Aristotle's wheel paradox

[merjet]

38 minutes ago, Jonathan said:

You're using the idea of a bet as an excuse to give yourself more time, or to avoid giving a solution completely. Regardless of how long negotiations were to take, you still would have had to produce results by your originally proposed deadline.

So, lets call off the bet. There, now you have no lame excuse to delay. Stop putting hours into writing posts and whining and making excuses, and put you time instead into producing and delivering the math. No more childish maneuvers to evade reality.

Oh, my. That was such a rational response. Not!! It was just another of your childish maneuvers to evade reality. 

You’re wrong again.

Who are you to dictate to me how much time I have?

I predicted your initial acceptance of the bet was nothing more than a bluff. I was right.   

 

[Brant Gaede]

I see two cats playing with a mouse that doesn't know its being played with and won't leave the playroom.

Without mouse Roger Bissell showing up the game's become quite old.

--Brant

mature?

[Brant Gaede]

2 minutes ago, merjet said:

Oh, my. That was such a rational response. Not!! It was just another of your childish maneuvers to evade reality. 

You’re wrong again.

Who are you to dictate to me how much time I have?

I predicted your initial acceptance of the bet was nothing more than a bluff. I was right.   

Never mind the paradox. I'll send you ten dollars if you can explain why a circle is an attribute of a physical wheel--why it has its own physicality.

You can paint a circle on a wheel but you can't put that wheel on a circle.

--Brant

[merjet]

This is a continuation of this post. The graph there is updated as shown below to better show what is depicted. 

Consider what happens as the wheel starts rolling. Relative to the center Pb trails Ps because of the longer path it’s taking. It continues to lag until the wheel rolls half-way although it's catching up with its extra speed. Then Pb starts pulling ahead and Ps lags. Despite moving slower, Ps catches up because it has a shorter path to go.

It seems easiest to describe this in the 1st, 2nd, 3rd, and 4th 90-degree moves.

          Phase            1st                    2nd                3rd                    4th

               Ps       pulls ahead      gets caught      falls behind       catches up

               Pb       falls behind      catches up       pulls ahead       loses lead

You might ask if this is sufficient for Ps and Pb starting at any other clock position. Let’s try 12:00.

         Phase           1st                       2nd               3rd                    4th

             Ps         falls behind       catches up      pulls ahead     gets caught

             Pb         pulls ahead       loses lead       falls behind      catches up

In addition to seeing the same descriptions but in different order, any pair of points on any radius is at 6:00 and 12:00 at some time during a rotation. Thus all points on the circles have been considered.

Are any of these terms a synonym for any common literal meaning of slips? Yes, falls behind, which Ps does once and Pb does once. Thus “the inner wheels slips” is correct 1/4th of the time and wrong 3/4ths of the time. J’s “solution” also completely misses the fact that the outer circle slips as well. Therefore, “the inner wheel slips” is misleading, confused, and wrong!

Are any of these terms a synonym for any common literal meaning of skids? Zero in my view.

So much for Jonathan’s flawed theory!  QED.  Boom!!     

On 9/17/2017 at 7:02 AM, merjet said:

A Jonathan is a variety of apple. So it seems I'm arguing pineapples. A pineapple is a variety of grenade. So maybe we will have an explosion on this thread in the near future.   Oh, my. That might be exciting! 

 

 

[Jon Letendre]

1 hour ago, merjet said:

This is a continuation of this post. The graph there is updated as shown below to better show what is depicted. 

Consider what happens as the wheel starts rolling. Relative to the center Pb trails Ps because of the longer path it’s taking. It continues to lag until the wheel rolls half-way although it's catching up with its extra speed. Then Pb starts pulling ahead and Ps lags. Despite moving slower, Ps catches up because it has a shorter path to go.

It seems easiest to describe this in the 1st, 2nd, 3rd, and 4th 90-degree moves.

          Phase            1st                    2nd                3rd                    4th

               Ps       pulls ahead      gets caught      falls behind       catches up

               Pb       falls behind      catches up       pulls ahead       loses lead

You might ask if this is sufficient for Ps and Pb starting at any other clock position. Let’s try 12:00.

         Phase           1st                       2nd               3rd                    4th

             Ps         falls behind       catches up      pulls ahead     gets caught

             Pb         pulls ahead       loses lead       falls behind      catches up

In addition to seeing the same descriptions but in different order, any pair of points on any radius is at 6:00 and 12:00 at some time during a rotation. Thus all points on the circles have been considered.

Are any of these terms a synonym for any common literal meaning of slips? Yes, falls behind, which Ps does once and Pb does once. Thus “the inner wheels slips” is correct 1/4th of the time and wrong 3/4ths of the time. J’s “solution” also completely misses the fact that the outer circle slips as well. Therefore, “the inner wheel slips” is misleading, confused, and wrong!

Are any of these terms a synonym for any common literal meaning of skids? Zero in my view.

So much for Jonathan’s flawed theory!  QED.  Boom!!     

 

 

Merlin, no one said anything about points drawn on a wheel skidding past other points drawn on a wheel, nor does Aristiotle ask us to envision that.

Aristotle asks us to imagine the "WHEEL" drawn on a real wheel and to imagine it rolling in contact with its imaginary "road."

[Jonathan]

1 hour ago, merjet said:

Are any of these terms a synonym for any common literal meaning of slips? Yes, falls behind, which Ps does once and Pb does once. Thus “the inner wheels slips” is correct 1/4th of the time and wrong 3/4ths of the time. J’s “solution” also completely misses the fact that the outer circle slips as well. Therefore, “the inner wheel slips” is misleading, confused, and wrong!

 

Um, dimwit, you’re conflating the motion of a wheel with the motion of a dot on the wheel. The wheel and the dot are not one and the same. See, the wheel is a larger thing than the dot, and has many other points on which we could choose to place a dot, or many dots, which are moving differently from the dot that you have arbitrarily chosen.

The premise of the alleged “paradox” is not to disregard the circles and pay attention to the motion of only one or two points at a time on the circle, but to pay attention to the motion of the entire circles and their relationships to the lines which they contact. The fact that a dot on one side of a wheel “falls behind” and then “catches up” at given moments does not mean that the circle as a whole does the same.

You are wrong that "the outer circle slips.” The large circle in the “paradox’s” premise is in contact with the lower line, and, as you said in your initial post on this thread, the wheel rolls one revolution. It rolls. It does not slip/skid. It maintains a consistent relationship with the line that it contacts. The smaller wheel slips/skids, but also maintains a consistent relationship with the line that it contacts: It does not fall behind and then "catch up.” It acts just as the animation that I posted illustrates: it rolls with slippage/skidding; it’s like a wheel on an icy surface which is being lightly braked. It is not rolling freely.

You’re still not even properly grasping the premise of the alleged “paradox.” You’re lost and confused, and incapable of holding the entirety of the setup in your mind at one time. You’re by far the most visually/spatially/mechanically inept person that I’ve ever met. You are demonstrating unfathomable depths of ineptitude. Your stupidity on this thread continues to multiply exponentially. It’s stunning.

J

[Jonathan]

1 hour ago, merjet said:

Are any of these terms a synonym for any common literal meaning of skids? Zero in my view.

Skid (automobile)

From Wikipedia, the free encyclopedia

 

 

For other uses, see Skid (disambiguation).

An automobile skid is an automobile handling condition where one or more tires are slipping relative to the road...

 

Slip (vehicle dynamics)

From Wikipedia, the free encyclopedia

 

 

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.(October 2016) (Learn how and when to remove this template message)

In (automotive) vehicle dynamics, slip is the relative motion between a tire and the road surface it is moving on. This slip can be generated either by the tire's rotational speed being greater or less than the free-rolling speed (usually described as percent slip), or by the tire's plane of rotation being at an angle to its direction of motion (referred to as slip angle).^[1]

In rail vehicle dynamics, this overall slip of the wheel relative to the rail is called creepage. It is distinguished from the local sliding velocity of surface particles of wheel and rail, which is called micro-slip.

 

Rolling

From Wikipedia, the free encyclopedia

Rolling is a type of motion that combines rotation (commonly, of an axially symmetric object) and translation of that object with respect to a surface (either one or the other moves), such that, if ideal conditions exist, the two are in contact with each other without sliding.

Rolling where there is no sliding is referred to as pure rolling. By definition, there is no sliding when the instantaneous velocity of the rolling object in all the points in which it contacts the surface is the same as that of the surface; in particular, for a reference plane in which the rolling surface is at rest, the instantaneous velocity of the point of contact of the rolling object is zero.

[Jonathan]

Hey, Jon, did you see the post where numbnuts revealed that he believes that the videos that we presented with our explanations are optical illusions?

4 hours ago, merjet said:

What physics of the solution have you presented here? What substance of your description of the motions of the objects have you presented? You haven’t presented such things. You have given no evidence from, say, a physics textbook to support what you believe you see. You have posted some videos and relied on a vague, confused, and misleading metaphor “slip/skid.” The videos were animation. What you see is not necessarily what is, e.g. the wagon-wheel effect.

Damn, I am so glad that I happened to have some time to come back to OL when I did! Good timing! What a gold mine of schadenfreude.

"...you see a very snooty, very well dressed dowager walking down the street, and then she slips on a banana peel . . . . What’s funny about it? It’s the contrast of the woman’s pretensions to reality. She acted very grand, but reality undercut it with a plain banana peel. That’s the denial of the metaphysical validity or importance of the pretensions of that woman."

Merlin is that dowager, and then some. He's slipping (define "slipping!" -- it's just a misleading metaphor!) on multiple banana peels, bloodying his face, and busting ribs and knees, all the while pointing at the rest of us and laughing, and telling us that we're the ones slipping on banana peels as we stand on solid ground and watch him flail and founder.

J

[Jon Letendre]

30 minutes ago, Jonathan said:

Hey, Jon, did you see the post where numbnuts revealed that he believes that the videos that we presented with our explanations are optical illusions?

Damn, I am so glad that I happened to have some time to come back to OL when I did! Good timing! What a gold mine of schadenfreude.

"...you see a very snooty, very well dressed dowager walking down the street, and then she slips on a banana peel . . . . What’s funny about it? It’s the contrast of the woman’s pretensions to reality. She acted very grand, but reality undercut it with a plain banana peel. That’s the denial of the metaphysical validity or importance of the pretensions of that woman."

Merlin is that dowager, and then some. He's slipping (define "slipping!" -- it's just a misleading metaphor!) on multiple banana peels, bloodying his face, and busting ribs and knees, all the while pointing at the rest of us and laughing, and telling us that we're the ones slipping on banana peels as we stand on solid ground all watch him flail and founder.

J

Yes.

[merjet]

3 hours ago, Brant Gaede said:

I see two cats playing with a mouse that doesn't know its being played with and won't leave the playroom.

Heh. You mistook a ratel for a mouse.  

[merjet]

21 hours ago, Jonathan said:

Um, dimwit, you’re conflating [snipped garbage]

You are the epitome of a dimwitted, stupid, ignoramus when it comes to math and physics. Bye. You and your buddy are both now on my ignore list. I am not going to waste any more time responding to your nonsense.

Oh, here's some "big math" for the obnoxious ignoramus.

http://mathworld.wolfram.com/Cycloid.html
http://mathworld.wolfram.com/CurtateCycloid.html

I'm sure he hasn't the foggiest notion of what it means. 

Regarding the Wikipedia stuff, I'll bet he hasn't the foggiest notion of how it applies and doesn't apply to the problem at hand. He copies some stuff from a Wikipedia page he poorly understands, then barks and howls like he has a PhD in physics or mechanical engineering.  

 

 

[Jonathan]

And still no big math from Merlin, or any other type of solution to the "paradox." Lot's of crying, and some distracted explanations which address something other than the premise of the "paradox," but still no solutions.

J

[BaalChatzaf]

1 hour ago, Jonathan said:

And still no big math from Merlin, or any other type of solution to the "paradox." Lot's of crying, and some distracted explanations which address something other than the premise of the "paradox," but still no solutions.

J

What paradox?  There is no paradox.  It is clear that the big wheel moves the hub a greater distance than the little wheel.

 

[Jonathan]

Correct. There is no paradox, thus the scare quotes around the word "paradox."

[Brant Gaede]

2 hours ago, merjet said:

Heh. You mistook a ratel for a mouse.  

Good comeback.

You finally scored one (in my eyes).

--Brant

your opinion obviously varies

[Jonathan]

2 hours ago, merjet said:

Oh, here's some "big math" for the obnoxious ignoramus.

http://mathworld.wolfram.com/Cycloid.html
http://mathworld.wolfram.com/CurtateCycloid.html

I'm sure he hasn't the foggiest notion of what it means. 

 

Retard, the point is not just to post some "big math," but to post math which represents your alleged solution to this really simple physics issue which you think is a difficult "paradox." Understand?

[Ellen Stuttle]

Me (here): "how would you guarantee that you did the math on your own?"

10 hours ago, merjet said:

How is that relevant?

How would you be providing your solution to what you seem to think is a very difficult problem if you just looked up and copied a solution?  (Why anyone would bet a cent that you couldn't provide a solution which you could just look up and copy, I don't know.)

Ellen

[merjet]

49 minutes ago, Ellen Stuttle said:

Me (here): "how would you guarantee that you did the math on your own?"

How would you be providing your solution to what you seem to think is a very difficult problem if you just looked up and copied a solution?  (Why anyone would bet a cent that you couldn't provide a solution which you could just look up and copy, I don't know.)

Please let me know when you discover where I copied it. I'd like to know, too. 

[Ellen Stuttle]

2 hours ago, merjet said:

Please let me know when you discover where I copied it. I'd like to know, too. 

So you have a reading problem too?

Ellen

[Brant Gaede]

9 hours ago, Jonathan said:

Retard, the point is not just to post some "big math," but to post math which represents your alleged solution to this really simple physics issue which you think is a difficult "paradox." Understand?

You missed the point. He is saying goodbye to you by giving you the finger. You and Jon are on his "ignore" list. Mostly he's left the room. I wonder for how long.

OL is a tough place for certain people wedded to agendas, especially the one of ego preservation.

--Brant

feeding frenzy in this case

[Jonathan]

8 hours ago, Brant Gaede said:

You missed the point. He is saying goodbye to you by giving you the finger. You and Jon are on his "ignore" list. Mostly he's left the room. I wonder for how long.

 

No, I got it.

He was lashing out by targeting what he hopes is a sore spot for me, which is my suckiness at math. The thing is, I freely admit that I suck at math, and that it takes a hell of a lot of effort for me to grasp math that others understand with ease. I understand and accept my deficiency. I don't have a psychological need to tell seasoned math professionals that their math which is way over my head is wrong. I don't identify with that type of insecurity, nor do I have any sympathy for it. 

Quote

 

OL is a tough place for certain people wedded to agendas, especially the one of ego preservation.

 

 I like carrying on the old Atlantis traditions. If someone's looking to play rough, he'll be accommodated. I'll be his huckleberry.

J

[merjet]

13 hours ago, Ellen Stuttle said:

So you have a reading problem too?

Ellen

"[It] is the recipient who communicates. The so-called communicator, the one who emits the communication, does not communicate. He utters. Unless there is someone who hears, there is no communication. There is only noise. The communicator speaks or writes or sings – but he does not communicate. Indeed, he cannot communicate. He can only make it possible, or impossible, for a recipient – or rather, "percipient" – to perceive." - Peter F. Drucker

[BaalChatzaf]

32 minutes ago, merjet said:

"[It] is the recipient who communicates. The so-called communicator, the one who emits the communication, does not communicate. He utters. Unless there is someone who hears, there is no communication. There is only noise. The communicator speaks or writes or sings – but he does not communicate. Indeed, he cannot communicate. He can only make it possible, or impossible, for a recipient – or rather, "percipient" – to perceive." - Peter F. Drucker

It takes two to tango....

 

[merjet]

Resolving the Paradox

Observing this video helped me to conclude that the paradox comes from confounding magnitudes of three different paths -- a straight one, a circular one, and a curved one. The curved one is a cycloid. Let Ps denote the point/dash at the  6:00 o'clock position on the circumference of the smaller circle with the disc at rest in the video. Let Pb denote the point/dash at the 6:00 o'clock position on the circumference of the bigger circle with the disc at rest. Roll the conjoined circles one revolution like in the video. How far either point moves is measurable in three ways. The straight path is the simplest one – as the crow flies -- but it is one that Ps or Pb does not travel. Its length is the same for Ps and Pb (all points really). The circular path's length is obviously the circumference and ignores the fact that the circle is moving. The curved paths do not ignore the circles moving. The curved paths show that Pb travels farther than Ps, the distance for Pb being 8 times its circle's radius (more than the circumference). Also, clearly Pb travels farther away from its horizontal line than Ps does (midway for both).

What I call 'straight', 'circular', and 'curved' correspond to translation, rotation, and rolling in this excellent video that Baal posted earlier.

The circular paths are not part of the rolling experiment and make a "red herring." Let Rs denote the radius of the smaller circle and Rb the radius of the bigger circle. The lengths of their paths follow, along with whether or not they are actually traveled. I show a range for Ps's curved path because the formula is very complicated. The closer Ps is to Pb, the closer the length is to 8*Rb. The closer Ps is to the center of the circles, the closer it is to Rb*2*pi.

       Point                      Straight                       Circular                                     Curved

         Ps                   No, Rb*2*pi                      No, Rs*2*pi                Yes, min= Rb*2*pi,  max= 8*Rb

         Pb                   No, Rb*2*pi                      No, Rb*2*pi                Yes, 8*Rb

       Center              Yes, Rb*2*pi                         n/a                                  n/a

The paradox, or oddity, is that the straight lengths are equal but the circular lengths are not, despite Pb's straight and circular lengths being equal. The different curved lengths, resulting from two different shaped paths, which many people don't consider, are the ones actually traveled. Many people know that circumference C = 2*pi*r, but few people know about the path lengths or even think about it. Considering them resolves the paradox.

If a vehicle wheel rolls many revolutions, common sense says that an inner circle travels the same distance as the outer circle. It's also true – based on the straight path. The same holds for a given point or arc on the inner circle based on the straight path, but it doesn't hold for the curved path.

There are many common objects that fit the wheel's basic shape, for example, a roll of duct tape. Saying the smaller circle formed by the duct tape's hole slips/slides/skids relative to the tape's largest circumference is bizarre to me.

 

 

[Brant Gaede]

On 9/22/2017 at 9:12 AM, merjet said:

"[It] is the recipient who communicates. The so-called communicator, the one who emits the communication, does not communicate. He utters. Unless there is someone who hears, there is no communication. There is only noise. The communicator speaks or writes or sings – but he does not communicate. Indeed, he cannot communicate. He can only make it possible, or impossible, for a recipient – or rather, "percipient" – to perceive." - Peter F. Drucker

Things no one has to know.

--Brant

and Drucker is full of it