Aristotle's wheel paradox

[merjet]

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.

[jts]

28 minutes ago, 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.

At the risk of stating the obvious, the inner circle is not not simply rolling, it is moving with the outer circle. Is this supposed to be difficult?

 

[merjet]

6 minutes ago, jts said:

At the risk of stating the obvious, the inner circle is not not simply rolling, it is moving with the outer circle. Is this supposed to be difficult?

The paradox is described in the link I gave.

[BaalChatzaf]

10 hours ago, 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.

the inner wheel slips

[merjet]

37 minutes ago, BaalChatzaf said:

the inner wheel slips

Huh?  Observe a wheel-rim-tire on a car roll. Does the rim slip relative to the tire?

[Jonathan]

6 hours ago, merjet said:

Huh?  Observe a wheel-rim-tire on a car roll. Does the rim slip relative to the tire?

Huh? Who said anything about the inner wheel slipping "relative to the tire"?!!!

One of the wheels must slip relative to the line that represents it's alleged "circumference" line.

J

 

[merjet]

34 minutes ago, Jonathan said:

Huh? Who said anything about the inner wheel slipping "relative to the tire"?!!!

Nobody. I asked if the rim slipped relative to the tire. The rim and the circumference of the tire where it meets the road is analogous to the two concentric circles of the paradox.

 

34 minutes ago, Jonathan said:

One of the wheels must slip relative to the line that represents it's alleged "circumference" line.

Mere assertion.

[Jonathan]

2 hours ago, merjet said:

 

Mere assertion.

Heh. It's a "mere assertion" which is correct.

Ah, by "resolve this paradox," you don't mean that we should identify what would happen in reality if we had concentric wheels which were in contact with separate planes, but instead you want a mathematical solution? If so, you should have stated that in the OP.

J

[merjet]

12 hours ago, Jonathan said:

Heh. It's a "mere assertion" which is correct.

Ah, by "resolve this paradox," you don't mean that we should identify what would happen in reality if we had concentric wheels which were in contact with separate planes, but instead you want a mathematical solution? If so, you should have stated that in the OP.

Heh. It's a mere assertion which is wrong and you give no reasons for why you believe it's correct.

LOL. I asked readers if they could resolve the paradox.  The paradox is mathematical, and the forum is science and mathematics. So why do believe the solution should not be mathematical? Also, the problem is very much about what happens in reality when a wheel rolls.

[merjet]

I found this video very helpful for thinking about the paradox. More hands-on is a roll of tape.

[jts]

Make the inner circle very small, a dot. The circumference of the dot is very small. But the dot travels the full distance. It is carried by the outer circle.Is that so hard to understand?

 

[merjet]

12 minutes ago, jts said:

Make the inner circle very small, a dot. The circumference of the dot is very small. But the dot travels the full distance. It is carried by the outer circle. Is that so hard to understand?

No. However, it does not resolve the paradox.

[jts]

58 minutes ago, merjet said:

No. However, it does not resolve the paradox.

What is the paradox?

 

[merjet]

25 minutes ago, jts said:

What is the paradox?

The paradox is described on the Wikipedia page in my first post. Better yet, watch this video as the wheel makes one full revolution. The marked point (at the 6:00 o'clock position at both start and end) on each circle appears to travel its circle's circumference, but the two circumferences are not the same length. On the other hand, the straight horizontal lines the two circles travel along are the same length. Thus the contradiction or paradox.  

 

[Jonathan]

17 hours ago, merjet said:

Heh. It's a mere assertion which is wrong and you give no reasons for why you believe it's correct.

LOL. I asked readers if they could resolve the paradox.  The paradox is mathematical, and the forum is science and mathematics. So why do believe the solution should not be mathematical?

I see the “paradox” as being physical, so I gave a physical solution since a mathematical one wasn’t specified.

Here's what wiki page that you linked to says:

Quote

Aristotle's wheel paradox is a paradox appearing in the Greek work Mechanica traditionally attributed to Aristotle.^[1] There are two wheels, one within the other, whose rims take the shape of two circles with different diameters. The wheels roll without slipping for a full revolution. The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. But the two lines have the same length, so the wheels must have the same circumference, contradicting the assumption that they have different sizes: a paradox.

I think the problem that you're having is that the premise begins with an assumption that not everyone makes. You, Merlin, apparently buy into the stated assumption that the lines "are apparently the wheels' circumferences." You seem to intuitively identify with that mistaken misperception. I don't. Apparently you can't visualize what would actually happen in reality when the surfaces contact each other. I can. The alleged "paradox" is only a paradox to those who lack visuospatial/mechanical reasoning abilities.

Quote

The paradox is mathematical, and the forum is science and mathematics.

The "paradox" is physical, and I described what happens in physical reality. I wrote:

"One of the wheels must slip relative to the line that represents it's alleged 'circumference' line."

That is correct. That's what happens in physical reality.

Quote

So why do believe the solution should not be mathematical?

I've said nothing to suggest that I think that the solution should not be mathematical. I've only said that if you wanted a mathematical solution, then you should have specified so. Do you understand that a mathematical solution is not the only possible solution? Physically constructing the objects and rolling them on their surfaces would be another possible solution, as would simply having a more effective brain for mechanical reasoning than you apparently do and describing what would happen in reality, just as I did.

J

[Jonathan]

When the orange wheel rolls without slipping on the surface of the bottom ruler, the smaller wheel slips (or skids) along the surface of the upper ruler.

 

[merjet]

4 hours ago, Jonathan said:

You seem to intuitively identify with that mistaken misperception. I don't. Apparently you can't visualize what would actually happen in reality when the surfaces contact each other. I can. The alleged "paradox" is only a paradox to those who lack visuospatial/mechanical reasoning abilities.

...

Physically constructing the objects and rolling them on their surfaces would be another possible solution, as would simply having a more effective brain for mechanical reasoning than you apparently do and describing what would happen in reality, just as I did.

Hogwash and ditto to you. You made an unwarranted assumption about the circles shown on the Wikipedia page and in the video that I linked. You assumed the circles can turn independently or slip or skid. The Wikipedia page even says the opposite: "the wheels roll without slipping." That the circles can turn independently or slip or skid in the video that I linked is clearly impossible. It is you who is misunderstood or needs a more effective brain for mechanical reasoning.

Of course, if the circles are independent, they could slip. But then there would be no paradox.

[Brant Gaede]

The closer you get to your destination the more you slip slide away.

--Brant

[Jonathan]

1 hour ago, merjet said:

Hogwash and ditto to you. You made an unwarranted assumption about the circles shown on the Wikipedia page and in the video that I linked. You assumed the circles can turn independently or slip or skid. The Wikipedia page even says the opposite: "the wheels roll without slipping." That the circles can turn independently or slip or skid in the video that I linked is clearly impossible. It is you who is misunderstood or needs a more effective brain for mechanical reasoning.

Of course, if the circles are independent, they could slip. But then there would be no paradox.

Heh. Wow, you're just not getting it. I made no assumption about the circles turning independently of each other. In fact, the video that I created and posted above follows the requirement of the circles being locked in alignment with each other. They roll together as if they are glued to each other. Can you seriously not see that?!!! Man, you're even more visuospatially/machanically limited than I had thought!

The wheels roll without slipping with each other, which means that, if the larger wheel is not slipping or skidding on its line, then the smaller wheel MUST slip or skid on its line. It's a very simple fact of reality that a circle which is rotating one full turn and covering more distance on a line than its circumference must, by definition, be slipping on the line (while not moving independently of the larger circle).

Have you never seen different sized gears sharing and locked to the same axle? The purpose of such gearing is to alter the amount of distance per rotation that is driven by the mechanism.

There is no paradox. There's only your limited thinking/visualizing capacity. Reality is reality, independent of your personal inability to grasp it.

[Jonathan]

14 minutes ago, Brant Gaede said:

The closer you get to your destination the more you slip slide away.

--Brant

Not if you're going to the Scarborough fair.

[Jonathan]

I just read the rest of the wiki article that Merlin linked to originally. Heh. Merlin, did you bother to read and comprehend it? Apparently not.

"One way to understand the paradox of the wheel is to reject the assumption that the smaller wheel indeed traces out its circumference, without ensuring that it, too, rolls without slipping on a fixed surface. In fact, it is impossible for both wheels to perform such motion. Physically, if two joined concentric wheels with different radii were rolled along parallel lines then at least one would slip; if a system of cogs were used to prevent slippage then the wheels would jam."

Duh!

[merjet]

7 hours ago, Jonathan said:

There is no paradox. There's only your limited thinking/visualizing capacity. Reality is reality, independent of your personal inability to grasp it.

There is a paradox. There's only your limited thinking/visualizing capacity. Reality is reality, independent of your personal inability to grasp it.

For example, what part of "the wheels roll without slipping" does Jonathan fail to understand? Also, he keeps referring to two wheels. However, in this video I posted earlier there is only one wheel with two circles drawn on it. Why does he insist there is no paradox and then quote "One way to understand the paradox of the wheel" to try to insult me? Duh! Note the quote refers to "the wheel." That's one wheel, not two wheels.

Given Jonathan's massive, unnecessary doses of personal insults and confusions and evasions along with any on-topic point he tries to make, it's a huge waste of time trying any useful dialogue with him. Bye, Jonathan. 

[Jonathan]

Hahahaha!

i can't wait to see where you take it from here!

[Brant Gaede]

2 hours ago, merjet said:

There is a paradox. There's only your limited thinking/visualizing capacity. Reality is reality, independent of your personal inability to grasp it.

For example, what part of "the wheels roll without slipping" does Jonathan fail to understand? Also, he keeps referring to two wheels. However, in this video I posted earlier there is only one wheel with two circles drawn on it. Why does he insist there is no paradox and then quote "One way to understand the paradox of the wheel" to try to insult me? Duh! Note the quote refers to "the wheel." That's one wheel, not two wheels.

Given Jonathan's massive, unnecessary doses of personal insults and confusions and evasions along with any on-topic point he tries to make, it's a huge waste of time trying any useful dialogue with him. Bye, Jonathan. 

Well, my untrained eyeballs have to say respecting "the wheel" is the lines are irrelevant to the implied proposition that something's goofy going on. Those lines aren't the wheel. In your video the wheel itself represents the actual physicality. Can we call the lines the (irrelevant parts of the) epistemology?

--Brant

at least J didn't call you an inverted prevaricating inside out thespian

[BaalChatzaf]

2 hours ago, Brant Gaede said:

Well, my untrained eyeballs have to say respecting "the wheel" is the lines are irrelevant to the implied proposition that something's goofy going on. Those lines aren't the wheel. In your video the wheel itself represents the actual physicality. Can we call the lines the (irrelevant parts of the) epistemology?

--Brant

at least J didn't call you an inverted prevaricating inside out thespian

In a physical instantiation of that scheme the inner wheel slip.  It has the same center as the outer wheel and is carried forward on outer circumference per revolution of the outer wheel.  The inner wheel is rigidly affixed to the outer wheel.  since it has a smaller radius its circumference is less than the circumference of the outer wheel so it slip on its rail by a distance equal to the difference of the circumferences.