Aristotle's wheel paradox

[Jonathan]

15 minutes ago, Jonathan said:

Merlin, do you think that you could see the outer circumference of the tire skidding or slipping while rolling on the ground? Do you think that your visual/spatial/mechanical cognitive abilities are sensitive and potent enough to recognize that a wheel with a circumference of ten feet has actually rolled twelve feet in one full rotation, or eight feet in one full rotation, therefore skidding or slipping?

 

8 minutes ago, merjet said:

More nonsense from the idiot.

Yes or no. Do you think that you would be able to see the skidding or slipping of a wheel on the ground, as I described it above?

I understand your evasion of the question, because obviously the next step might be for me to test your ability to see it.

Or perhaps I already have? Heh. And not only can't you see it, but you can't even measure it and check its math.

Thanks for the entertainment!

J

[Jon Letendre]

On 9/28/2017 at 10:47 AM, merjet said:

Heh. Where's the math? Drawing little colored lines for small arcs doesn't suffice.

It seems to me you are confusing there being a paradox and resolving a paradox.

The outermost circle's/wheel's circumference is a maximum, not a minimum.

He can't understand my simple diagram.

[merjet]

12 minutes ago, Jon Letendre said:

He can't understand my simple diagram.

LOL. He believes a 45 degree arc suffices for a 360 degree rotation even when the center has moved! 

[Jon Letendre]

2 minutes ago, merjet said:

LOL. He believes a 45 degree arc suffices for a 360 degree arc even when the center has moved! 

See? Doesn't understand what the diagram shows. Imagines something, what I don't know, something different, will occur if it keeps rolling.

Lost and confused, lashing out in anger.

[merjet]

29 minutes ago, Jon Letendre said:

Lost and confused, lashing out in anger.

Heh. Laughing while angry. What a concept!

[Jon Letendre]

2 minutes ago, merjet said:

Heh. Laughing while angry. What a concept!

You fail to grasp my simple diagram.

[Jon Letendre]

You don't want to talk about the simple diagram because you know that when you do, you expose your failure to grasp it.

So you would rather correct me about laughing, versus not yet today lashing out like an infant.

Youre so transparent.

[Ellen Stuttle]

4 hours ago, Jonathan said:

Does the “solution” that Merlin found online actually solve the “paradox”?

No.

You're right, it doesn't, although I haven't yet read your explanation of why not,  but I was thinking yesterday and didn't have time to post that the supposed paradox pertains to the linear distance traveled by the respective 6:00 o'clock points not to the total distance.

I'm interjecting quickly in hopes that you're still tracking posts although it's Friday and often you take the weekend off from posting.

I thought of something which might help anyone see that there isn't a real paradox :

Use a different regular figure besides a circle as the inner figure - a square, a triangle, a star.

For instance, an equilateral triangle with one angle pointing downward and on the same diagonal as the 6:00 o'clock point of the outer circle.

Now revolve the whole figure. Would anyone think that the triangle wasn't being dragged along by the revolution of the whole figure?

When the initial 6:00 o'clock point of the outer circumference gets to the 12:00 o'clock point, where will the formerly down-pointing angle of the triangle be?

Maybe J could provide a graphic - I don't know how to use any computer graphics programs.

Ellen

[Ellen Stuttle]

1 hour ago, Jon Letendre said:

He can't understand my simple diagram.

Besides which, your diagram wasn't what I'd linked to.  I think that Merlin didn't click the link I provided, just asked for some math - which isn't even needed to get what the diagram shows.

Ellen

[BaalChatzaf]

Wheels are utterly charming.  When a wheel  a wheel is happily rolling along half the points on the rim are moving in the direction of the hub  and half are moving in a direction opposite to that of the hub.  So if half the points are going forward and half backward,  how does it get anywhere?

[Jon Letendre]

14 minutes ago, Ellen Stuttle said:

Besides which, your diagram wasn't what I'd linked to.  I think that Merlin didn't click the link I provided, just asked for some math - which isn't even needed to get what the diagram shows.

Ellen

He can't grasp what the diagram shows.

Puffing up and emitting some synthetic confidence about the reason being absence of math is a face-saving, protection-of-weak-and-fragile-ego, move.

[Ellen Stuttle]

On September 28, 2017 at 12:47 PM, merjet said:

Heh. Where's the math? Drawing little colored lines for small arcs doesn't suffice.

It seems to me you are confusing there being a paradox and resolving a paradox.

The outermost circle's/wheel's circumference is a maximum, not a minimum.

Re Jon's diagram, which I suppose you're referring to, that isn't what I linked to - and math isn't needed to understand Jon's diagram.

I think you added the line "The outermost circle's/wheel's circumference is a maximum, not a minimum" on an edit.  At any rate, I don't recall its being there when I first read your reply.

Your statement is backward to your own presentation - here:

[MJ] "Yes, min= Rb*2*pi,  max= 8*Rb"

Ellen

[Max]

Holy Jesus... I started to read this thread, but gave up somewhere halfway, as the solution is so trivial. Jonathan c.s. are of course right: the origin of the paradox lies in the supposition that *both* wheels are moving without slipping. That is of course impossible: if the wheels are part of a rigid body, they rotate with the same angular velocity. After one revolution the small wheel travels a smaller distance than the large wheel (2 pi r vs. 2 pi R; r < R). If the outer wheel rotates without slipping and travels a distance of 2 pi R, the inner wheel *must* also travel 2 pi R during one revolution (it's a rigid body). However, its own translational movement due to rotation is only 2 pi r, so it *must* be slipping to make up the difference.

Of course we could also suppose that it is the inner wheel that moves without slipping, in that case the outer wheel must slip (moving slower than its "natural" movement), and finally it's also possible that both  wheels are slipping (one going faster, the other going slower than their "natural"movement). 

What will happen in reality depends on other mechanical conditions, such as the magnitude of the respective friction forces. 

 

[Jon Letendre]

1 hour ago, Max said:

Holy Jesus... I started to read this thread, but gave up somewhere halfway, as the solution is so trivial. Jonathan c.s. are of course right: the origin of the paradox lies in the supposition that *both* wheels are moving without slipping. That is of course impossible: if the wheels are part of a rigid body, they rotate with the same angular velocity. After one revolution the small wheel travels a smaller distance than the large wheel (2 pi r vs. 2 pi R; r < R). If the outer wheel rotates without slipping and travels a distance of 2 pi R, the inner wheel *must* also travel 2 pi R during one revolution (it's a rigid body). However, its own translational movement due to rotation is only 2 pi r, so it *must* be slipping to make up the difference.

Of course we could also suppose that it is the inner wheel that moves without slipping, in that case the outer wheel must slip (moving slower than its "natural" movement), and finally it's also possible that both  wheels are slipping (one going faster, the other going slower than their "natural"movement). 

What will happen in reality depends on other mechanical conditions, such as the magnitude of the respective friction forces. 

 

Hi Max, it is clear that you get it.

But in the 2nd para, I believe you meant to say the larger will be forced to spin-out, or, spin faster than...

Also, what happens in reality is there is friction and rolling of the actual wheel at the road. There cannot be friction between circles and their imaginary roads. Nevertheless, we can imagine friction, or gears on the smaller road, and then, as you said, the larger, actual wheel will be forced to spin at a rate different than consistent with the road.

I also agree with the final point of para 2. If the inner circle skids, but not as much as natural, then the outer wheel will have to spin-out. One will skid (inner) and one will over-spin, or spin-out (outer.)

[Jon Letendre]

3 hours ago, BaalChatzaf said:

Wheels are utterly charming.  When a wheel  a wheel is happily rolling along half the points on the rim are moving in the direction of the hub  and half are moving in a direction opposite to that of the hub.  So if half the points are going forward and half backward,  how does it get anywhere?

And that goofy point at the center - it goes forward only and isn't even able to rotate!!

[Jon Letendre]

4 hours ago, BaalChatzaf said:

Wheels are utterly charming.  When a wheel  a wheel is happily rolling along half the points on the rim are moving in the direction of the hub  and half are moving in a direction opposite to that of the hub.  So if half the points are going forward and half backward,  how does it get anywhere?

Hey, serious question.

Is there any reason for 360 degrees?

Using 1,000 or something else would also work, right? Of course, all the computational techniques would have to be adjusted, and they would be amenable to that, right? 

Like how many days in a week, or even having the week, it is entirely optional, yes?

[Brant Gaede]

On 9/27/2017 at 11:52 AM, Jon Letendre said:

Jon, you need a drink.

--Brant

 

[BaalChatzaf]

36 minutes ago, Jon Letendre said:

Hey, serious question.

Is there any reason for 360 degrees?

Using 1,000 or something else would also work, right? Of course, all the computational techniques would have to be adjusted, and they would be amenable to that, right? 

Like how many days in a week, or even having the week, it is entirely optional, yes?

360 is Babylonian  We inherited their angles.  The real honest to goodness measure of an angle is the radian.  Take a  section of the circumference that is equal to the radia. That subtends an angle of one radian.  There are 2 pi  radians to the full circle (360 degrees).  pi radians is a straight angle. The ends of a diameter subtend on arc of pi radians.  pi/2 radians is a right angle.  4 x pi/2 = 2pi.   so pi/2 is a quarter of the way around the circumference of the circle.

 

[Jon Letendre]

2 minutes ago, BaalChatzaf said:

360 is Babylonian  We inherited their angles.  The real honest to goodness measure of an angle is the radian.  Take a  section of the circumference that is equal to the radia. That subtends an angle of one radian.  There are 2 pi  radians to the full circle (360 degrees).  pi radians is a straight angle. The ends of a diameter subtend on arc of pi radians.  pi/2 radians is a right angle.  4 x pi/2 = 2pi.   so pi/2 is a quarter of the way around the circumference of the circle.

 

Thanks. It really is elegant how radians come directly from circles, rather than being forced by preexisting counting systems.

[merjet]

20 hours ago, Ellen Stuttle said:

I think you added the line "The outermost circle's/wheel's circumference is a maximum, not a minimum" on an edit.  

I should have said, "A point on the outermost circle's/wheel's circumference sets a maximum, not a minimum." That is for a curved path. I was mislead by your saying, "all points of all circles of the "paradox" setup ..... travel farther than the outermost circle's/wheel's circumference."  Mid-sentence you changed focus from a curved path to the straight horizontal path.  The straight horizontal path is a minimum.

[merjet]

14 hours ago, BaalChatzaf said:

Wheels are utterly charming.  When a wheel  a wheel is happily rolling along half the points on the rim are moving in the direction of the hub  and half are moving in a direction opposite to that of the hub.  So if half the points are going forward and half backward,  how does it get anywhere?

[BaalChatzaf]

14 hours ago, Jon Letendre said:

Thanks. It really is elegant how radians come directly from circles, rather than being forced by preexisting counting systems.

The radian for plane angles and the steradian for solid angles are the "natural"  measure for the angles.  They do not involve an arbitrary measure set by custom or habit.  That is why the second (measure of time)  is a certain number of oscillations of the cesium atom.  The is nature talking.  See:  https://en.wikipedia.org/wiki/Caesium_standard

[Ellen Stuttle]

Un-raveling the supposed paradox

 

14 hours ago, merjet said:

[below]

Me - here:  "I think you added the line 'The outermost circle's/wheel's circumference is a maximum, not a minimum' on an edit."

 

Merlin - here:  "I should have said, "A point on the outermost circle's/wheel's circumference sets a maximum, not a minimum." That is for a curved path. I was mislead by your saying, 'all points of all circles of the "paradox" setup ..... travel farther than the outermost circle's/wheel's circumference.'  Mid-sentence you changed focus from a curved path to the straight horizontal path.  The straight horizontal path is a minimum."

 

Your insertion clarifies what you meant, but, no, I didn't change focus from the curved path to the straight horizontal path.  According to your own chart, "all points of all circles of the 'paradox' setup ..... travel farther than the outermost circle's/wheel's circumference."  The only point which travels exactly the distance of the outermost circle's/wheel's circumference is the center, which, as only one point, isn't a circle.

I'll re-post your chart in full.  (I'm presuming, btw, that you're correct in giving the distance traveled by a point on the outermost circumference as 8*Rb.  I haven't checked that out.)

 

On September 23, 2017 at 9:05 AM, merjet said:

 

       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

 

 

I've subsequently realized in so many words what I've "sensed" all along: The only path which is relevant to the supposed paradox is the straight horizontal path.  Seeing why requires distinguishing each position of the total configuration from the particular point occupying that position at a given time.

At each instant, however far laterally the instantaneous point of tangency - which is to say, the 6:00 o'clock position of the outer circumference - has traveled is how far laterally every other position has traveled.  Which point instantaneously occupies each position (except the center) keeps changing, but the relationship of all positions to one another remains constant.

For instance, the 9:00 o'clock position of the outer circumference - whatever point is occupying that position at a given instant - is always a quarter distance around the circumference from the 6:00 o'clock position of the outer circumference and is always a radius distance from the center.  Similarly, every other position, whether on the outer circumference or internal to the figure, is always in the same relationship to all other positions of the figure and is always a constant distance from the center.

Thus, the distance the center - the only position always occupied by the same point - has traveled laterally is the distance every position has traveled laterally.  There's only one lateral distance involved, uniform for each position of the whole figure.

The supposed paradox arises from having the false presumption that a position on the internal circle should have moved a different distance laterally from the distance moved laterally by a comparable position on the outer circumference (e.g., the 6:00 o'clock position of the inner circle compared to the 6:00 o'clock position of the outer circumference).  The false presumption results from thinking of the internal circle as if it were an independent figure rolling on its own.  But it isn't an independent figure. It's a fixed-relationship subconfiguration of the total configuration.

Ellen

 

[Brant Gaede]

On 9/11/2017 at 6:23 PM, merjet said:

Can you resolve this paradox?  

It may help to imagine a point at the 6:00 o'clock position on each circle and then rolling the wheel one revolution. 

I will give my solution later.

Okay.

--Brant

hit us with it (after you answer Ellen)

[merjet]

12 hours ago, Ellen Stuttle said:

[1] I didn't change focus from the curved path to the straight horizontal path.  

[2] The only path which is relevant to the supposed paradox is the straight horizontal path.  

[3] Similarly, every other position, whether on the outer circumference or internal to the figure ... is always a constant distance from the center.

[4] Thus, the distance the center - the only position always occupied by the same point - has traveled laterally is the distance every position has traveled laterally.  There's only one lateral distance involved, uniform for each position of the whole figure.

[5] The supposed paradox arises from having the false presumption that a position on the internal circle should have moved a different distance laterally from the distance moved laterally by a comparable position on the outer circumference (e.g., the 6:00 o'clock position of the inner circle compared to the 6:00 o'clock position of the outer circumference).  The false presumption results from thinking of the internal circle as if it were an independent figure rolling on its own.  But it isn't an independent figure. It's a fixed-relationship subconfiguration of the total configuration.

[1] You did. Otherwise, you made a false claim. Take your pick.

[2] I totally disagree. If everyone thought that, then there would be no paradox.

[3] Whoop-de-do!

[4] Whoop-de-do! Also, I said that already. “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).”

[5] You show the same haughty attitude as Jonathan and Jon. You behave like there is one and only one correct perspective or solution to a problem – yours. Anything else must be wrong, and you are obsessed with attacking it and/or the person making it. However, there is more than one way to crack an egg. There is more than one way to prove the Pythagorean Theorem.