Aristotle's wheel paradox

[Brant Gaede]

So I'm right: the "paradox" is mental, not physical.

--Brant

[merjet]

16 minutes ago, BaalChatzaf said:

In a physical instantiation of that scheme the inner wheel slip.  It has the same center as the outer wheel and is carried forward on outer circumference per revolution of the outer wheel.  The inner wheel is rigidly affixed to the outer wheel.  since it has a smaller radius its circumference is less than the circumference of the outer wheel so it slip on its rail by a distance equal to the difference of the circumferences. 

If the inner wheel is rigidly affixed to the outer wheel, how does it slip? In this video how does the inner circle "slip"?   

[Jonathan]

Jesus H. Christ.

[Jonathan]

Maybe it will help if we divide up the circles and add some colors to both it and the lines. In  this video, I've also made the small circle even smaller in order to make what's happening even more visible. So as to address Merlin's irrelevant objections, I've constructed it so that there are two black circles on the same disc.

Watch it a few times if you have to. Can you see how the large black circle properly "rides" the bottom line, but the small black circle skids along the top line?
 

And it might also help if we focus on a closeup of just one of the colored sections.

Look at the orange space as being like a slice of pie. If we focus for a moment on the large black circle, and look at it as the crust of a slice of pie, and we compare it to the orange ruler section below it, then we can see that the length of the orange slice’s crust and the orange section of the ruler appear to be the same.

In contrast, if we focus on the small black circle, and look at it as the crust of a slice of pie, and we compare it to the orange ruler section below it, then we can see that the length of the small orange slice’s crust is much smaller than the orange section of the ruler.

That length on the ruler is the distance that small orange slice of pie’s crust travels. In other words, the crust slips along the line.

Now, with this in mind, go back and watch the video a few more times.

Get it now? Understand?

No? Still not grasping it?

How about thinking about a bike wheel and sprocket.

 

See, the way that a bike works is that a chain (a line of a certain length) engages the small diameter gear at the center of the wheel, and then when the wheel moves, it covers much more distance on the ground (a line just like in the "paradox" scenario!) than the length of the chain.

This is, like, kindergarten physics. It's not that hard. No need to egghead it out.

J

[BaalChatzaf]

4 hours ago, merjet said:

If the inner wheel is rigidly affixed to the outer wheel, how does it slip? In this video how does the inner circle "slip"?   

The outer wheel move farther right when it rolls without slipping than the inner wheel could. Conclusion -- the inner wheel is dragged along.

[Brant Gaede]

11 hours ago, merjet said:

If the inner wheel is rigidly affixed to the outer wheel, how does it slip? In this video how does the inner circle "slip"?   

How can it not? If it doesn't nothing is going anywhere. (The video only shows one wheel, btw. Your first question has no video. And a "circle" is not a wheel.)

Jonathan has got you.

--Brant

[BaalChatzaf]

27 minutes ago, Brant Gaede said:

How can it not? If it doesn't nothing is going anywhere. (The video only shows one wheel, btw. Your first question has no video. And a "circle" is not a wheel.)

Jonathan has got you.

--Brant

The motion of the inner circle or inner while is a combination of translation and rolling. 

[BaalChatzaf]

See https://www.quora.com/What-is-a-satisfying-explanation-to-Aristotles-wheel-paradox

[merjet]

7 hours ago, Brant Gaede said:

Jonathan has got you.

Heh. You are entitled to your opinion, no matter how shaky it is. What exactly did he say that allegedly "got me" in your opinion?

[merjet]

23 hours ago, BaalChatzaf said:

The motion of the inner circle or inner while wheel is a combination of translation and rolling. 

I'll accept that. What I don't accept is that the smaller wheel/circle slips or skids.

[merjet]

10 hours ago, BaalChatzaf said:

See https://www.quora.com/What-is-a-satisfying-explanation-to-Aristotles-wheel-paradox

It doesn't satisfy me. In the math section he says "pure rotation," and his math is based on that.  "Pure rotation occurs when a body rotates about a fixed non-moving axis" (link).  Aristotle's wheel has a moving axis. In other words, there is translational motion and rotation. The author's math doesn't hold for the smaller wheel/circle, so his alleged proof is flawed. Moreover, Section 20.2 -- Constrained Motion: Translation and Rotation -- here shows mathematics for a rolling wheel. Note that it is way more complicated than the math in the alleged proof. It also uses Cartesian coordinates for the translation motion and polar coordinates for the motion about the center of the circle. The alleged proof uses only polar coordinate math!

 

[Jonathan]

10 hours ago, Brant Gaede said:

Jonathan has got you.

--Brant

I "got" Merlin on this issue of the "paradox," but I also "got" him on the issue of visual/spatial/mechanical reasoning. He took my comments to be "personal insults" when they were actually just descriptions of reality. In fact, I think it should be clear now, after  all of the evidence that has been presented and he still doesn't get it, that his visual/spatial/mechanical reasoning abilities are worse than I had initially suspected.

Merlin is a very bright boy, generally speaking. He's much brighter than I am in many areas. My interest here is that he apparently shares a specific deficiency with some great minds of the past. They also couldn't see what he can't, and it is fascinating to observe the creative yet erroneous methods in which they spun their wheels while being stuck due to their specific deficiency.

 

[merjet]

If one holds that the smaller wheel or circle slips/skids, then it logically follows that there is some place on the circumference that slips/skids. Any point on the circumference traces a curve, a cycloid, as described and graphically depicted here. Pick a point on the smaller circle in this video (posted earlier), e.g. the marked one at the 6:00 o'clock position when the wheel is at the starting position on the left side. Then watch it move as the wheel rolls. Does it ever slip/skid similar to a spot on a tire that slips/skids while in contact with the road it's rolling on? Maybe with Jonathan's distorting glasses and deficient visual/spatial/mechanical reasoning it does, but not according to my eyes. Watch it some more but imagine that the (top) wire is a little lower, creating a gap between the circle and the wire. Even call the wire a "rail", Jonathan's term. So I must ask: how does a point on the circle slip/skid on the "rail" when it is not even in contact with it?    Duh!

[Brant Gaede]

4 hours ago, merjet said:

I'll accept that. What I don't accept is that the smaller wheel/circle slides or slips or skids.

It doesn't--if nothing moves. You are simply conflating without surcease a circle and a wheel. They are as different as can be. The concept "wheel" is one step from physicality while "circle" is further away and more abstractive. I've never before run into anyone who thinks of a wheel as a circle. My car doesn't run on circles. It runs on wheels.

--Brant

[Brant Gaede]

24 minutes ago, merjet said:

If one holds that the smaller wheel or circle slips/slides/skids, then it logically follows that there is some place on the circumference that slips/slides/skids. Any point on the circumference traces a curve, a cycloid, as described and graphically depicted here. Pick a point on the smaller circle in this video (posted earlier), e.g. the marked one at the 6:00 o'clock position when the wheel is at the starting position on the left side. Then watch it move as the wheel rolls. Does it ever slip/slide/skid similar to a spot on a tire that slips/slides/skids while in contact with the road it's rolling on? Maybe with Jonathan's distorting glasses and deficient visual/spatial/mechanical reasoning it does, but not according to my eyes. Watch it some more but imagine that the (top) wire is a little lower, creating a gap between the circle and the wire. Even call the wire a "rail", Jonathan's term. So I must ask: how does a point on the circle slip/ slide/skid on the "rail" when it is not even in contact with it?    Duh!

Epistemology never trumps metaphysics, but here you think it does.

--Brant

[Brant Gaede]

4 hours ago, merjet said:

Heh. You are entitled to your opinion, no matter how shaky it is. What exactly did he say that allegedly "got me" in your opinion?

It's what you are saying. He's basically repeating it with different words.

--Brant

[Jon Letendre]

I love this stuff.

I have not read any responses, but I did see someone wrote "it's a mental paradox, not a physical one."

The setup description is a lie, red and blue lines are not same length,  the red lines should be dashed to represent that they cannot be continuous like the blue lines.

The larger wheel pulls the smaller one along, so that the smaller skips across its "ground."

If the red line is string, it will snap.

[Jon Letendre]

Ok, I read the threat. Well done Jonathan, Brant, Bob.

Merlin. Oh, Merlin.

[merjet]

1 hour ago, Jon Letendre said:

The setup description is a lie, red and blue lines are not same length,  the red lines should be dashed to represent that they cannot be continuous like the blue lines.

The larger wheel pulls the smaller one along, so that the smaller skips across its "ground."

If the red line is string, it will snap.

I assume you are talking about the Wikipedia graphic in the first post. Another model of Aristotle's wheel is this video (posted earlier). Do you have the same comments about it?

[Jon Letendre]

Hi Merlin,

I watched the video.

The smaller wheel is sliding along its "ground", not adhering to it.

If we forced the smaller wheel to adhere to its ground, them the bigger wheel would spin-out, go faster than the ground underneath it.

[Jonathan]

Here are some more videos which, hopefully, will help Merlin, and anyone else like him, see what’s going on in the “paradox.”

This first video is just a straight-on view of the scenario in motion:

 

Now, here’s exactly the same thing, but with the area of the small circle isolated. The same disc and the large circle are still there, but they’ve been colored black so as not to be a visual distraction:

 

Here is a closeup of the small circle passing the midpoint:

 

Here is the same closeup of the small circle passing the midpoint, but this time it’s isolated (the disc and large circle are still there, but now they are painted black so as not to be a visual distraction):

 

Can you see the “skidding” now?

J
 

[Jon Letendre]

I snapped a still from your video, Merlin.

At beginning, the screw at center of wheels and the 6o'clock marks and the white start marks are all lined up.

[Jon Letendre]

Shortly into rotation, another still...

Can you see the disparity between the two black dots?

The big dot looks like if you let it fall down, it would reach its start at its white dot.

But the small wheel's black dot will never reach, because it is not making honest, adhered travel, but rather sliding along.

[BaalChatzaf]

7 hours ago, merjet said:

It doesn't satisfy me. In the math section he says "pure rotation," and his math is based on that.  "Pure rotation occurs when a body rotates about a fixed non-moving axis" (link).  Aristotle's wheel has a moving axis. In other words, there is translational motion and rotation. The author's math doesn't hold for the smaller wheel/circle, so his alleged proof is flawed. Moreover, Section 20.2 -- Constrained Motion: Translation and Rotation -- here shows mathematics for a rolling wheel. Note that it is way more complicated than the math in the alleged proof. It also uses Cartesian coordinates for the translation motion and polar coordinates for the motion about the center of the circle. The alleged proof uses only polar coordinate math!

 

There is rotation about a fixed center, there is translation, purely straight line motion and there is rolling which is instantaneous   infinitesimal rotation around a point of tangent contact.

[BaalChatzaf]

3 hours ago, Brant Gaede said:

It doesn't--if nothing moves. You are simply conflating without surcease a circle and a wheel. They are as different as can be. The concept "wheel" is one step from physicality while "circle" is further away and more abstractive. I've never before run into anyone who thinks of a wheel as a circle. My car doesn't run on circles. It runs on wheels.

--Brant

Euclidean geometric concepts are static.  No motion....  Circles don't roll and points don't move.  The injection of dynamics into geometry came much later than Euclid (and therefore much later than Aristotle).  Dynamics was put into math around the time of Galileo, and Galileo was one of the pioneers of dynamics.  You can date  mathematical dynamic  to about 500 years ago.  Give or take.  Many besides Galileo came up with dynamical rules of thumb but Galileo published,  so he is credited.