Aristotle's wheel paradox

[Peter]

Wheels schmiels. How about that rotating hurricane? And clouds. What do you see in clouds? I remember a kid who could get you to see just about anything in a fluffy white cloud if you lay down on your back and stared at it long enough.

[Jonathan]

4 hours ago, merjet said:

You call it skid. I call it "goes along for the ride." The common reference of skid is loss of traction, which doesn't apply to an inner circle.

Wrong. It DOES apply to an inner circle since the premise of the “paradox” is to make that inner circle contact its own line! The inner circle loses traction compared to the line which it contacts in Aristotle’s setup of the “paradox."

Quote

It also involves friction, and an inner circle doesn't resist the horizontal movement.

False. The inner circle DOES resist the horizontal movement. The smaller the inner circle, the greater the friction between it AND THE LINE ALONG WHICH IT TRAVELS!

Let’s review now, because you, Merlin, have great difficulty keeping the entire setup in your mind at one time. You keep forgetting about the fact that the setup includes TWO lines — one on which the larger circle is rolling, and one on which the smaller circle is said to be rolling, but on which it is actually slipping/skidding. Got that? Okay, the smaller circle loses traction on the line that it contacts as described in the "paradox's" setup.

Quote

Physics makes a distinction between rolling without slipping and rolling with slipping.

Yes, physics does indeed make a distinction between rolling without slipping and rolling with slipping. Rolling with slipping is known as a type of "skidding." The smaller circle rolls with slipping. In other words, it skids. The smaller the circle on the wheel, the greater the friction and skidding. I’ve presented videos which visually show the skidding. You, being a stubborn moron, refused to consider them, and brushed them aside as optical illusions!

J

[Jonathan]

4 hours ago, merjet said:

That's a humongous IF which changes the context to where it is no longer Aristotle's wheel or a rolling roll of duct tape. 

The sad thing is that Merlin is not embarrassed at all about how stupid he is showing himself to be. After everything that has been said and demonstrated on this thread, reality is still not sinking in, and it hasn't even occurred to him yet to explore the possibility that he should have some self-doubt, and that he should listen to others' arguments a little more carefully. Nope. It's just full throttle idiocy.

[Peter]

Jonathan wrote, "Nope. It's just full throttle idiocy." Deep thoughts but this thread is starting to sound like robo calls from Omaha Steaks. If you listen to their speeches suddenly a live person is on the line to complete the sale.  

[merjet]

2 hours ago, BaalChatzaf said:

Any translation of the center (hub) that is not accompanied by an instantaneous roll round the point of tangency  at the rim   is by definition a skid.  Of course, we are assuming the rigidity of the wheel/circle.  At no point does the wheel/circle become deformed.

Suppose the point on the rim you refer to is at 6:00 o'clock before the roll begins. The wheel rotates 90 degrees clockwise to put the point at 9:00. Where is the point's tangent line?

[BaalChatzaf]

45 minutes ago, merjet said:

Suppose the point on the rim you refer to is at 6:00 o'clock before the roll begins. The wheel rotates 90 degrees clockwise to put the point at 9:00. Where is the point's tangent line?

each point on the rim is instantaneously a tangent point for an infinitesimal interval of time.  The only way to deal with this stuff in a mathematically rigorous manner is to use parametric co-ordinates with time as the driving parameter.  So for each you have a hub position  and a point of tangency on the rim.

[Jon Letendre]

5 hours ago, BaalChatzaf said:

Any translation of the center (hub) that is not accompanied by an instantaneous roll round the point of tangency  at the rim   is by definition a skid.  Of course, we are assuming the rigidity of the wheel/circle.  At no point does the wheel/circle become deformed.

Yes, agreed.

[Jon Letendre]

3 hours ago, merjet said:

Suppose the point on the rim you refer to is at 6:00 o'clock before the roll begins. The wheel rotates 90 degrees clockwise to put the point at 9:00. Where is the point's tangent line?

It doesn't have one.

Only the point, the one and only point, the one on the outside circumference of a rolling wheel, the one point currently in contact with the road, has a tangent line. And the tangent line is the road.

A moment later, that point's neighbor is in contact with the road and now this neighboring point is the point of tangency, meaning the point where wheel touches road.

[Brant Gaede]

12 hours ago, Jonathan said:

You’re referring to curved paths that don’t exist in reality, at least not yet. You’re only imagining these curved paths. You have yet to demonstrate any tracings of their actual paths. You’re just making assertions without backing them up.

And your imagination of physical entities and the motions isn't reliable. You're rather inept. A good example is when you stupidly asserted that the wheel in the first video that you posted wasn’t riding on it’s actual edge, but was riding on a smaller invisible wheel which was behind the main wheel and which was riding on a surface that you misperceived as being a ledge. You haven’t shown that you’ve actually traced the paths, but instead you’ve just made empty assertions, where others here, like Jon and I, have actually visually demonstrated our positions.

So, dicknibbler, it’s time for you to prove your assertions. Demonstrate that you can actually trace the paths of specific points. Actually physically trace the paths in the video that you initially posted, as well as the paths in the videos that I posted, and the ones that Jon posted.

--Brant

[Brant Gaede]

12 hours ago, Jonathan said:

Correct. I would not have lost the bet. I would have won it. The bet included the condition that the douchelord would have to present a "big math" solution which would back up his claims that our explanations of what was happening with the motion of the circles were wrong. In fact, we have accurately described the physical reality of the motions and relationships. Merlin is still not getting it!

--Brant

[Brant Gaede]

11 hours ago, Jonathan said:

Wrong. It DOES apply to an inner circle since the premise of the “paradox” is to make that inner circle contact its own line! The inner circle loses traction compared to the line which it contacts in Aristotle’s setup of the “paradox."

False. The inner circle DOES resist the horizontal movement. The smaller the inner circle, the greater the friction between it AND THE LINE ALONG WHICH IT TRAVELS!

Let’s review now, because you, Merlin, have great difficulty keeping the entire setup in your mind at one time. You keep forgetting about the fact that the setup includes TWO lines — one on which the larger circle is rolling, and one on which the smaller circle is said to be rolling, but on which it is actually slipping/skidding. Got that? Okay, the smaller circle loses traction on the line that it contacts as described in the "paradox's" setup.

Yes, physics does indeed make a distinction between rolling without slipping and rolling with slipping. Rolling with slipping is known as a type of "skidding." The smaller circle rolls with slipping. In other words, it skids. The smaller the circle on the wheel, the greater the friction and skidding. I’ve presented videos which visually show the skidding. You, being a stubborn moron, refused to consider them, and brushed them aside as optical illusions!

J

--Brant

[Brant Gaede]

11 hours ago, Jonathan said:

The sad thing is that Merlin is not embarrassed at all about how stupid he is showing himself to be. After everything that has been said and demonstrated on this thread, reality is still not sinking in, and it hasn't even occurred to him yet to explore the possibility that he should have some self-doubt, and that he should listen to others' arguments a little more carefully. Nope. It's just full throttle idiocy.

--Brant

[Ellen Stuttle]

Me (here): "There isn't a real paradox, since it isn't the case that a point on an inner circle's circumference both is and isn't traveling a distance equal to that circle's circumference."

On September 24, 2017 at 9:53 AM, merjet said:

You say it as if a paradox requires there be a real contradiction. That mischaracterizes a paradox. “A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion” (Wikipedia, my bold).

My understanding of "paradox" is as you quote from Wikipedia, but note "apparently sound reasoning from true premises."  The initial supposed paradox attributed to Aristotle includes as part of the setup the false premise that the inner circle is rolling in true contact with its imagined road.  It couldn't be doing that.

 

On September 24, 2017 at 9:53 AM, merjet said:

Pages 3-9 of Mathematical Fallacies and Paradoxes is about the paradox we are addressing. Said pages can be seen on Google Books. The author's "wheel" consists of an axle, dime, and half-dollar, such that the axle passes through the centers of both the dime and half-dollar glued together. On page 9 he presents his solution to the paradox as follows. By the way, this is the source cited on Wikipedia to justify "slipping."

"[The] "point" tracing the line segment at the foot of the dime is both rotating and slipping all the time. It is slipping with respect to the tabletop. Since the dime does not touch the tabletop, you do not notice the slipping."

That describes the "slipping" relative to the "real road," not an "imaginary road" tangent to the dime. So pardon my pun, but it doesn't make a dime's worth of difference whether the "slipping" is relative to the horizontal tangent of the larger circle or the horizontal tangent of the smaller circle. Moreover, he says that it is not noticeable, as opposed to you and others here that it is clearly visible.

My contention from day 1 has been that "slipping" or "skidding" is a metaphor, and a very poor one as a substitute for translation or horizontal motion (ref.1, ref. 2).

There's "slipping" in regard to both lines, or "roads" or roads (which are equal in length).  I agree that the "slipping" of the inner circle, or wheel, is metaphoric, if we're talking about an imagined circle or about a physical object where the inner wheel makes no physical contact with an actual track, but not that it isn't visibly obvious that the inner circle, or wheel, in the diagrams or demos which have been offered is moving farther than it would move in a true roll.  If the inner circle or wheel were made close to the same circumference as the outer one, seeing that it's being translationally dragged in the rolling movement would be more difficult.

Ellen

[Ellen Stuttle]

17 hours ago, merjet said:

[....]

But I will say regarding the main stipulation of the bet I offered [Jonathan] - a bet he was too chicken to take -- has now been settled in my favor. In other words, if he hadn't been such a chicken, he would have lost. It's obvious to a rational being.

Transparent argument from intimidation.

As you proposed the terms - see - you wanted to make Bob the judge.  But Bob has been saying from the start that there isn't a paradox, so he'd have been unlikely to judge that you'd provided a resolution to a paradox which he was saying doesn't exist.

Jonathan wanted to change the terms of the bet - see - in a way such that you couldn't possibly have won, since he wanted to include the requirement that you refute his physics demos.

Ellen

[merjet]

13 hours ago, Jon Letendre said:

It doesn't have one.

Only the point, the one and only point, the one on the outside circumference of a rolling wheel, the one point currently in contact with the road, has a tangent line. And the tangent line is the road.

A moment later, that point's neighbor is in contact with the road and now this neighboring point is the point of tangency, meaning the point where wheel touches road.

There being only one tangent line is a figment of your imagination. It takes less than 12 seconds of this video to prove it. There is an unlimited number of tangent lines to the cycloid. The center of the circle in the video is what Baal called the hub. What Baal refers to as "skid" – a metaphor -- is how the slope of the tangent lines changes along different points on the curve . "Shift" describes it better than "skid." He used "skid" in a much different way than you did. The two usages are about as much alike as day and night. So his using it does not support your notion of "skid" one iota. Your comment and alleging Baal agrees with you doesn't even begin to pass muster.

I'd bet the measurement implications of all this is not even on your radar screen. 

[merjet]

2 hours ago, Ellen Stuttle said:

The initial supposed paradox attributed to Aristotle includes as part of the setup the false premise that the inner circle is rolling in true contact with its imagined road.  It couldn't be doing that.

Where did you get that notion? It's not on the Wikipedia page for Aristotle's wheel paradox following "The problem is then stated."

[merjet]

4 hours ago, Ellen Stuttle said:

Jonathan wanted to change the terms of the bet - see - in a way such that you couldn't possibly have won, since he wanted to include the requirement that you refute his physics demos.

The obnoxious numbskull also "bet" that I would never present a solution to the "non-paradox." That "bet" has been proven dead wrong.

[merjet]

5 hours ago, Brant Gaede said:

--Brant

 

4 hours ago, Brant Gaede said:

--Brant

 

4 hours ago, Brant Gaede said:

--Brant

 

4 hours ago, Brant Gaede said:

--Brant

Brant shows how ignorant he is 4 times in only a few minutes.

[BaalChatzaf]

15 hours ago, Jon Letendre said:

It doesn't have one.

Only the point, the one and only point, the one on the outside circumference of a rolling wheel, the one point currently in contact with the road, has a tangent line. And the tangent line is the road.

A moment later, that point's neighbor is in contact with the road and now this neighboring point is the point of tangency, meaning the point where wheel touches road.

That is an instantaneous point of tangency.  Now you see why Newton and Leibniz had to invent calculus  to deal with motion.  Poor old Aristotle did not have a chance and even Archimedes who was smart enough to do it,  did not invent a form a calculus for dealing with motion.  Archimedes developed a theory for static balancing forces  but he never invented dynamics.  That came much, much later.  It is "problems"  like the  rolling wheel and falling bodies that  indicate just how essential differential calculus and differential equations are for the development of physical science.   

It all comes down to grasping the instantaneous, infinitesimal  details  of sustained motion.  Motion as a unity over time  and grasped instantaneously.  Quite a trick!

[BaalChatzaf]

18 hours ago, Peter said:

Wheels schmiels. How about that rotating hurricane? And clouds. What do you see in clouds? I remember a kid who could get you to see just about anything in a fluffy white cloud if you lay down on your back and stared at it long enough.

The cyclonic storm is a marvelous beautiful instance  of the so-called "Coriolis Force"  which a manifestation of a mechanical references system in accelerated motion.  The "force"  which is manifested by the curved paths of the eye of the cyclones  is not a "true force". So called centrifugal forces is also an artifact of a reference from in motion,  the motion of rotating around a fixed point.  It is not the result of two bodies interacting dynamically. Newton capture one of the great conservation principles in his third law.  If body A pushes against body B (this is an "action") with a force F  then B  push back against A  with force -F.   Contact forces come in pairs, reaction pairs.   This "law" of motion is equivalent to the conservation of linear momentum.  The path of cyclonic storms are instances  of conservation of  angular momentum.  Again,  even the great genius Archimedes did not have a handle on this.  Galileo was among the first  to  grasp  inertia and momentum   He plowed the road for Newton. 

[Brant Gaede]

3 hours ago, merjet said:

Brant shows how ignorant he is 4 times in only a few minutes.

All I want from this thread is your explanation of wheel-circle equivalence.

--Brant

nada, zip, nothing

[Jonathan]

9 hours ago, Ellen Stuttle said:

Jonathan wanted to change the terms of the bet - see - in a way such that you couldn't possibly have won, since he wanted to include the requirement that you refute his physics demos.

 

No, I wanted to clarify the terms of the bet within the full context of the discussion. The context was that Merlin was claiming that my description and animations were "wrong." Any proposed solution that he would offer would therefore have to, by definition, refute my position.

[Jon Letendre]

10 hours ago, Ellen Stuttle said:

The initial supposed paradox attributed to Aristotle includes as part of the setup the false premise that the inner circle is rolling in true contact with its imagined road.  It couldn't be doing that.

Yes, thank you, that is correct, Ellen. Resolving the paradox is as simple as grasping those two sentences. Amazing as it seems, Merlin does not yet grasp them.

[Jon Letendre]

5 hours ago, BaalChatzaf said:

That is an instantaneous point of tangency.  Now you see why Newton and Leibniz had to invent calculus  to deal with motion.  Poor old Aristotle did not have a chance and even Archimedes who was smart enough to do it,  did not invent a form a calculus for dealing with motion.  Archimedes developed a theory for static balancing forces  but he never invented dynamics.  That came much, much later.  It is "problems"  like the  rolling wheel and falling bodies that  indicate just how essential differential calculus and differential equations are for the development of physical science.   

It all comes down to grasping the instantaneous, infinitesimal  details  of sustained motion.  Motion as a unity over time  and grasped instantaneously.  Quite a trick!

Yes.

[Jon Letendre]

8 hours ago, merjet said:

There being only one tangent line is a figment of your imagination. It takes less than 12 seconds of this video to prove it. There is an unlimited number of tangent lines to the cycloid. The center of the circle in the video is what Baal called the hub. What Baal refers to as "skid" – a metaphor -- is how the slope of the tangent lines changes along different points on the curve . "Shift" describes it better than "skid." He used "skid" in a much different way than you did. The two usages are about as much alike as day and night. So his using it does not support your notion of "skid" one iota. Your comment and alleging Baal agrees with you doesn't even begin to pass muster.

I'd bet the measurement implications of all this is not even on your radar screen. 

A figment of my imagination. You feel compelled to talk about me. You could skip that sentence, your geometric point is made by the following sentences. But you can't keep personality out of it, because you don't seem to have the discipline. One of your many character flaws. Shall we discuss them at length? Or geometry? I'll have your answer soon, I suppose.

Baal is on YOUR side, not mine, you say. I'm glad you have something important to you going. Maybe you're right, but I don't care who else does or does not see truth ,I'll just keep aiming for truth and ignore personalities. You do whatever seems good for you.

You sure do like making bets.