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1 hour ago, BaalChatzaf said:

The small wheel mounted on the same  hub as the big (outer wheel)   slips and drags.. For an angle theta that a radius turns  the center is  moved  2*pi*theta*R  horizontally  where R is the radius of the larger outer wheel.   This  exceeds   2*pi*theta*r   where r is the radius of the smaller innerwheel.   R > r  so the distance that the center goes during a turn of angle theta is greater than the smaller inner wheel would have gone if it did NOT SLIP.  The resolution of the so-called paradox is that the inner wheel slips  by the quantity  2*pi*theta* (R-r).   As the article I quoted states  the appearance of a paradox is based on the  false assertion that the existence of a continuous one to one function between the points on the circumference of the inner and outer wheels in implies the length of the arc on the inner wheel corresponding to a turn  of   d theta (in infinitesimal turn) equal  the length of the corresponding arc on the outer wheel.   Not so.   the length of the outer arc is to the length of the inner arc  as R is to r.   Problem solved.

It is unnecessary to fall into the philosophical   tar pit  of Logical Positivism which denies  an external reality  and asserts all we have  are relations  between data, i.e. perceptions of the outer reality.  This is sometimes  called phenomenalism.   It says  that either there is no outer (external) reality  or all that our minds can ever get are the experience of the outer reality (if it exists).  This is also the premise that Kant used. He said there is an external reality,  but we only get what the mind filters in (of it).   To be truthful, I do not know of a satisfactory resolution of the disconnect between the external or "real" reality  and the  perceived reality that our intellects can deal with.  I do not resolve the paradox  (I am sorry to say). I  AVOID the paradox  by resorting to the "shut up and calculate" tactic in which I get an answer that conforms to what I experience each and every time I make a measurement and calculate an assertion of what I will measure.  This approach is never wrong, but it is totally unsatisfying to those who insist that the :"real" reality is there  and can be experienced or sensed.  Kant  denied this.  Me?   I avoid the disagreement.

I beg you to forgive me for not resolving the question you raised.  The best I can do for you is clarify the problem,  but I still leave you with the problem.  Sorry about that.

Huh? (Hi Bob) There is "no paradox", indeed -- for the reason that there is NO "slip and drag".

One turn of the large circle = one turn of the small circle. Otherwise, a tyre on your car will be slipping on its wheel rim.

The false implication drawn is made clear in your link - by the inner horizontal line tracing the passage of a point on the small wheel.

Differing velocities, would you agree?

Phenomenalism, yup.

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2 hours ago, Max said:

It seems it's no longer a question of physics and mathematics, but of language and the meaning of sentences.

Right.

On the issue of the workings of bicycle wheels and sprockets, we've already explained it to Merlin, but in response to my having brought it up again, and to your jumping in on the issue, Merlin bombards with questions that have already been addressed ad nauseam: "Can you be more discerning? What is this alleged principle anyway? Is it about gear ratios? Can you identify it clearly? How is it relevant to the paradox?"

Merlin doesn't understand the explanations and answers that we've given. He doesn't get it. And one of the tactics that he's using to avoid accepting his mental limitations is playing word games as you noticed above. His mind is attempting to twist reality to conform to his personal inability to grasp it. He is changing the meanings of words and sentences in order to avoid accepting reality.

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21 minutes ago, anthony said:

Huh? (Hi Bob) There is "no paradox", indeed -- for the reason that there is NO "slip and drag".

One turn of the large circle = one turn of the small circle. Otherwise, a tyre on your car will be slipping on its wheel rim.

The false implication drawn is made clear in your link - by the inner horizontal line tracing the passage of a point on the small wheel.

Differing velocities, would you agree?

Phenomenalism, yup.

Wow, Merlin's apparently not alone. We have another one who doesn't get it.

J

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4 hours ago, anthony said:

Huh? (Hi Bob) There is "no paradox", indeed -- for the reason that there is NO "slip and drag".

One turn of the large circle = one turn of the small circle. Otherwise, a tyre on your car will be slipping on its wheel rim.

The false implication drawn is made clear in your link - by the inner horizontal line tracing the passage of a point on the small wheel.

Differing velocities, would you agree?

Phenomenalism, yup.

No slip or skid or drag between the large and small wheel, that’s correct. They are fixed one to the other, just like a rubber tire is fixed to a steel wheel. That’s the setup the paradox prescribes and Bob, Jonathan, Max and myself understand that well. None of us are suggesting slippage between the two wheels. Any claim that one of us did suggest it is pure misinterpretation by someone who understands neither the paradox nor our comments on it.

The Paradox says there is no slip or skid between the large wheel and what?

The Paradox also says there is no slip or skid between the small wheel and what?

If one cannot answer both questions correctly, then one still does not understand the paradox to begin with.

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"The small wheel ... slips and drags". I am going only on Bob's quote.

No slip, relative from one wheel to the other, nor either wheel to the surface. (And distance travelled is dependent on the circumference of the large wheel).

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4 minutes ago, anthony said:

No slip, relative from one wheel to the other, nor either wheel to the surface. (And distance travelled is dependent on the circumference of the large wheel).

Ok, good, you answered both questions, indirectly, by stating “nor either wheel to the surface.”

Almost.

The Paradox says the large wheel experiences no slip with the surface (of the road.)

But the small wheel doesn’t reach the road, so the paradox states no slip of the small wheel relative to what?

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whoops. Tried to link the original  'paradox', unsuccessfully. Anyhow, there's no "slip", any which way.

I reckon this false paradox 'works' by power of suggestion.

It is 'suggested' (by having a small wheel and a mere addition of a line from 1st position to 2nd position, at the bottom of the small wheel - which equals the lower line of the large wheel's travel - that the two wheels must have "the same circumference".

But the travel is ~only~ determined by the large wheel circumference; that contains the small one, which is irrelevant and a red herring.

Equally, and proving the point, if one took the centre point of the wheels, the length of line would be the same (and for any diameter of inner circle etc..)

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28 minutes ago, anthony said:

whoops. Tried to link the original  'paradox'. Anyhow, there's no "slip", any which way.

I reckon this false paradox 'works' by power of suggestion.

It is 'suggested' (by the small wheel and a mere addition of a line from 1st position to 2nd position, at the bottom of the small wheel - which equals the lower line of the large wheel's travel - that the two wheels must have "the same circumference".

But the travel is ~only~ determined by the large wheel circumference; that contains the small one, which is irrelevant and a red herring.

Equally, if one took the centre point of the wheels, the length of line would be the same (and for any diameter of inner circle etc..)

Well, maybe Tony does get it?

J

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1 hour ago, anthony said:

It is 'suggested' (by having a small wheel and a mere addition of a line from 1st position to 2nd position, at the bottom of the small wheel - which equals the lower line of the large wheel's travel - that the two wheels must have "the same circumference".

Not just suggested, the paradox states, asserts, explicitly claims, that the small wheel does not skip or slip during rotation, relative to what?

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47 minutes ago, Jon Letendre said:

Not just suggested, the paradox states, asserts, explicitly claims, that the small wheel does not skip or slip during rotation, relative to what?

Can we eliminate "slip" etc. from debate? The 'paradox' disallows it, there obviously won't be any, and I had no cause to raise it until someone else did.

If there's no slippage, the question - "relative to what?" - then, is superfluous.

The power of "suggestion" was not verbally stated, claimed, and I didn't say so. What I described is more of a "visual suggestion" by way of a ploy with geometry, lines etc.., effective in fooling really smart over-analytical guys, too clever by half.

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Perhaps you should read the thread first, before pontificating and making dumb remarks like a true randroid. There are animations, videos, graphs and mathematical descriptions that all show clearly that the small wheel does slip. Go study those contributions first, before saying "evidence? I don't need no stinking evidence, I know the wheel doesn't slip, because A = A!"

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2 hours ago, anthony said:

Can we eliminate "slip" etc. from debate? The 'paradox' disallows it, there obviously won't be any, and I had no cause to raise it until someone else did.

If there's no slippage, the question - "relative to what?" - then, is superfluous.

The power of "suggestion" was not verbally stated, claimed, and I didn't say so. What I described is more of a "visual suggestion" by way of a ploy with geometry, lines etc.., effective in fooling really smart over-analytical guys, too clever by half.

Here is the question again:

The paradox asserts that the small wheel does not slip during rotation, relative to what?

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4 hours ago, anthony said:

No slip, relative from one wheel to the other, nor either wheel to the surface.

Here you correctly state the paradox.

Now, regarding the small wheel. It doesn’t reach the surface of the road. It therefore has no opportunity, strictly speaking, to roll on the road, with or without slip, for it is always above the road, never in contact at all with the road.

Yet the paradox does say what you say it says -  that the small wheel rolls over its ______________ without slipping.

What is the small wheel in contact with, and rolling over, without slipping?

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9 hours ago, anthony said:

Huh? (Hi Bob) There is "no paradox", indeed -- for the reason that there is NO "slip and drag".

One turn of the large circle = one turn of the small circle. Otherwise, a tyre on your car will be slipping on its wheel rim.

The false implication drawn is made clear in your link - by the inner horizontal line tracing the passage of a point on the small wheel.

Differing velocities, would you agree?

Phenomenalism, yup.

wrong.  The small wheel slips and drags because the center is carried horizontally by the outer wheel.  Do the math and you will see your error.  Or as I sometimes say  "shut up and calculate".

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The answer is its imaginary road. The small (red) wheel is rolling over, without slipping, it’s imaginary (red) road (according to the setup of the paradox.)

Some versions of the paradox state the above explicitly. Others just point to the above drawing and say, “oh look, the lines are the circumferences.” Both are the same paradox statement, both versions require both wheels be rolling without slipping, on their road.

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To clarify for the reader, what’s depicted above is one blue wheel rolling from left to right (or right to left, doesn’t matter.)

There are not two blue wheels, but one. On one side we see it in it’s starting position, on the other in it’s ending position, following one rotation over road, depicted as blue line.

The red circle is permanently fixed, or drawn onto, the blue wheel.

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10 minutes ago, BaalChatzaf said:

The small wheel slips and drags because the center is carried horizontally by the outer wheel.

Slips and drags relative to what?

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With credit to JTS, who made the suggestion to make the small wheel extremely small on the first page, very early in the comments. I think it works very well.

Each wheel, blue, red and green, rotates just once and they go from the left position to the right, in that one rotation.

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Gears are not directly essential to the paradox or thinking through the paradox. It is true however, that anyone who truly grasps what’s going on with gears will not be pulled in by the paradox for more than a few mere moments. Gears can also help a person to “see” what rolling without any slipping means.

Rolling, point-to-point true rolling, without skidding or over-spinning, is what the blue wheel is doing on the blue road. If the blue wheel had 48 teeth then the blue chain has to have 48 links. We could wrap the chain around the blue wheel and satisfy ourselves that indeed the circumference = blue line length.

The green wheel would have only about 4 teeth. So if one tried to rotate the wheel, the green wheel would be asked to traverse 48 links of green chain, but only have opportunity to rotate through 4 teeth. It wouldn’t mesh with its chain and such a setup could never rotate at all. When all of this is uncontroversial to you, you probably really do get it and you see there is no paradox to explain. There is an alleged equality of things that are plainly not equal. That’s it.

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Jonathan’s videos on this thread accurately depict the motions of the wheel and small wheel. I am sure there is much misunderstanding about what his videos depict, so, to clarify, they depict a small wheel as it traverses it’s line, they depict the action happening at the area circled in yellow. His videos accurately depict how the small wheel is skidding, slipping, relative to the road, its imaginary road.

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6 hours ago, Jon Letendre said:

Slips and drags relative to what?

little wheel  slips and drags over its track, the purple line in your diagram.  By the way, the center of both wheels is one and the same point..

From the wiki article on Aristotle's wheel "paradox"  --- "One way to understand the paradox of the wheel is to reject the assumption that the smaller wheel indeed traces out its circumference, without ensuring that it, too, rolls without slipping on a fixed surface. In fact, it is impossible for both wheels to perform such motion. Physically, if two joined concentric wheels with different radii were rolled along parallel lines then at least one would slip; if a system of cogs were used to prevent slippage then the wheels would jam. A modern approximation of such an experiment is often performed by car drivers who park too close to a curb. The car's outer tire rolls without slipping on the road surface while the inner hubcap both rolls and slips across the curb; the slipping is evidenced by a screeching noise"

Here is the entire article:  https://en.wikipedia.org/wiki/Aristotle's_wheel_paradox#Analysis

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Here is a thorough analysis of the Wheel "paradox"  problem.   It addresses itself to the issue of the relation between a thought experiment and a real physical experiment. Have a look:  https://www.humanities.mcmaster.ca/~rarthur/articles/aristotles-wheelfinal.pdf

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Ask yourself what does it mean for a wheel to roll without slipping.  It means  that if a wheel of radius R  turns through an angle A,  the center moves  R*A.  So when A = 2*pi (one turn)  the center moves a distance  equal to the circumference of  the wheel.  (A is measured in radians).

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Everyone is aware that a turning wheel has *rotational* velocity which increases proportionately from the centre (theoretically, zero m/sec) to maximum speed at the outer rim? Right...? Therefore, the rim of the inner wheel will always  move slower - rotationally - than the larger rim (the forward velocity equal for both). So, no hop, skip and jump - drag, slide, whatever - between multiples of wheels of any radius on the same hub - and no gears needed to adjust the relative speeds -

the two wheels conform to each other as if one wheel, however the number of revolutions, in precise alignment.

Besides, this distracts and gets us nowhere. The "implication" of the diagram is that given two equal lines, "representing" two wheel circumferences, then the two circles are equal circumference. Evidently some clever placement of lines suggestibly tricks the mind into this obvious fallacy.

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1 hour ago, anthony said:

Everyone is aware that a turning wheel has *rotational* velocity which increases proportionately from the centre (theoretically, zero m/sec) to maximum speed at the outer rim? Right...? Therefore, the rim of the inner wheel will always  move slower - rotationally - than the larger rim (the forward velocity equal for both). So, no hop, skip and jump - drag, slide, whatever - between multiples of wheels of any radius on the same hub - and no gears needed to adjust the relative speeds -

the two wheels conform to each other as if one wheel, however the number of revolutions, in precise alignment.

Besides, this distracts and gets us nowhere. The "implication" of the diagram is that given two equal lines, "representing" two wheel circumferences, then the two circles are equal circumference. Evidently some clever placement of lines suggestibly tricks the mind into this obvious fallacy.

So, yes, hop, skip and jump -drag and slide, of the small wheel over its “track”, line, or imaginary road, while the large wheel rolls without slipping.

If the small wheel rolls without slipping, then the large wheel spins-out, or over-spins the road.

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