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.

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.

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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.

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. 

1 hour ago, anthony said:

Do we all agree this far? I will be as unambiguous as I can: A wheel inside a wheel travels a distance in excess of its own circumference when the large wheel turns its own circumference. (And they both rotate, once, together doing a "true roll" and no slippage. 

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

1 hour ago, anthony said:

A circle in a circle, ditto.

This is a fact about wheels and circles. Immediately, at this stage, the 'paradox' can be dispensed with. It is accurate to reality, non-contradictory.

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

1 hour ago, anthony said:

The large circumference is the *only* determinant of lateral distance. (Very odd, any other way...)

----

Then a 'second track' gets brought in, and the fun starts. This track - apparently - makes the wheel in a wheel behave differently, according to some.

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.

29 minutes ago, Ellen Stuttle said:

Some yes, some no.  Long subject, which I don't have time for.

The definition of "paradox" I used in the post you're referring to - "an apparent contradiction between two true premises" - is the definition which I thought Jon gave in a post somewhere up the thread, but I couldn't find where on searching.  Maybe Jon was quoting a dictionary source.  The Search function (very irritatingly to me) does not pick up material enclosed in a quote box.  Or maybe I misremembered the definition Jon gave.  Either way, I think it's a good definition, and that it doesn't cover the "Aristotle's Wheel" problem.

I'm curious:  Have you or Jon objected to Jonathan's saying that the problem isn't a genuine paradox?  (The post of his I was agreeing with isn't the first time he's said that.)

Ellen

I can't remember, but you stated your position again so explicitly, that I was wondering why it should be so important. As you'll have seen, there are many different definitions of a paradox, and also many different kinds of paradoxes. No problem for me, I don't believe so much in the "one and only" correct definition à la Rand (e.g. her definition of altruism). My viewpoint is, that so many of those classic "paradoxes" are known as "paradoxes", that I see no reason not to use that term for that kind of "paradoxes", genuine or not.

For myself I use the definition: an apparent contradiction in an argument caused by a more or less hidden error in the argument or in the premises. In general it isn't difficult to move the error from a false premise to an error in the argument, and an error in the argument can always be thought of as the result of an implicit false premise, there is no sharp distinction between the two options. Changing the formulation a bit can change the formal expression of a paradox, without really changing its essence. Therefore I think my definition isn't that much different from yours, only less restricting, while I also admit false premises. But as I said, I find definitions not that important (the only correct one!) as long as you state them clearly.

9 hours ago, Darrell Hougen said:

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.

But that doesn't solve Aristotle's paradox, it's avoiding it by changing the conditions. That there is no paradox when two concentric wheels can rotate independently from each other, Aristotle no doubt could have also figured out, but that wouldn't have helped him solving his paradox. The two wheels forming one rigid body can easily be realized in a physical system, no contradiction there. The contradiction emerges when you suppose that both wheels can roll without slipping/both circles can trace out their circumference at the same time. That is the essence of the paradox.

10 hours ago, anthony said:

First, go see a wheel and tyre in motion. Observe and establish that an 'internal' wheel does, indeed, "traverse a distance greater than its own circumference". Always.

Why should I? Did I deny anywhere that the internal wheel traverses a distance greater than its own circumference? That has always been part of my argument.

10 hours ago, anthony said:

I.e. it traverses the distance of the outer tyre's circumference, which exceeds its own. Without slipping. 

It can only traverse the distance of the outer tyre's circumference if it slips. Otherwise it is impossible.

10 hours ago, anthony said:

There's this "objectivist buzzword" called "reality" - what it is (and does). HOW it happens is something further. I suggested the differing tangential speeds as the cause.

No vague suggestions, show the calculations, just as I've done. I've also shown that it is just the differing tangential speeds that explain the slipping. Reality is not what you think it is, but what it is.

1 minute ago, anthony said:

That is quite disingenuous. Of course - the small wheel rolls its own circumference. It, too, revolves - once. BUT, the ~distance travelled~ is greater than its own circumference, and you know what I meant, in my brief way of stating that.  So, you fail: "that is by definition slipping"... 

No, it is by definition "rolling".

Wrong, it is rolling and slipping.

1 minute ago, anthony said:

Repeat: it does not "get past its own circumference". It ~traverses a distance~ greater than its own circumference. Geez. Poor attempt. 

And that is the definition of slipping. If it wouldn't slip, it could not traverse a distance greater than its own circumference. Show me how the wheel could traverse a distance greater than its own circumference without slipping, and without resorting to pure magic or meaningless objectivist buzzwords.

25 minutes ago, anthony said:

The entire wheel rolls forwards, on two tracks, where there was before just one surface. Assuming the weight on each track is carefully and evenly distributed, the outcome will be what the diagram denotes: the large wheel (tyre) rolls its circumference; the inner wheel (rim) rolls one revolution--but far past its circumference. It does not 'slip', it doesn't need to.

Really? Rolling one revolution, far past its circumference: that is by definition slipping, it is definitely not rolling without slipping, because then it couldn't get past its own circumference.

2 hours ago, anthony said:

haha. A problem with the young guys is to not identify before they calculate. Floating math abstractions.

Peikoff parrot. He also started to talk about "floating abstractions" when things went way over his head.

30 minutes ago, Jonathan said:

I hadn't planned on it. Perhaps I will. I don't know. Diagrams, animations and videos of real things seem to have no effect on those who aren't cognitively suited to visuals. Showing such people reality seems to be a waste of time. "Look, this is what happens in reality." "No, that's a trick. Anyone can do optical tricks. I'm right cuz I know I am."

I think you're right. We have now posted so many diagrams, animations, videos and mathematical derivations, that anyone who is seriously committed to finding the solution to this puzzle has material enough for studying the problem, to either accept the slippage solution, or to come up with a valid counterargument. showing  what would be wrong in our examples and derivations and to give an alternative solution, and not a "solution" that isn't. 

30 minutes ago, Jonathan said:

The only thing that might succeed is showing certain people the math of the cables' forces. I wouldn't know where to begin with that. Besides, even that would come down to their first recognizing and accepting what the cables are doing, and how the smaller wheel's cable affects the movement of the entire rig.

The people who don't get the original problem, won't get this version either. Bob had directly the solution of the original version. I think he's still more or less thinking of that one, not yet seeing that the new puzzle is different (though a variation on the old theme). So I wouldn't bother too much about this version, he'll get it sooner or later.

30 minutes ago, Jonathan said:

I think that those who don't get it pretty quickly on their own are very unlikely to ever be convinced by others' explanations, diagrams, math, or direct reality.

Indeed, this is where the men are separated from the boys.

9 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.

What highly abstract reasoning? It's all rather simple and the math is also quite elementary. I think that a big problem for those old guys was that they didn't know calculus, they didn't for example have a notion of the concept "instantaneous speed" (Zeno's arrow problem!), while that is now a piece of cake for us. So they were puzzling about wheels jumping over gaps in their supports, trying to make sense of it all.

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:

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...

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? 

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.

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? 

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.

1 hour ago, BaalChatzaf said:

If the outer wheel does not slip, the inner wheel does.

It's the other way around. The shortest cable determines the movement, as it can't be lengthened, but the large cable can be loosened. The small wheel rolls without slipping, generating a proper cycloid. The large wheel is slipping backwards, loosening its cable, generating part of a prolate cycloid.

1 hour ago, anthony said:

Right. I see it. Good effort. One helluva investment for so little return. 

(I do not think Aristotle was looking for solutions to the phenomenon, it puzzled him, that's all).

We don't know for sure that Aristotle wrote that text at all, perhaps it was Archytas of Tarentum, as has been suggested. But whoever it was, we can't know whether he was looking for solutions or not. I think he was (it would be rather unnatural for such a person not to try to solve the puzzle), but that he couldn't find the solution. After all, after him people like Galileo, Mersenne, Fermat and Boyle also tried to solve the puzzle. 

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.

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! 

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.

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

Quote

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.

7 minutes ago, Jonathan said:

Look again. I've bolded it for you:

 

In antiquity, the wheel problem was described in the Aristotelian Mechanica, as well as in the Mechanica of Hero of Alexandria.^[1] In the former it appears as "Problem 24", where the description of the wheel is given as follows.

For let there be a larger circle ΔZΓ a smaller EHB, and A at the centre of both; let ZI be the line which the greater unrolls on its own, and HK that which the smaller unrolls on its own, equal to ZΛ. When I move the smaller circle, I move the same centre, that is A; let the larger be attached to it. When AB becomes perpendicular to HK, at the same time AΓ becomes perpendicular to ZΛ, so that it will always have completed an equal distance, namely HK for the circumference HB, and ZΛ for ZΓ. If the quarter unrolls an equal distance, it is clear that the whole circle will unroll an equal distance to the whole circle, so that when the line BH comes to K, the circumference ZΓ will be ZΛ, and the whole circle will be unrolled. In the same way, when I move the large circle, fitting the small one to it, their centre being the same, AB will be perpendicular and at right angles simultaneously with AΓ, the latter to ZI, the former to HΘ. So that, when the one will have completed a line equal to HΘ, and the other to ZI, and ZA becomes again perpendicular to ZΛ, and HA to HK, so that they will be as in the beginning at Θ and I.^[2]

The problem is then stated:

Now since there is no stopping of the greater for the smaller so that it [the greater] remains for an interval of time at the same point, and since the smaller does not leap over any point, it is strange that the greater traverses a path equal to that of the smaller, and again that the smaller traverses a path equal to that of the larger. Furthermore, it is remarkable that, though in each case there is only one movement, the center that is moved in one case rolls a great distance and in the other a smaller distance.^[1]

-----

J

You beat me to it...

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.

6 hours ago, anthony said:

"This is trivial". Untrivial, I'd think,, when I have had to argue the self-evident with somebody - that the "tangential" (thanks for the reminder)speeds are dissimilar for different points in a circle. This could be no more than a sidebar to the main 'paradox', but not unimportant.

But the tangential speed for different points on a circle is the same. It is the speed in the rest frame of the circle, no translation. Of course it is different for points on circles with different radius, perhaps that's what you mean.

6 hours ago, anthony said:

(If it is not recognized, for only one example, that in a race around a circular track, runners have to start in "staggered" lane positions, one needs to restate the obvious). 

Your mathematics do not convince. They strike me as reverse justification. For I have seen, or can envisage, an archery target rotate-roll, and like anybody, many more kinds of round objects, and I have "more faith" in observation than seeing facts forced to conform to math i.e. to prove "slippage".

The original paradox was stated in terms of rollig circles. Any problem with circles is a geometrical/mathematical problem, so I don't see why a mathematical treatment of the paradox would not be the ideal method to solve it. Those circles are tracing out their circumference, a corresponing physical object would be a wheel, rolling without slipping it is the equivalent of a circle tracing out its circumference. Any objections so far? Now a wheel is a very good object in this case, as wheels are meant to roll without slipping, and a wheel concentric in a wheel (just as a circle within a circle in the original description) is easily realised (flange, hub), so a practical test of Aristotle's paradox is fairly easy to realize.

Now about your archery target: perhaps it isn't difficult to rotate it, but that isn't yet rolling. For that you'd have to accurately cut out the target at the outer circle and roll it over the ground or some other support. But then you still haven't one of those smaller circles rolling. Rotating, yes. But rolling needs a support and that circle has to be raised from the rest of the target to allow contact with that support. Now I seriously doubt that you've done that. Probably you just imagined that doing, but that is not good, reality-based evidence! Especially as you apparently already have great difficulty in observing the slipping in the animations and videos that we've seen here, and where many people clearly see the slippage. Some objects are just much better to visualize some effect than other object. The iris and pupil of the eye for also two concentric circles, but they are not well suited for a demonstration of Aristotles paradox. How would you roll an iris and a pupil? Yes in you imagination, but then you'd better concentrate on the mathematical solution. Experiment and mathematical analysis show definitively that slippage occurs on the smaller wheel, if the large wheel rolls without slipping. It that is not basing it on observing reality...

6 hours ago, anthony said:

I think this is the crux of the matter I ask of everyone: To comply with the "wheel within a wheel" - if one superimposed on any ring in the archery target, a fixed, protruding, (inner)wheel, would there be any change? 

How could there be? A target may be revolved like a circle or wheel, and one can *see* all its components rotate correspondingly. No slip. Why should "circles" and "wheels" act at all differently? What is there so remarkable about a 'track' and 'slippage'  which we don't need with circles -- but have to have to 'correct' wheels? Why do many here separate the practice from the theory?

But you do need them with circles, those are the equivalents of the tangents that form an essential part of the original problem. You may take them away, but then you take the problem also away. Child and bathwater!