Aristotle's wheel paradox

[Jon Letendre]

17 minutes ago, Brant Gaede said:

The Earth rotates at, what, 24,000 mph at the equator.

How come at the true North Pole you seem to be as standing still as anywhere else on the planet if the North Pole doesn't slip or slide?

--Brant

Could you restate?

[Brant Gaede]

Just now, Jon Letendre said:

Could you restate?

I'm imagining the earth as a wheel with an inner wheel we call the North Pole. If the two are rigidly attached to each other the inner wheel will spin at the same speed as the most outer--or 1000mph.

--Brant

[Jon Letendre]

5 minutes ago, Brant Gaede said:

I'm imagining the earth as a wheel with an inner wheel we call the North Pole. If the two are rigidly attached to each other the inner wheel will spin at the same speed as the most outer--or 1000mph.

--Brant

I'm trying to imagine as you are.

Do you mean the view of earth I would get from directly above it? So that it looks like a soccer ball about ten feet from me, with the North Pole in the exact center?

[BaalChatzaf]

43 minutes ago, Jon Letendre said:

The Ascent can't go fast like that. But she will climb up steep terrain, at a fraction of 1 mile per hour.

Its great that you ride still - keep it up!

Every nice day I ride.  I can't do it in the winter.  Slippery streets are deadly for two wheel vehicles.  In the winter I go to the gym and use a boooooring  stationary bike.

[Brant Gaede]

8 minutes ago, Jon Letendre said:

I'm trying to imagine as you are.

Do you mean the view of earth I would get from directly above it? So that it looks like a soccer ball about ten feet from me, with the North Pole in the exact center?

I'm imagining an almost infinite number of wheels the first being formed at the equator projected as high as the North Pole. Now, go halfway down from the North Pole to the equator and imagine another wheel projected up. The North Pole wheel rigidly attached to the equator wheel  will turn at the same 1000 MPH. The real North Pole doesn't do this. If it's attached to the halfway up wheel it will turn at 500 MPH assuming that that wheel isn't in turn attached to the equator wheel. Attached to nothing in one day the NP wheel will not rotate at all unless it has momentum from inertia.

--Brant

[Jon Letendre]

21 minutes ago, Brant Gaede said:

I'm imagining an almost infinite number of wheels the first being formed at the equator projected as high as the North Pole. Now, go halfway down from the North Pole to the equator and imagine another wheel projected up. The North Pole wheel rigidly attached to the equator wheel  will turn at the same 1000 MPH. The real North Pole doesn't do this. If it's attached to the halfway up wheel it will turn at 500 MPH assuming that that wheel isn't in turn attached to the equator wheel. Attached to nothing in one day the NP wheel will not rotate at all unless it has momentum from inertia.

--Brant

 I'm sorry, I'm not following your thought.

[BaalChatzaf]

38 minutes ago, Jon Letendre said:

Could you restate?

He is ask if there can be some twisting or torsion in the earth's crust.  The crust is going with a 1000 mph tangential velocity at the equator and 0 mph at the poles.  If the Earth's crust was completely rigid there would be no torsion.  By the way the sun's equator has a higher angular velocity than the sun's poles  so there is twisty torsion on the surface of the sun. That twists the magnetic field lines of force so much that eventually the magnetic lines of force give way hurl very large chunks of sun outward. This is a mass ejections and if it hits the earth it can cause all sorts of nasty things to happen with power lines and transformers. 

In 1859 the Mother of Mass Ejections hit Earth (it was described by a British astronomer Carrington).  This mass ejections cause telegraph lines to sparks and blew up batteries powering the telegraphs.  Hundreds of telegraphers were burned by sparks that jump from their  key sets.  Fortunately the world was not wired the way it is now,  so civilization did not come to a screeching halt because the onlyh electrical things at that time were the telegraph systems.  If a Carrington even hit the earth now  it would destroy our power grids and fry the transformers.  It would also destroy many of our satellites.  Kiss GPS goodbye for a while. It would take years for the world to put itself back together again. 

[merjet]

10 hours ago, Jonathan said:

I can see the paths, but you misidentified one as a cycloid and the other as a curtate cycloid. They are actually both curtate. The only true, common cycloid would be one created from a point on the outer rim of the wheel which contacts the table on which it rolls.

Apparently you didn't fully grasp the conditions required to create common cycloids versus curtate ones (and presumably prolate ones) before claiming to "see" them in the video. Oops! Busted again!

......

Can you not see the skidding of the small circle in the video in which I visually isolated it?

You are wrong again. The larger circle on the wheel in the video was designed to represent the outer circumference of a real wheel. The lower horizontal string/wire represents the road surface a real wheel would roll on. If you watch the video closely enough, it is abundantly clear that the lower dot rises from its starting point by the lower string/wire more quickly than does the higher dot from its starting point by the lower string/wire. After only a quarter rotation the lower dot reaches the same altitude as the higher dot even though its starting position was much lower. The lower dot traces travels a cycloid shape. The higher dot traces traces a curtate cycloid shape.

There is no dot for illustrating the path of the wheel for obvious reasons, except maybe to you. Apparently you didn't fully grasp the conditions of the demonstration. You misidentified a regular cycloid as a curtate cycloid and the table as representing a road. Oops! Busted again! It must be your deficient visual/spatial/mechanical reasoning skills.

I can see the "skidding" of the small circle in the video.

[merjet]

1 hour ago, Jonathan said:

Bike gears can't work according to Merlin's little theories, so that picture has to be fake!

Try not to be such an obnoxious jerk.

[Jon Letendre]

To the extent I think I follow you, all those wheels are in fact attached to each other, as they are projections of one solid globe.

One goes faster as one goes farther from the center of a spinning wheel because one must now cover a larger circumference in the same period of rotation.

[Jon Letendre]

Only the pink dot traces a cycloid.

Both black dots trace curtate cycloids.

[Jon Letendre]

Here I have added the white dot where the pink dot originated from.

The smallest wheel's non-adherence to its black line is evident.

[Jon Letendre]

The pink dot traces a cycloid and accordingly the pink line and pink arc are of equal length. Their equal length tell us that the producer of the video was honest and correct in telling us that the outside of this wheel is indeed staying in contact with the bottom of the groove in the wood deck.

The greenblack dot traces a curtate cycloid and accordingly the green line is noticeably longer than the green arc.

The blueblack dot traces a very curtate cycloid and the obvious disparity in segment lengths demonstrates that the blueblack "wheel" is not keeping traction to its line, but is skidding, slipping, sliding over it.

[Jon Letendre]

[Jon Letendre]

Next, let's observe the paths the dots trace through space..

The yellow dot has moved a greater length than any of the others.

[Jon Letendre]

The purple dot has gone really, really far.

Has the purple dot had occasion to make many revolutions? It sure looks that way, given all the ground it has covered. But we can see that the wheel has only rotated about an eighth of a turn, so that means the purple "wheel" (the circle around the screw) has only rotated about an eighth of a turn. This proves that the purple "wheel" and the yellow, blue and green "wheels" are all sliding over their imaginary horizontal lines. Thus there is no paradox.

[Brant Gaede]

2 hours ago, merjet said:

You are wrong again. The larger circle on the wheel in the video was designed to represent the outer circumference of a real wheel. The lower horizontal string/wire represents the road surface a real wheel would roll on. If you watch the video closely enough, it is abundantly clear that the lower dot rises from its starting point by the lower string/wire more quickly than does the higher dot from its starting point by the lower string/wire. After only a quarter rotation the lower dot reaches the same altitude as the higher dot even though its starting position was much lower. The lower dot traces travels a cycloid shape. The higher dot traces traces a curtate cycloid shape.

There is no dot for illustrating the path of the wheel for obvious reasons, except maybe to you. Apparently you didn't fully grasp the conditions of the demonstration. You misidentified a regular cycloid as a curtate cycloid and the table as representing a road. Oops! Busted again! It must be your deficient visual/spatial/mechanical reasoning skills.

You are still mixing up wheel and circle. That's your essential fallacy.

--Brant

[merjet]

6 hours ago, Brant Gaede said:

You are still mixing up wheel and circle. That's your essential fallacy.

--Brant

Jeesh. A wheel is a circular object that revolves on an axle fixed below a vehicle or other object to enable it to move easily or roll over the ground. A wheel is 3-dimensional with a 2-dimensional aspect of it being a circle. I'm not the only one using circle and wheel sometimes interchangeably. Why don't you tell others they are committing a fallacy?

[merjet]

22 hours ago, anthony said:

I've just tuned in but I gather you've all been arguing apples against pineapples, Merjet.

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! 

[merjet]

12 hours ago, Jonathan said:

False. I've used scare quotes when referring to the alleged "paradox." See, I just did it there again. Do you know what scare quotes are?

Yes. I don't always catch their significance or always use them as appropriate. Do you? I believe not. You have not been using them with slip and skid.

[Brant Gaede]

2 hours ago, merjet said:

Jeesh. A wheel is a circular object that revolves on an axle fixed below a vehicle or other object to enable it to move easily or roll over the ground. A wheel is 3-dimensional with a 2-dimensional aspect of it being a circle. I'm not the only one using circle and wheel sometimes interchangeably. Why don't you tell others they are committing a fallacy?

You can't do your stuff without a circle. It only works with a circle. Or circles. Your original video shows a wheel with two circles to illustrate the "paradox." Sans circle there isn't a paradox. That's because there isn't an inner wheel attached to the outer wheel (which, of course, would slip or slide as the outer wheel turns). Sans wheel, where are the circles? No matter how hard you try you cannot change a wheel into a circle or a circle into a wheel. There is no "aspect" of a circle in a wheel except as a thought rendered as an illustration.

Like I keep saying, you keep mixing up the physical with the mental by slapping the latter onto the former pretending--now I know it's effectively to yourself--that the mental is also the physical here. Thinking makes the paradox. Qua thinking thinking unmakes it too. No real, physical wheel is affected.

--Brant

[merjet]

1 hour ago, Brant Gaede said:

You can't do your stuff without a circle. It only works with a circle. Or circles. Your original video shows a wheel with two circles to illustrate the "paradox." Sans circle there isn't a paradox. That's because there isn't an inner wheel attached to the outer wheel (which, of course, would slip or slide as the outer wheel turns). Sans wheel, where are the circles? No matter how hard you try you cannot change a wheel into a circle or a circle into a wheel. There is no "aspect" of a circle in a wheel except as a thought rendered as an illustration.

Like I keep saying, you keep mixing up the physical with the mental by slapping the latter onto the former pretending--now I know it's effectively to yourself--that the mental is also the physical here. Thinking makes the paradox. Qua thinking thinking unmakes it too. No real, physical wheel is affected.

--Brant

The wheels -- each with 2 concentric circles -- in Jonathan's video and my video are isomorphic. The difference in inner circle sizes is irrelevant. So whatever your objection is, tell it to Jonathan. Maybe he can understand your mumbo-jumbo.

[BaalChatzaf]

We are talking about  wheels  (not  static circles)  which means there are three motions:  rotation (turning),  translation (sliding/skidding),  rolling which is rotation + translation

Please see https://en.wikipedia.org/wiki/Rolling#/media/File:Rolling_animation.gif

[Jonathan]

13 hours ago, merjet said:

You are wrong again. The larger circle on the wheel in the video was designed to represent the outer circumference of a real wheel. The lower horizontal string/wire represents the road surface a real wheel would roll on.

Heh. It doesn't matter what the larger circle was "designed to represent." It is not the outer circumference, despite what anyone wishes to label it. The wheel's actual outer circumference is its actual outer circumference, and any point on a concentric circle on the wheel which is smaller than that circumference will trace a curtate cycloid. Both circles in the video that you posted are smaller than the circumference of the wheel, and both of their points trace curtate cycloids, despite your misperceptions due to your massive cognitive deficiencies.

See, the way that reality works is that wishing doesn't make things so. If you arbitrarily wish a random line to "represent" the line on which a wheel rolls, any points on the wheel don't magically obey your wish. They move in retaliation to the actual rolling circumference, not to the one that you've falsely labeled as the circumference. You've hilariously misidentified reality once again.

But you don't have to take my word for it. You can test it yourself. The method would be for you to try to wish that a different line represents the surface on which the wheel rolls. Let's go with the smaller circle in the video. Now that you wish it to represent the line on which the wheel rolls, is the path traced by the larger circle's dot suddenly a prolate cycloid? Do you now imagine that you can see a prolate cycloid just as you earlier imagined that you could see a common cycloid when it was actually a curtate cycloid the you were looking at?

Quote

If you watch the video closely enough, it is abundantly clear that the lower dot rises from its starting point by the lower string/wire more quickly than does the higher dot from its starting point by the lower string/wire. After only a quarter rotation the lower dot reaches the same altitude as the higher dot even though its starting position was much lower. The lower dot traces travels a cycloid shape. The higher dot traces traces a curtate cycloid shape.

False. Both are curtate cycloids. Neither is a common cycloid. Just do a motion-track with a multiplier filter on the dots and sustain them from frame to frame of video. Then you'll see reality rather than the fantasy that you're believing in.

Quote

There is no dot for illustrating the path of the wheel for obvious reasons, except maybe to you. Apparently you didn't fully grasp the conditions of the demonstration.

The absence of a dot on the wheel's circumference doesn't change the fact of reality that it is the circumference which is rolling on the table. The large circle inside of the circumference doesn't magically become the circumference upon which the wheel rolls just because you've labeled it that way!

 

Quote

I can see the "skidding" of the small circle in the video

Yay!!! Finally!!! And what does that skidding tell you? What does it mean? Think it through! You can do it!

J

[Jon Letendre]

The outside circumference of the wheel is rolling on the table.

Only dots such as the pink one, at the outside edge of the wheel, will trace a cycloid.

The black dots are not at the edge of the wheel where rolling contact with the table is made, so they trace curtate cycloids.