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Aristotle's wheel paradox

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37 minutes ago, anthony said:

Anyhow, an intervention clearly was not what Aristotle was after, he was puzzled that's all...

An "intervention"? Hahahahaha!

J

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24 minutes ago, anthony said:

It should be easily understood that placing the two attached wheels on two surfaces ~spreads~ their combined weight equally, and therefore each experiences the same friction and drag -- or lack of.  

Skid one, the other skids too.

False. Their nature allows that one can skid while not the other. Their nature does not allow for both to not skid.

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Just now, Jon Letendre said:

False. Their nature allows that one can skid but not the other. Their nature does not allow for both to not skid.

Yes. One must slip/skid. Both might slip/skid. Only one can roll without slipping/skidding, at which time the other must slip/skid.

Tony will respond with a statement which forgets/leaves out at least one aspect of the original setup of the "paradox."

J

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It's like a chess match.

Tony is in check, and he can't hold all of the threats in mind at one time.

No, you can't move there, Tony, because of the bishop.

No, not there either, because now you're forgetting about the knight again.

Okay, now you're focusing on the bishop and knight, and you've forgotten about the rook again.

Oops, now that you're thinking of the rook, you've lost track of the bishop again.

And now that we've reminded you yet again of the bishop, you've needed to make room in your mind for it by dropping the knight once again.

Ad infinitum.

J

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43 minutes ago, anthony said:

It should be easily understood that placing the two attached wheels on two surfaces ~spreads~ their combined weight equally, and therefore each experiences the same friction and drag -- or lack of.  

Skid one, the other skids too.

Yes, I think that's what is being badly missed. 

In everyone's imagination the bigger wheel is still dictating proceedings - well, because it is "big", and because it once did, pre-inner track.

Except - now that there are two tracks which carry both wheels' weight simultaneously, there is NO dominant wheel. They are equal, inter-dependent, the action of one will affect the other. Imaginings... 

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4 minutes ago, anthony said:

Yes, I think that's what is being badly missed. 

In everyone's imagination the bigger wheel is still dictating proceedings - well, because it is "big", and because it once did, pre-inner track.

Except - now that there are two tracks which carry both wheels' weight simultaneously, there is NO dominant wheel. They are equal, inter-dependent, the action of one will affect the other. Imaginings... 

Still an idiot.

 

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20 minutes ago, Jonathan said:

Tony will respond with a statement which forgets/leaves out at least one aspect of the original setup of the "paradox."

Just as predicted.

J

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20 minutes ago, anthony said:

Yes, I think that's what is being badly missed. 

In everyone's imagination the bigger wheel is still dictating proceedings - well, because it is "big", and because it once did, pre-inner track.

Except - now that there are two tracks which carry both wheels' weight simultaneously, there is NO dominant wheel. They are equal, inter-dependent, the action of one will affect the other. Imaginings... 

There is still a dominant wheel, the large one rolling on the road.

Simply make the friction of the small wheel less than the friction of the big wheel, then the big wheel will remain dominant one.

Or, make friction of the small wheel zero. Its road is just a line, a merely imagined surface anyway, and then we analyse the small wheel’s interaction with that line and find that it skids that line. It’s just that you don’t find that. You can’t see it, no matter how it is pointed out to you.

You could switch to student, for one discussion, and stand a chance of seeing it, but you always switch back to professor before you get there. You must stay convinced there is something you are not seeing, until you get there.

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49 minutes ago, Jon Letendre said:

There is still a dominant wheel, the large one rolling on the road.

Simply make the friction of the small wheel less than the friction of the big wheel, then the big wheel will remain dominant one.

Or, make friction of the small wheel zero. Its road is just a line, a merely imagined surface anyway, and then we analyse the small wheel’s interaction with that line and find that it skids that line. It’s just that you don’t find that. You can’t see it, no matter how it is pointed out to you.

Either you have a track or you don't. If a track, it must be the same track as a "road", with the identical frictional qualities (or you should maybe specify that the track must be liberally oiled... or, the wheel must be ~just enough~ in contact with its surface to allow slippage, but ~not too much~ to cause drag - (for examples)).

As this set-up stands, on face value, since each wheel is traveling on its own road, bearing the same weight, neither is dominant.

If the "road is just a line" (from the start, I suggested that this is a cognitive challenge, before being a mechanical problem. I don't believe you supported this at the time), and you want to "make friction of the small wheel zero", you're trying to have things both ways. Abstract -or- physical, whenever it suits you.

You're going to find that 'track and slippage', in reality, create bigger problems, require compromises and raise contradictions, and to top it all, can't work practically.

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17 minutes ago, Jon Letendre said:

There is still a dominant wheel, the large one rolling on the road.

Simply make the friction of the small wheel less than the friction of the big wheel, then the big wheel will remain dominant one.

True, but this is all totally beside the point: Tony seems to think that the fact that in practice there are seldom such smaller wheels slipping against a second support would somehow be a refutation of the paradox. Now of course such slipping is in general undesirable, as it causes wear and costs extra energy. But that is with regard to Aristotle's paradox completely irrelevant. Aristotle describes a thought experiment about rolling circles, this was not an exercise in good wheel design or a discussion about the merits of different existing wheel designs. That is not to say that you cannot demonstrate it with a physical model, it is perfectly possible to realize such a wheel with a smaller wheel slipping while the outer wheel rolls without slipping and it is a good illustration of the principle of the paradox. But Aristotle was interested in the question how it is possible to map the circumference of the small circle 1-1 onto a longer track, without skipping or temporarily stopping of one of the circles. That was his problem, and the mechanism of slipping and modern insights about infinite sets are the solution that escaped him. Don't try to change his subject and turn it into you own hobby horse!

We now know that the small wheel cannot roll without slipping if the large wheel rolls without slipping. If you take away the support of the small wheel, it still makes exactly the same movement, which you could call slipping against an imaginary support. After all, circles and lines are also imaginary constructions, and that is what Aristotle was talking about. Talk about forces and friction is irrelevant, perhaps useful for practical applications, but we are not talking about a consumer test of car wheels, we're talking about mapping circles onto line segments.  

I'll repeat what I wrote earlier: this reminds me of Einstein's twin paradox: solving that is not arguing that in practice we don't send a twin with huge speeds through space, so that when he returns he's still a young man, while his brother in the meantime has become an old man. Anyway, that would be far beyond our current technical possibilities. Yet no one doubts that the twin paradox is about a very real effect.
 

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2 hours ago, anthony said:

But the only point of the paradox is the different circumferences and their identical travel distance! You're being difficult, you know what I mean.

The notion all you are expressing is the artificial lengthening of the small wheel's travel -- by sliding-- in order to - effectively - equalize the two wheels' circumference-lengths.

Anyhow, an intervention clearly was not what Aristotle was after, he was puzzled that's all, and was looking for an explanation for the phenomenon.

The smaller wheel's travel distance is lengthened (though not artificially) by its slipping relative to a horizontal tangent at the 6 o'clock position.  This is not the same statement as that the smaller wheel's circumference is lengthened.

Ellen

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14 hours ago, Max said:

 

We now know that the small wheel cannot roll without slipping if the large wheel rolls without slipping. If you take away the support of the small wheel, it still makes exactly the same movement, which you could call slipping against an imaginary support. After all, circles and lines are also imaginary constructions, and that is what Aristotle was talking about. Talk about forces and friction is irrelevant, perhaps useful for practical applications, but we are not talking about a consumer test of car wheels, we're talking about mapping circles onto line segments.  


 

 

The usual repetition from me. As long as there is nothing for the inner wheel to slip *on*, or in relation *to*, both wheels-circles behave normally. The large circumference rotates once, the inner circumference, likewise. I definitely won't call this "slipping against an imaginary support", the cause of the action lies somewhere else. 

What seems to me the missing piece of the puzzle, comes in - because the inner has to rotate slower. Tangential velocity plays a central role.

In staying away from the purely mechanical option, you can't get over-theoretical and purely abstract - there's a risk of denying the practicality of real wheels. Abstraction without causal identification makes for "Pi in the sky", :).

So then put in a *physical* track and both wheels are supported, but instead of rotating "normally" and in synch as previous, the inner one's slower tangential speed (now on its own surface) must cause it to drag. Not slip: drag. The braking effect of this action imparts drag onto the large wheel as well. So either they both instantly come to a stop, or far greater power has to be applied to keep them moving. It's likely then both will slide, not roll.

(Of course, imagining the (impossible) theory of 'zero friction' on the inner track, would eliminate the need for a track altogether - simply, we'd be back where the paradox started, without progress. I believe by your latest comments you recognised this dead-end contradiction ahead.)

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5 hours ago, anthony said:

The usual repetition from me. As long as there is nothing for the inner wheel to slip *on*, or in relation *to*, both wheels-circles behave normally. The large circumference rotates once, the inner circumference, likewise. I definitely won't call this "slipping against an imaginary support", the cause of the action lies somewhere else. 

What seems to me the missing piece of the puzzle, comes in - because the inner has to rotate slower. Tangential velocity plays a central role.

In staying away from the purely mechanical option, you can't get over-theoretical and purely abstract - there's a risk of denying the practicality of real wheels. Abstraction without causal identification makes for "Pi in the sky", :).

So then put in a *physical* track and both wheels are supported, but instead of rotating "normally" and in synch as previous, the inner one's slower tangential speed (now on its own surface) must cause it to drag. Not slip: drag. The braking effect of this action imparts drag onto the large wheel as well. So either they both instantly come to a stop, or far greater power has to be applied to keep them moving. It's likely then both will slide, not roll.

(Of course, imagining the (impossible) theory of 'zero friction' on the inner track, would eliminate the need for a track altogether - simply, we'd be back where the paradox started, without progress. I believe by your latest comments you recognised this dead-end contradiction ahead.)

Hey Tony,

Check out the Where Are You? thread:

We could use your input over there. Help us out.

Do you agree with Bob, or with the stupid gang of slippists who have migrated from here to that thread?

J

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15 hours ago, anthony said:

The usual repetition from me. As long as there is nothing for the inner wheel to slip *on*, or in relation *to*, both wheels-circles behave normally. The large circumference rotates once, the inner circumference, likewise. I definitely won't call this "slipping against an imaginary support", the cause of the action lies somewhere else. 

What seems to me the missing piece of the puzzle, comes in - because the inner has to rotate slower. Tangential velocity plays a central role.

In staying away from the purely mechanical option, you can't get over-theoretical and purely abstract - there's a risk of denying the practicality of real wheels. Abstraction without causal identification makes for "Pi in the sky", :).

[....]

What is there for the inner wheel to have tangential velocity in relation to?  You won't accept the idea of an imaginary support, but then you turn around and rely on the idea of a tangent which doesn't physically exist.

Ellen

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6 hours ago, Ellen Stuttle said:

What is there for the inner wheel to have tangential velocity in relation to?  You won't accept the idea of an imaginary support, but then you turn around and rely on the idea of a tangent which doesn't physically exist.

He seems to have some vague notion that the fact that the tangential velocity of the smaller wheel is smaller than that of the large wheel "somehow" compensates for the fact that the small wheel after one revolution has to move over a larger distance than its own circumference. He always comes back with that argument. Where is the proof? Well, I've shown with quite simple math that this is not the case, but, on the contrary, that the math shows that the small wheel must slip to keep up with the large one. But aww, mathematics, that is just all about floating abstractions, you should of course consider the causal identity of the wheel! I give up...

 

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4 hours ago, Max said:

But aww, mathematics, that is just all about floating abstractions, you should of course consider the causal identity of the wheel!

Yes, math is a crutch, geometry is a crutch, and reality is a crutch. Holding all of the elements in one's mind at once is a crutch. Being able to visuospatially track the motion and relationships is a crutch.

J

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5 hours ago, Max said:

He seems to have some vague notion that the fact that the tangential velocity of the smaller wheel is smaller than that of the large wheel "somehow" compensates for the fact that the small wheel after one revolution has to move over a larger distance than its own circumference.

 

One of his little theories was that the surface which the smaller wheel rolls on (during the moments when he decides to accept the existence of that surface rather than deny it) is a ledge with material beneath it which acts as a sort of infinite series of stilts for the smaller wheel, thus making it equal in height to the larger wheel, which in his mind cancels out the differences in circumference. He treats the ledge under the smaller wheel as being a part of the smaller wheel, which he believes makes the smaller wheel's circumference equal to the larger wheel's.

That's the level of stupidity that we're dealing with.

J

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2 hours ago, Jonathan said:

Yes, math is a crutch, geometry is a crutch, and reality is a crutch. Holding all of the elements in one's mind at once is a crutch. Being able to visuospatially track the motion and relationships is a crutch.

J

Here's where the "crutch" charge was started, in a diatribe by Merlin on November 19, 2018.  Merlin has subsequently made the "crutch" charge nine times in close to the identical wording (viz., that the track is your crutch without which "you are too lame-brained to deal with reality").

Ellen

 

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CALCULATING TANGENTIAL VELOCITY ON A CURVE

By Steven Holzner

When an object moves in a circle, if you know the magnitude of the angular velocity, then you can use physics to calculate the tangential velocity of the object on the curve.

 

At any point on a circle, you can pick two special directions: The direction that points directly away from the center of the circle (along the radius) is called theradial direction, and the direction that’s perpendicular to this is called the tangential direction.

When an object moves in a circle, you can think of itsinstantaneous velocity (the velocity at a given point in time) at any particular point on the circle as an arrow drawn from that point and directed in the tangential direction. For this reason, this velocity is called the tangential velocity. The magnitude of the tangential velocity is the tangential speed, which is simply the speed of an object moving in a circle.

Given an angular velocity of magnitude

image0.png

the tangential velocity at any radius is of magnitude

image1.png

The idea that the tangential velocity increases as the radius increases makes sense, because given a rotating wheel, you’d expect a point at radius r to be going faster than a point closer to the hub of the wheel.

A ball in circular motion has angular speed around the circle.

 

A ball in circular motion has angular speed around the circle.

Take a look at the figure, which shows a ball tied to a string. The ball is whipping around with angular velocity of magnitude

image3.png

You can easily find the magnitude of the ball’s velocity, v, if you measure the angles in radians. A circle has

image4.png

the complete distance around a circle — its circumference — is

image5.png

where r is the circle’s radius. In general, therefore, you can connect an angle measured in radians with the distance you cover along the circle, s, like this:

image6.png

where r is the radius of the circle. Now, you can say that v= s/t, where v is magnitude of the velocity, s is the distance, and t is time. You can substitute for s to get

image7.png

In other words,

image8.png

Now you can find the magnitude of the velocity. For example, say that the wheels of a motorcycle are turning with an angular velocity of

image9.png

If you can find the tangential velocity of any point on the outside edges of the wheels, you can find the motorcycle’s speed. Now assume that the radius of one of your motorcycle’s wheels is 40 centimeters. You know that

image10.png

so just plug in the numbers:

image11.png

Converting 27 meters/second to miles/hour gives you about 60 mph.

 

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13 hours ago, Ellen Stuttle said:

What is there for the inner wheel to have tangential velocity in relation to?  You won't accept the idea of an imaginary support, but then you turn around and rely on the idea of a tangent which doesn't physically exist.

Ellen

I did your research work for you, above. Tangential velocity is real, apparent and logical. Also mathematical.

Note: "The idea that the tangential velocity increases as the radius increases makes sense..." 

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27 minutes ago, anthony said:

I did your research work for you, above. Tangential velocity is real, apparent and logical. Also mathematical.

Note: "The idea that the tangential velocity increases as the radius increases makes sense..." 

You haven't answered the question.

Ellen

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20 minutes ago, Ellen Stuttle said:

You haven't answered the question.

Ellen

He hasn't grasped the question.

He saw some words, and then posted something that was related to them.

J

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6 hours ago, Max said:

He seems to have some vague notion that the fact that the tangential velocity of the smaller wheel is smaller than that of the large wheel "somehow" compensates for the fact that the small wheel after one revolution has to move over a larger distance than its own circumference. He always comes back with that argument. Where is the proof? Well, I've shown with quite simple math that this is not the case, but, on the contrary, that the math shows that the small wheel must slip to keep up with the large one. But aww, mathematics, that is just all about floating abstractions, you should of course consider the causal identity of the wheel! I give up...

 

No. You misrepresent me. The "small wheel after one revolution moves a larger distance than its own circumference." -- is "a given", I have kept saying. A fundamental. The large wheel exceeds the size of the small one; and the small one - or any point or circle in the large one - is merely along for the ride.

"Where is the proof?"

That doesn't need proof or argument or math, anyone with sight can observe what it does.

Then separately, is an explanation for the above, NOW, citing the relative tangential velocities of the inner, to the outer wheels. But with the same translational (forward) velocity and same angular speed.

That the inner wheel rotates exactly once, to the outer's once, is evidently a property of tangential velocity. I.E., it turns less quickly than the larger one.

Therefore -- it does not "slip".

But IF - you wrongly accept the premise of equal rotational speeds, then and only then could it be imagined to slip.

 

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14 hours ago, Ellen Stuttle said:

What is there for the inner wheel to have tangential velocity in relation to?  You won't accept the idea of an imaginary support, but then you turn around and rely on the idea of a tangent which doesn't physically exist.

Ellen

Did you read the article Ellen? Tangential velocity is an objectively measurable and calculable velocity. It doesn't need to be in relation to anything. 

But - every point/circle, from the outer rim, inwards, rotates slower. Less Vt.  Depending on its radius. (That's the distance from the circle center).

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8 hours ago, Max said:

 But aww, mathematics, that is just all about floating abstractions, you should of course consider the causal identity of the wheel! I give up...

 

Well, Max - when "the causal identity" is not the beginning point, it seems to me a lot of things go wrong.

For the record, I have kept maintaining that math and experiment are also critical. But not prematurely. And I'm sure your math is flawless, but lacking the right premises it too, like philosophical principles, can be "floating abstractions" detached from reality. And not in themselves prove anything.

"I give up...". Yup, perhaps me too! ;)

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