back, rotates more than its "rolling distance", causing its own cable to become slack.

Exactly. Right on the money.

J

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1 hour ago, BaalChatzaf said:

the inner wheel will slip until its cable is taught at which point the entire rig stops. 

False. The inner wheel will roll. It will not slip at all. It is not physically possible for it to slip. Its cable prevents any slippage. The inner wheel will roll without slippage until it reaches the end of its cable, which is one rotation.

1 hour ago, BaalChatzaf said:

The outer wheel will have done only part of one turn. 

 

False. The two wheels are affixed to one another. When one completes a full rotation, the other does as well. However, since the large wheel must over-spin in comparison to its surface (any point on its perimeter will create a prolate cycloid during the wheels' motion), it will have travelled a distance shorter that its circumference, and its cable will have let off slack (the length of the slack will be equal to the length of the large wheel's circumference minus the length of the small wheel's circumference).

Bob, you're not properly envisioning the scenario, especially the effects of the cables. I would suggest building a model and observing how the reality of it differs from your mistaken imagining of what happens. A couple of spools of thread with different diameters, glued together and a nail for an axle would work.

J

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18 hours ago, 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.

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.

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

It's all a wonderful study in a variety of psychological issues, needs, fragilities and defenses. My current curiosity: How much has Rand's work played in hindering these two? Had they not been exposed to Rand's notions of "ideal men" and the grand, Romantic Realist Ego, would they be as pigheaded? As inwardly blind? As heroically self-certain (to the point of rejecting obvious reality)? 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.

J

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I want to clarify that my criticism of the schematic drawing for being too vague to represent one reality situation only does not preclude that the paradox was stipulated in words by Aristotle.

I wasn't aiming at the paradox per se, but at what prompts people to think of a paradoxical situation (or refute it for that matter).

21 hours ago, Ellen Stuttle said:

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.

Quite agreed - and I've said before - that the problem isn't a genuine paradox - defined as "an apparent contradiction between two true premises" - but instead an actual contradiction between a false assumption and reality.

This is perfectly put.

Michael

 

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6 hours ago, 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? 

 

I agree, Max. “Aristotle’s Wheel Paradox” is fine. Semantics aside, Ellen clearly gets all of it. We could easily show her that the paradox can be stated her preferred way, with the slightest adjustments, as a contradiction between two apparently true premises.

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5 hours ago, Jonathan said:

False. The inner wheel will roll. It will not slip at all. It is not physically possible for it to slip. Its cable prevents any slippage. The inner wheel will roll without slippage until it reaches the end of its cable, which is one rotation.

False. The two wheels are affixed to one another. When one completes a full rotation, the other does as well. However, since the large wheel must over-spin in comparison to its surface (any point on its perimeter will create a prolate cycloid during the wheels' motion), it will have travelled a distance shorter that its circumference, and its cable will have let off slack (the length of the slack will be equal to the length of the large wheel's circumference minus the length of the small wheel's circumference).

Bob, you're not properly envisioning the scenario, especially the effects of the cables. I would suggest building a model and observing how the reality of it differs from your mistaken imagining of what happens. A couple of spools of thread with different diameters, glued together and a nail for an axle would work.

J

I am computing the transverse of the center of a circle of radius r. If it doesn't slip  and it turns through angle theta then it will traverse   r*theta  (theta measured in radians).   Now look at the outer wheel radius R  where R > r.  If the little wheel rigidly affixed to the outer wheel turns an angle theta so does the outer wheel.  But the outer will will bring the common center R * theta to the right  which exceeds  r*theta  hence the inner wheel must have been dragged for a distance of   (R - r)*theta.  Attaching a wire or cable to the inner wheel does not change the geometry.

L.L.A.P \\//

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5 hours ago, 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.

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

It's all a wonderful study in a variety of psychological issues, needs, fragilities and defenses. My current curiosity: How much has Rand's work played in hindering these two? Had they not been exposed to Rand's notions of "ideal men" and the grand, Romantic Realist Ego, would they be as pigheaded? As inwardly blind? As heroically self-certain (to the point of rejecting obvious reality)? 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.

J

It's even more interesting when they go off the Rand track not knowing it as with open borders. (Not applying that to N and T, but to B and B.)

--Brant 

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47 minutes ago, BaalChatzaf said:

I am computing the transverse of the center of a circle of radius r. If it doesn't slip  and it turns through angle theta then it will traverse   r*theta  (theta measured in radians).   Now look at the outer wheel radius R  where R > r.  If the little wheel rigidly affixed to the outer wheel turns an angle theta so does the outer wheel.  But the outer will will bring the common center R * theta to the right  which exceeds  r*theta  hence the inner wheel must have been dragged for a distance of   (R - r)*theta.  Attaching a wire or cable to the inner wheel does not change the geometry.

L.L.A.P \\//

The point is that the inner wheel cannot be dragged, as it is held back by its own cable (that is fixed to the support). It cannot roll further than r*theta,  its cable is taut. From its attachment point on surface 1 then a piece of string with length r*theta lies stretched on surface 1, the rest ist still wound around the wheel. Perhaps it helps to look at the figure:

45381043674_93c81ac422_b.jpg

Therefore it is now the small wheel that determines the movement, the large wheel must "follow", that is, it is held back, slipping, while it rotates together with the small wheel. The large wheel has unrolled R*theta of its own cable, while it has only traveled over a distance of r*theta, there is "too much" unrolled cable from wheel 2, therefore it is slack, and lies there like a dead snake when you roll far enough.

I suppose Jonathan is now making a new picture or animation to make it even clearer...

 

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I have no problem seeing all the physicalities involved and only wonder at the need for any math and highly abstract reasoning. I guess because it all started with such in those ancient days. It's a mind trap.

--Brant

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7 hours ago, Brant Gaede said:

It's even more interesting when they go off the Rand track not knowing it as with open borders. (Not applying that to N and T, but to B and B.)

--Brant 

Who are N, T, B and B? one B is Bins?

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15 hours ago, Brant Gaede said:

I have no problem seeing all the physicalities involved and only wonder at the need for any math and highly abstract reasoning. I guess because it all started with such in those ancient days. It's a mind trap.

--Brant

Right. But a minimum of mechanical mind is required to see the trap. Less than that and no trap can be perceived, only jumbles. Result: non-specific criticisms of perfectly accurate depictions (Merlin) and downright comical long-list criticisms of purely non-essential elements of all the accurate depictions (Tony.)

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