Aristotle's wheel paradox

[Jonathan]

11 hours ago, anthony said:

You mean this gear train "cannot roll at all"? I've seen gears fixed solidly onto larger gears, and run on racks.

Double reduction gear[edit]

 

Double reduction gears

A double reduction gear comprises two pairs of gears, as single reductions, in series.^[3] In the diagram, the red and blue gears give the first stage of reduction and the orange and green gears give the second stage of reduction. The total reduction is the product of the first stage of reduction and the second stage of reduction.

It is essential to have two coupled gears, of different sizes, on the intermediatelayshaft. If three gears were used, the overall ratio would be simply that between the first and final gears, the intermediate gear would only act as anidler gear: it would reverse the direction of rotation, but not change the ratio

Right over his head.

This is what we're dealing with. He read something about gears, and thinks that it applies to the discussion, and even refutes the argument that he's trying to address. He thinks that he's teaching us, giving us new knowledge.

J

[Jules Troy]

Should have called this the “How to beat a dead horse to death “ Paradox.

[Brant Gaede]

50 minutes ago, merjet said:

LOL. A jeer from the peanut gallery. Your double standard is clear. You are upset that I vastly improved Wikipedia. But you have no complaints about an obnoxious ignoranus, lying, contradictory, reality-faking jackass named Jonathan. And apparently you were also duped by Jonathan and believe that gratuitously adding a second surface isn’t “beyond the pale.”

I bet you couldn’t found any errors in my solutions!

Water is wet?

--Brant

[Jonathan]

45 minutes ago, merjet said:

...

There's still the unanswered question, gramps. You've been asked -- challenged -- to state what is paradoxical when you remove the line under the small wheel. You've evaded.

For example, on Friday, I asked:

"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?"

Gramps, you responded with the lie that you've answered it many times, to which I replied:

"Where? When? Cite the post in which you identified what is paradoxical about a wheel rolling over a surface while an inner wheel attached to it is not rolling on its own surface."

Still no answer.

J

[Jonathan]

10 minutes ago, Jules Troy said:

Should have called this the “How to beat a dead horse to death “ Paradox.

We have a long history of enjoying discovering how thick people's skulls are, how stubborn and how willing to self-immolate.

J

[Jonathan]

56 minutes ago, merjet said:

It is telling that the people who claim slippage apparently feel compelled to gratuitously add elements to its original formulation. Why? With their crutch, they have a “solution.” Without their crutch, they are helpless.

 

We've already covered this. We've all reviewed the original description. You're wrong, Merlin, and you're lying and cheating, and dragging Wikipedia into your fragile ego driven obsession.

its both comical and sad what you're doing to yourself

J

[Brant Gaede]

It's interesting how Tony is head and shoulders above Merlin on this subject. I think Merlin knows and Tony doesn't.

--Brant

[anthony]

17 hours ago, Darrell Hougen said:

Hi Tony,

You are correct that the glass will roll off to one side. But, there is nothing special about a level surface. If the surface were tilted, what do you think would happen? Imagine that the surface is tilted so that the ends of the cup are straight up and down. Won't the glass continue to curve in the same direction as long as it doesn't slip?

You can create the same effect by just putting the small end of the cup on a book of the appropriate height. Try it. You don't need any fancy scientific equipment.

Darrell

Hi Darrell, believe me, I put in some time playing around with books and bottles and canisters!

What I assume we are doing here is re-creating a wheel+wheel, experimentally,  by means of other objects. I understand that a cone also has a greater and smaller diameter. However, it is unsuitable for experiment in that it has an inherent bias. You try to force it to roll it straight - but the only way is by inducing slippage. The built-in "bias" biases the effects. So to say.

So the cylinder is the closest to an "extruded" wheel, which rolls straight and true. One cylinder will roll on a surface after it's pushed, in a straight line. Connect another cylinder, of lesser diameter, and repeat - the same outcome. (No slip)

Now place the small cylinder on a 'ledge' of sorts so that both cylinders are supported on surfaces.  "All things being equal" - all the factors I've mentioned have to be precisely right -  there is no reason whatsoever why the cylinder combination(e.g. a wine bottle) will not roll as it did without a second platform--straight and slip-less. The 'rule of tangential velocity' equally applies here, to both 'wheels'.

All it is is an accurate reproduction of what we all know a rolling bottle does when without a 'track'. Bottle plus neck rolls - one rotation - with no slippage. Add a track, and albeit some friction/drag which has to be equalised on both wheels, one can reproduce the same scenario. 

By that standard of reality, IF one finds slippage, IF the bottle rolls skew, we know the setup of the experiment is imprecise.

 

[anthony]

6 hours ago, Max said:

Good example, but I'm sure Tony doesn't understand what's happening here.

Question is: do you understand? 

Two gears, of different diameters, are being forced into two chainlinks. You see the problem yet?

Does it occur to anyone that the teeth of the two gears are unequally spaced, because they have differing circumferences, while the links in the chain are the same constant? 

-->jamming and/or slippage

[Max]

5 minutes ago, anthony said:

Does it occur to anyone that the teeth of the two gears are unequally spaced

Are they?

[anthony]

12 hours ago, Darrell Hougen said:

 

I think Max was thinking of something more like this:

But with two concentric circular gears and two linear gears or "pinions."

Darrell

 

Better still. And why wouldn't an inner, concentric cog or gear revolve on its own track or pinion? "Slippage"?

I have been around some weird and wonderful machinery in factories. There are much more complex set-ups than TWO combined gears on TWO pinions.

As long as the teeth in each match and align.

[anthony]

8 minutes ago, Max said:

Are they?

Evidently, or else they would both mesh. 

[anthony]

It has to be explained further. An identical teeth-frequency, spacing and size, on two gears of unequal circumference, will not match. You just need to look and see.

Where are all those spatio-visual proclaimed experts when you need them?

[Jon Letendre]

57 minutes ago, anthony said:

Evidently, or else they would both mesh. 

Evidently not. The wheels have the same tooth spacing and the chains are also dimensionally the same.

This video proves it...

 

[Jon Letendre]

This time the large wheel rolls without slip

 

[anthony]

Not getting this, still. The "same tooth-spacing" on two different circumferences, is non-identical. 'The curves of these two gears are dissimilar, being smaller and larger 'wheels'. And of course, one or other can engage with 'no slip'. Not both at once.

Where are the mechanical proclaimed "experts", also?

[Jonathan]

4 minutes ago, anthony said:

Not getting this, still. The "same tooth-spacing" on two different circumferences, is non-identical. 'The curves of these two gears are dissimilar, being smaller and larger 'wheels'. And of course, one or other can engage with 'no slip'. Not both at once.

Where are the mechanical proclaimed "experts", also?

We're right here, laughing at an ignoramus.

J

[Jonathan]

Much has been made, by the moron twins, of the second line -- the line at the base of the small wheel which they say is not there and isn't a part of the formulation of the "paradox."

Purely for the sake of argument, let's momentarily accept the position that that line that Aristotle (or whomeverthefuck) referred to as "HK that which the smaller unrolls" does not exist. Okay? So line HK is not there.

And then we hear someone say that he thinks it's a freaking paradox that the small circle travels the length of the large circle to which it is attached. He tells us that he expected the smaller circle to only travel the length of its own circumference. Naturally, we look at him in disbelief. Dumbfounded. Tardfounded. But then we perhaps feel some pity and generosity, and decide to try to help him grasp simple reality.

So, it occurs to us to add line HK (which is actually already there, but, remember that, for the sake of argument, we're accepting the morons' assertion that it is not there), and we explain that using line HK, and placing marks on it as well as the small wheel which rolls on it, is a means of measuring, tracking and identifying what is happening, and showing that the presumptions that the morons had made do not match reality.

Good so far? Okay. So, the question is, how is adding line HK "cheating," or a "crutch" or "scam"? By what standard are certain means of solving such "paradoxes" out of bounds or unacceptable? Is there any logic to it? And I'm not asking the moron twins. I'm asking the others here. Do any of you have an inkling of where they're coming from? The mindset or psychology behind it?

J

[Jon Letendre]

42 minutes ago, Jonathan said:

Much has been made, by the moron twins, of the second line -- the line at the base of the small wheel which they say is not there and isn't a part of the formulation of the "paradox."

Purely for the sake of argument, let's momentarily accept the position that that line that Aristotle (or whomeverthefuck) referred to as "HK that which the smaller unrolls" does not exist. Okay? So line HK is not there.

And then we hear someone say that he thinks it's a freaking paradox that the small circle travels the length of the large circle to which it is attached. He tells us that he expected the smaller circle to only travel the length of its own circumference. Naturally, we look at him in disbelief. Dumbfounded. Tardfounded. But then we perhaps feel some pity and generosity, and decide to try to help him grasp simple reality.

So, it occurs to us to add line HK (which is actually already there, but, remember that, for the sake of argument, we're accepting the morons' assertion that it is not there), and we explain that using line HK, and placing marks on it as well as the small wheel which rolls on it, is a means of measuring, tracking and identifying what is happening, and showing that the presumptions that the morons had made do not match reality.

Good so far? Okay. So, the question is, how is adding line HK "cheating," or a "crutch" or "scam"? By what standard are certain means of solving such "paradoxes" out of bounds or unacceptable? Is there any logic to it? And I'm not asking the moron twins. I'm asking the others here. Do any of you have an inkling of where they're coming from? The mindset or psychology behind it?

J

They are Roark, Galt, geniuses. We riffraff would live in caves if it wasn’t for them.

[Darrell Hougen]

6 hours ago, merjet said:

LOL. A jeer from the peanut gallery. Your double standard is clear. You are upset that I vastly improved Wikipedia. But you have no complaints about an obnoxious ignoranus, lying, contradictory, reality-faking jackass named Jonathan. And apparently you were also duped by Jonathan and believe that gratuitously adding a second surface isn’t “beyond the pale.”

I bet you couldn’t find any errors in my solutions!

Hi Merlin,

I agree that Jonathan is sometimes an obnoxious jackass, but the fact of the matter is that his analysis of Aristotle's paradox is correct. He has also been very patient at times, going out of his way to produce illustrative videos. We all owe him a debt of gratitude for that. And, so far as I know, he hasn't taken this dispute outside of OL. You really should put the Wikipedia page back the way it was or let us do it.

Since you laid down the gauntlet, I'll take a look at your solutions later, when I get the chance.

Cheers,

Darrell

[Jonathan]

15 minutes ago, Jon Letendre said:

They are Roark, Galt, geniuses. We riffraff would live in caves if it wasn’t for them.

Heh, imagine a building or a generator built with Tony or Merlin's understanding of mechanics!

J

[Jon Letendre]

10 minutes ago, Jonathan said:

Heh, imagine a building or a generator built with Tony or Merlin's understanding of mechanics!

J

 

[Jonathan]

It wouldn't even be that good.

J

[BaalChatzaf]

Behold!

 

[Ellen Stuttle]

Slippage / Spin-out

 

I thought of a simple way to explicate both "slippage" and "spinning-out" without making reference to tracks.

Maybe this will help Tony.

(The following would be easy to see though it's a bit cumbersome to describe verbally.  If someone with graphics abilities would like to draw the figures at reduced scale, I'd be grateful.)

---

Imagine a circle with a circumference of exactly 1 meter.

Now imagine within that circle a smaller concentric circle with a circumference of exactly 50 centimeters.

Thus the larger circle's circumference is exactly twice that of the smaller circle.

Now make a mark at the 6 o'clock position of the larger circle.

Then make marks at 5-centimeter intervals going counterclockwise around the  larger circle's circumference.

Do the same procedure with the smaller circle.

You now have 20 evenly-spaced marks around the circumference of the outer circle and 10 equally evenly-spaced marks around the circumference of the smaller circle.

Now imagine a radius drawn from the center of the figure through the 6 o'clock position of the smaller circle and terminating at the 6 o'clock position of the outer circle.

Next imagine a radius drawn from the center of the figure through the 1st-to-the-right 5-centimeter mark of the smaller circle and terminating at the outer circle's circumference

That radius will intersect the outer circle at its 1st-to-the-right 10-centimeter mark.

Now revolve the figure around the outer circle's circumference until the second radius is lined up with the 6 o'clock positions.

The outer circle will have revolved through 10 centimeters of its circumference.

The inner circle will have revolved through only 5 centimeters of its circumference.

However, the inner circle's 6 o'clock position will have moved laterally a distance of 10 centimeters.

The difference between the distance around the smaller circle's circumference and the lateral distance traversed is what's being called "slippage."

-

Then, instead of revolving around the outer circle's circumference, revolve around the inner circle's circumference.

When the second radius is lined up with the 6 o'clock positions, the inner circle will have revolved through 5 centimeters of its circumference.

The outer circle will have revolved through 10 centimeters of its circumference but will have traveled laterally only a distance of 5 centimeters.

The difference between the distance around the larger circle's circumference and the lateral distance traversed is what's being called "spinning-out."

Ellen