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Wheels schmiels. How about that rotating hurricane? And clouds. What do you see in clouds? I remember a kid who could get you to see just about anything in a fluffy white cloud if you lay down on your back and stared at it long enough.

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4 hours ago, merjet said:

You call it skid. I call it "goes along for the ride." The common reference of skid is loss of traction, which doesn't apply to an inner circle.

Wrong. It DOES apply to an inner circle since the premise of the “paradox” is to make that inner circle contact its own line! The inner circle loses traction compared to the line which it contacts in Aristotle’s setup of the “paradox."

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It also involves friction, and an inner circle doesn't resist the horizontal movement.

False. The inner circle DOES resist the horizontal movement. The smaller the inner circle, the greater the friction between it AND THE LINE ALONG WHICH IT TRAVELS!

Let’s review now, because you, Merlin, have great difficulty keeping the entire setup in your mind at one time. You keep forgetting about the fact that the setup includes TWO lines — one on which the larger circle is rolling, and one on which the smaller circle is said to be rolling, but on which it is actually slipping/skidding. Got that? Okay, the smaller circle loses traction on the line that it contacts as described in the "paradox's" setup.

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Physics makes a distinction between rolling without slipping and rolling with slipping.

Yes, physics does indeed make a distinction between rolling without slipping and rolling with slipping. Rolling with slipping is known as a type of "skidding." The smaller circle rolls with slipping. In other words, it skids. The smaller the circle on the wheel, the greater the friction and skidding. I’ve presented videos which visually show the skidding. You, being a stubborn moron, refused to consider them, and brushed them aside as optical illusions!

J

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4 hours ago, merjet said:

That's a humongous IF which changes the context to where it is no longer Aristotle's wheel or a rolling roll of duct tape.

The sad thing is that Merlin is not embarrassed at all about how stupid he is showing himself to be. After everything that has been said and demonstrated on this thread, reality is still not sinking in, and it hasn't even occurred to him yet to explore the possibility that he should have some self-doubt, and that he should listen to others' arguments a little more carefully. Nope. It's just full throttle idiocy.

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Jonathan wrote, "Nope. It's just full throttle idiocy." Deep thoughts but this thread is starting to sound like robo calls from Omaha Steaks. If you listen to their speeches suddenly a live person is on the line to complete the sale.

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2 hours ago, BaalChatzaf said:

Any translation of the center (hub) that is not accompanied by an instantaneous roll round the point of tangency  at the rim   is by definition a skid.  Of course, we are assuming the rigidity of the wheel/circle.  At no point does the wheel/circle become deformed.

Suppose the point on the rim you refer to is at 6:00 o'clock before the roll begins. The wheel rotates 90 degrees clockwise to put the point at 9:00. Where is the point's tangent line?

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45 minutes ago, merjet said:

Suppose the point on the rim you refer to is at 6:00 o'clock before the roll begins. The wheel rotates 90 degrees clockwise to put the point at 9:00. Where is the point's tangent line?

each point on the rim is instantaneously a tangent point for an infinitesimal interval of time.  The only way to deal with this stuff in a mathematically rigorous manner is to use parametric co-ordinates with time as the driving parameter.  So for each you have a hub position  and a point of tangency on the rim.