Aristotle's wheel paradox

[Jonathan]

5 hours ago, merjet said:

“All circles have the same circumference!” Oh, my! What an authoritative site you and J appeal to.  

Hmmm. So, you're not receptive to things like irony, parody or sarcasm, huh?

Um, see, what's going on with that quote is that he is mocking the view that a wheel can unroll freely without slipping and end up after one rotation traveling a length other than its circumference. Get it now? No?

Man, you're sure revealing a lot of your deficiencies on this thread.

J

 

[Max]

7 hours ago, merjet said:

Mr. Wessen didn’t have a better explanation, as given in my previous post.

Why isn’t 'forced along by the rolling of the wheel' enough of an explanation?

Because it doesn't address the problem of the paradox, namely how the points of the "forced" wheel are mapped onto the longer distance on its track. For the "forcing" wheel it's rather obvious: circumference and traveled distance are equal, so it's easy to construct an 1-1 map, while for the "forced" wheel the circumference and traveled distance are unequal. What the old guys didn't know, is that it is also possible to map a smaller segment 1-1 onto a larger segment, but if you do that, you'll have to take into account that the smaller wheel is slipping (supposing that the large wheel is the "forcing" wheel).

7 hours ago, merjet said:

He used an incorrect title, Aristotle’s Wheels. He also says, “All circles have the same circumference!” Oh, my! What an authoritative site you and J appeal to.  

Well, that was obviously irony, the apparent result of Aristotle's paradox. Does that invalidate the article? And if Aristotle writes about two circles, the mechanical implementation would be two wheels, so nothing wrong with that either.

[Brant Gaede]

19 hours ago, Ellen Stuttle said:

Too late to tell Newton he goofed.  The planets, not having been informed that they can't orbit, are still orbiting.

Ellen

The planets--absent atmosphere--are not in flight  As far as I know, flight is for birds and insects and airships only.

--Brant

[Brant Gaede]

On 12/13/2018 at 3:09 AM, anthony said:

 An  object in flight is either level, rising or falling.       

This is a correct statement.

--Brant

[Ellen Stuttle]

1 hour ago, Brant Gaede said:

The planets--absent atmosphere--are not in flight  As far as I know, flight is for birds and insects and airships only.

--Brant

 

Tony referenced Max's talking about vectors.

 

On December 13, 2018 at 5:09 AM, anthony said:

[bold added]

Arguments are being made by analogy, metaphor, simile. That's unsatisfying. All good and fine as instruments toward understanding and communication, but they can't substitute for reality, nor reasoning. As with Max's comparative allusion to vectors and your sliding car. An object in flight is either level, rising or falling. A car tyre is either biting or sliding. At any instant in time. An entity cannot do all/both "at the same time and in the same respect". (That both have forward velocity has to be taken as the given).

 

Max was talking about something thrown - a ball - not something self-propelled. 

 

On December 12, 2018 at 8:22 AM, Max said:

There is no contradiction. It is perfectly possible to add independent motions like rolling and slipping to a combined motion, that is a characteristic of vector quantities. If you throw a ball to someone, its motion is a combination of a forward motion (in the x-direction) and an upward motion (in the z-direction). (In that case those vectors change continuously due to effects from gravity and air resistance). Rolling itself is a combination of rotation and translation, such that the contact point with the support has speed zero. Add an extra translation movement and you get a combination of rolling and slipping.

 

Tony's saying "an object" and "falling" sounds like he meant something passive despite the word "flight," which could mean something self-propelled.

(And even with something self-propelled, there are competing forces.)

Ellen

[Brant Gaede]

You quoted too little, Ellen. I quoted it back. Flight is flight and I'm a pilot.

--Brant

[Ellen Stuttle]

13 minutes ago, Brant Gaede said:

You quoted too little, Ellen. I quoted it back. Flight is flight and I'm a pilot.

--Brant

And a context ignorer.  See the post above.

Any bets that Tony thinks that a thrown ball is "either level, rising, or falling"?

Ellen

[Brant Gaede]

1 minute ago, Ellen Stuttle said:

And a context ignorer.  See the post above.

Any bets that Tony thinks that a thrown ball is "either level, rising, or falling"?

Ellen

You created the new context. Not I. Your put down was thus gratuitous.

--Brant

[Brant Gaede]

 

5 minutes ago, Brant Gaede said:

deleted  

 

[Brant Gaede]

This is a slice and dice thread. Not fair to poke and choke.

--Brant

[Brant Gaede]

On 12/13/2018 at 5:08 AM, anthony said:

Gets to the point I have to blow my own horn, if only to put some of these condescending presumptions down. I had a good education in physics, enough to be employed at 21 by one of the (recognized) leading nuclear research institutes in the world at the time with some of the best minds in their field, in the Physical Metallurgy dept. Yup, in South Africa!  Albeit, without a Degree, as a lab assistant, for almost 2 years I made up the samples and equipment which were inserted into the reactor, experiments I assisted. I was invited by my Dept. heads to continue higher education and declined. While not well up on Physics any longer,  I have some clue about the scientific method and simple mechanics. Enough trumpeting.

But you have no understanding of or true engagement here of your critics? Please pardon the rhetorical, but the boat is pulling you, not you the boat.

--Brant

[Darrell Hougen]

On 12/13/2018 at 3:11 PM, anthony said:

Very nice! Lucid illustrations we could have used earlier.

I've questions about veering "toward the small end because the small end doesn't go as far as the big end".

1. As I see it, R and r are now equivalent. [R] You have raised the small end to compensate for the diameters, so aren't they equal?

2. The neck's lesser rotating speed Vt compensates for its smaller diameter. If the speed was constant, - i.e. higher - yes, we'd have veering in that direction.

3. The contact made at all points (for large and small diameters between a plane and a circle) is theoretically the same.

So, will it veer?

Hi Tony,

No, raising the small end doesn't compensate for the difference in the diameters. It doesn't make them equal. The big end still has diameter = 2R and the small end still has diameter = 2r.

Think about it --- you're still calling them the "big end" and the "small end". If you can't discern the difference between the two ends, why do you have two different names for them? The fact is that the two ends are different. That fact is both perceptually obvious and logically required. Simply raising up one end doesn't change its size.

Now, what happens when the bottle rolls? Every time the big end rotates by one complete rotation the small end does too. I think you agree with that. In the case I pictured, the bottle will probably fall off the rails before it rotates one complete time, but the same logic applies to part of a rotation.

If the big end rotates by a tenth of a rotation, so does the small end. Now, without slipping, that means that in one tenth of a rotation, the big end will travel 2 * pi * R / 10 inches and the small end will travel 2 * pi * r / 10 inches. And, if R > r, then 2 * pi * R / 10 >  2 * pi * r / 10. It has to be that way. That is what logic demands.

The tangential speed of the small end during rotation is less than the tangential speed of the big end. That is true. But, the fact that the tangential speed of the small end is less than that of the big end means that the small end is traveling more slowly than the big end. That's because the speed of the center of each end relative to the rail on which it rides is equal to the tangential speed of a point on the circumference relative to the center of that end. If that is hard to understand, don't worry about it right now. Concentrate on what I said above. If the big end is bigger than the small end, then it must travel farther in the same amount of time if neither end slips.

So, yes, the bottle will veer off to the side.

Darrell

 

 

 

[Darrell Hougen]

On 12/2/2018 at 5:07 PM, Jonathan said:

False. The inner wheel will roll. It will not slip at all. It is not physically possible for it to slip. Its cable prevents any slippage. The inner wheel will roll without slippage until it reaches the end of its cable, which is one rotation.

False. The two wheels are affixed to one another. When one completes a full rotation, the other does as well. However, since the large wheel must over-spin in comparison to its surface (any point on its perimeter will create a prolate cycloid during the wheels' motion), it will have travelled a distance shorter that its circumference, and its cable will have let off slack (the length of the slack will be equal to the length of the large wheel's circumference minus the length of the small wheel's circumference).

Bob, you're not properly envisioning the scenario, especially the effects of the cables. I would suggest building a model and observing how the reality of it differs from your mistaken imagining of what happens. A couple of spools of thread with different diameters, glued together and a nail for an axle would work.

J

Hi Jonathan,

It's funny how adding the cables actually made the problem more difficult for some people to understand. I would have thought it would have made things simpler.

Darrell

[Jon Letendre]

13 minutes ago, Darrell Hougen said:

Hi Jonathan,

It's funny how adding the cables actually made the problem more difficult for some people to understand. I would have thought it would have made things simpler.

Darrell

I think we will all have to accept that our discussants cannot process what we are saying, cannot cognitively manage all the parts, conditions and motions at once and that there is no teaching innovation that will change that.

[Darrell Hougen]

On 12/13/2018 at 10:09 AM, anthony said:

Arguments are being made by analogy, metaphor, simile. That's unsatisfying. All good and fine as instruments toward understanding and communication, but they can't substitute for reality, nor reasoning. As with Max's comparative allusion to vectors and your sliding car. An object in flight is either level, rising or falling. A car tyre is either biting or sliding. At any instant in time. An entity cannot do all/both "at the same time and in the same respect". (That both have forward velocity has to be taken as the given).

Hi Tony,

Mathematically, it is possible for a tire to roll and slide at the same time. What happens at the molecular level is another question. It is quite possible that the molecules of the tire temporarily adhere to the road surface and then jump to a new location.

Consider what happens when you drive your car. If you turn a corner, then different parts of each tire are simultaneously moving at different speeds relative to the surface of the road. Assume you turn left and consider the left front tire. The left side of the tire travels a shorter distance than the right side of the tire as you go around the corner. Since the left side and right side of the tire have the same diameter, either one side or the other or both must be simultaneously rolling and sliding. Whether it is true simultaneity or the molecules in the tire are temporarily adhering to the surface before jumping to a new location is anyone's guess at this point. Perhaps there is some information online that would answer the question. But, if it is easier for you to grasp the idea of high-speed microscopic deformations of the tire than the mathematical explanation using the continuum, then I would have no argument against your point of view.

Darrell

[Jon Letendre]

On 11/19/2018 at 4:35 PM, Jonathan said:

Wicked.

It is time now to ride my V4 1986 VFR750F INTERCEPTOR up Squaw Pass.

Our course is outlined in red on the maps. We travel uphill and west, away from Denver, from the right to the left.

We begin at 8,430' elevation, just northwest of Evergreen, Colorado, and climb 315' in one minute and 54 seconds.

The speed limit is 30mph but we will cover 1.63 miles, averaging 52mph.

25mph is the slowest we will go, in the tightest three turns.

Then 70 and 80mph in the straights. For example after passing the car we're in a long straight and at the end of it, when we are directly beside the yellow sign advising an imminent "15mph" hairpin right, we are still traveling at 80mph. (Funny thing about brakes, they work supernaturally well when you're going steeply uphill!)

The GoPro™ video camera is Chesty™ mounted at about my sternum. I'm crunched leaned over, so my eyes are only a few inches higher.

Pretend you are really there, my passenger, and see if you can feel some of what the terror and adrenaline is like. This will help: We are in elk, bear and deer country. If an elk walks out on a straight we are going to die. If a bear is in the road in a corner, we will crash and he is going to try to kill us when he gets back on his feet. We will try to shoot him when he comes to kill us, but it's a 50/50 thing who emerges from crash-stun first.

 

 

[Jon Letendre]

Now let's ride world-class auto racing track, High Plains Raceway, a 2.55 mile track developed and owned by a handful of Colorado car clubs.

I am in the white helmet on the same INTERCEPTOR, up ahead. We are following me onboard a modern 1050cc Triumph Speed Triple. Unlike my 750cc VFR, this bike has major power for the straights, modern wide tires, fantastic suspension.

The Triumph's sound is cleaner and a higher pitched whizzing. You can see the Triumph's rpm needle, her sound is the one going up and down with that needle. Mine is the foghorn sound.

The gauge displays mph. The long straight is only 0.65 miles long and I can consistently reach about 130mph on the old VFR once the tires are warmed up and I get in the groove.

Here I am pushing harder than on Squaw Pass, a public road. On the street, about 75-80% is my maximum. This is 90-95%. I want to keep my vintage bike, I don't want to crash, but it is acceptable to crash. There is an ambulance on site on the track days that I attend.

 

 

Clockwise ... 

[Jonathan]

Ugh. I'm getting 'video unavailable' notices. 

Lokkinh forward to seeing them.

J

[Michael Stuart Kelly]

4 minutes ago, Jonathan said:

Ugh. I'm getting 'video unavailable' notices. 

Lokkinh forward to seeing them.

J

Jonathan,

Ditto.

Maybe there is a privacy setting active on Jon's YouTube channel?

Something's off on the YouTube side.

Michael

[Jon Letendre]

43 minutes ago, Michael Stuart Kelly said:

Jonathan,

Ditto.

Maybe there is a privacy setting active on Jon's YouTube channel?

Something's off on the YouTube side.

Michael

OK, thanks. Fixing them ...

[Jonathan]

18 hours ago, Darrell Hougen said:

Hi Jonathan,

It's funny how adding the cables actually made the problem more difficult for some people to understand. I would have thought it would have made things simpler.

Darrell

Ditto. Mechanical visualization/reasoning ability is not as common as one might think. And it's also complex -- someone who seems to easily grasp one scenario might suddenly have great difficulty with just a slight new twist to it.

It's fun to explore these cognitive differences.

J

[anthony]

We gotta cue in a dose of reality to puncture the slippist balloon.

I ask that anyone goes look at a car wheel. At least for once do some visualization of the original entity: an everyday wheel in motion.  See the car maker's badge in the center of the rim which is at the center of a tire? Nominate that to be the inner wheel. Same exercise, just that this tiny circle makes it more graphic.

Observe carefully: as the tire makes one revolution, so does the badge make one revolution. But the badge is +/- 6 cms. in diameter, the tire diameter +/-  60 cms. Further, the badge can be calculated to revolve about 18cms. and the tire's track is 1.8meters --their relevant circumferences!. Strange!  What's going on here?

Therefore the outer wheel traverses a distance 1.8m. But equally does the badge/inner wheel travel this far, despite a differential of 1:10 circumference. Strange!

Any slippage observed - or needed to be induced?

Nope. The wheel (and Aristotle's diagram of a wheel) behave as their reality dictates. The single driver and cause is the motion of the outer wheel, the inner wheel conforms.

Perhaps if one imagines a wheel containing an infinite quantity of inner concentric circles, each of which has a 'turn speed' - i.e. tangential velocity - which decreases as their radii decreases, one can imagine a physical, inner wheel transposed onto any circle - and not relevant if a very small wheel or one almost the full size of the outer wheel.

These varying tangential velocities of the imagined 'concentric circles' are what 'hold' (so to speak) the turning wheel together. Mess with them at any point, and THEN you will get slip/skid inside the wheel, causing jamming and breakage.

While a few respondents have appeared to have taken Vt into account, they don't seem to get the ramifications of it  - or - have argued that Vt is precisely what causes/necessitates track and slip. (Placing the cart before the horse). 

From the get-go, the entire premise of inducing track/slippage, clearly has been built on the wheel - only - possessing *angular* velocity. Thereby, erroneously assuming upon a velocity which is identical at any point inside a wheel (and of a wheel within the wheel).

I.e. True, the rpm's are a constant--but the Vt is variable, the product of angular velocity -and- radius..

If angular velocity were all there is, a wheel self-evidently could not function, not to mention this denies the observable identity of a wheel. From one faulty premise, all subsequent exploratory stages will be faulty also. ("Cognition", mechanics, experiment, etc.)

[Jonathan]

31 minutes ago, anthony said:

We gotta cue in a dose of reality to puncture the slippist balloon.

I ask that anyone goes look at a car wheel. At least do some visualization of an original entity, for once: an everyday wheel in motion.  See the car maker's badge in the center of the rim, which is at the center of a tire? Nominate that to be the inner wheel. Same exercise, just that this tiny circle makes it more graphic.

Observe carefully: as the tire makes one revolution, so does the badge make one revolution. But the badge is +/- 6 cms. in diameter, the tire diameter +/-  60 cms. Further, the badge can be calculated to revolve about 18cms. and the tire's track is 1.8meters --their relevant circumferences!. Strange!  What's going on here?

Therefore the outer wheel traverses a distance 1.8m. But equally does the badge/inner wheel, despite a differential of 1:10 circumference. Strange!

Any slippage observed - or needed to be induced?

Nope. The wheel (and Aristotle's diagram of a wheel) behave as their reality dictates. The single driver and cause is the motion of the outer wheel.

Perhaps if one imagines a wheel containing an infinite quantity of inner concentric circles, each of which has a 'turn speed' - i.e. tangential velocity - which decreases as their radii decreases, one can imagine a physical, inner wheel transposed onto any circle - and not relevant if a very small wheel or one almost the full size of the outer wheel.

These varying tangential velocities of the imagined 'concentric circles' are what 'hold' (so to speak) the turning wheel together. Mess with them at any point, and THEN you will get slip/skid inside the wheel, causing jamming and breakage.

While a few respondents have appeared to have taken Vt into account, they don't seem to get the ramifications of it  - or - have argued that Vt is precisely what causes/necessitates track and slip. (Placing the cart before the horse). 

From the get-go, the entire premise of inducing track/slippage, clearly has been built on the wheel - only - possessing *angular* velocity. Thereby, erroneously assuming upon a velocity which is identical at any point inside a wheel (and of a wheel within the wheel).

I.e. The rpm's are a constant--while the Vt is variable, the product of angular velocity -and- radius..

If angular velocity were all there is, a wheel self-evidently could not function, not to mention this denies the observable identity of a wheel. From one faulty premise, all subsequent exploratory stages will be faulty also. ("Cognition", mechanics, experiment, etc.)

The above is yet more confirmation.

As Jon just said:

"I think we will all have to accept that our discussants cannot process what we are saying, cannot cognitively manage all the parts, conditions and motions at once and that there is no teaching innovation that will change that."

J

[Jonathan]

6 hours ago, Jon Letendre said:

Now let's ride world-class auto racing track, High Plains Raceway, a 2.55 mile track developed and owned by a handful of Colorado car clubs.

I am in the white helmet on the same INTERCEPTOR, up ahead. We are following me onboard a modern 1050cc Triumph Speed Triple. Unlike my 750cc VFR, this bike has major power for the straights, modern wide tires, fantastic suspension.

The Triumph's sound is cleaner and a higher pitched whizzing. You can see the Triumph's rpm needle, her sound is the one going up and down with that needle. Mine is the foghorn sound.

The gauge displays mph. The long straight is only 0.65 miles long and I can consistently reach about 130mph on the old VFR once the tires are warmed up and I get in the groove.

Here I am pushing harder than on Squaw Pass, a public road. On the street, about 75-80% is my maximum. This is 90-95%. I want to keep my vintage bike, I don't want to crash, but it is acceptable to crash. There is an ambulance on site on the track days that I attend.

 

 

Clockwise ... 

Very cool vids. I especially like it when there are more riders out front. With the wide lens, that gives a better sense of scale, distance and speed.

Tanx fer sharing.

J

[Jonathan]

And I'm not too proud to admit that I started leaning into the corners by the end of the second vid.

J