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i doubt Aristotle had a grandfather clock in his den.

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Thanks for noticing, Max. It’s easy for me, to be honest. I don’t mind stupid, it doesn’t rub me the wrong way at all. It’s only when snippy gets added to stupid that I have to explode or walk aw

These are not solutions. "Solution" 1 is nothing else than a short recapitulation of the paradox: Aristotle: If I move the smaller circle I am moving the same centre, namely Α; now let the larger

No, it isn't. Artistotle wrote: "nowhere does the greater stop and wait for the less in such a way as to remain stationary for a time at the same point" and "the smaller does not skip any point", so h

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32 minutes ago, anthony said:

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.

Straw man. No one denies that the small wheel moves a larger distance that its own circumference, that is part of the paradox.

32 minutes ago, anthony said:

"Where is the proof?"

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

Read better, I didn't ask proof of the triviality you mention here, I asked for a proof of your opinion that the fact that the tangential velocity of the small wheel is smaller than that of the large wheel explains the paradox without slipping of the small wheel, something you've repeatedly asserted. You cannot give such a proof, as this proposition is false, I've given proof of the opposite.

32 minutes ago, anthony said:

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".

There, now you say it again!

Further, your terminology is wrong: the tangential velocities differ, but the wheels turn equally fast. Tangential velocity is a linear velocity (m/s), "turning" of "rotation" is an angular velocity (rad/s)'

32 minutes ago, anthony said:

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

The rotational speeds are equal, in contrast to the tangential speeds.

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35 minutes ago, Max said:

There, now you say it again!

Further, your terminology is wrong: the tangential velocities differ, but the wheels turn equally fast. Tangential velocity is a linear velocity (m/s), "turning" of "rotation" is an angular velocity (rad/s)'

The rotational speeds are equal, in contrast to the tangential speeds.

Forget my occasionally lax terminology. I use a word loosely at times for explication. "Rotational" instead of "tangential".

Geez, the nit-picking here.

In your words, you have just conceded the differing 'tangential" velocity (= m/s).

So. What do YOU think is the only cause and driver of the small wheel's and large wheel's equal rotation - i.e. 1: 1 ?

By trial and error, consider and remove both the other, equal, velocities of a circle and what are you left with?

Right. The varying tangential velocity! Which is, commonsensically, how a wheel can function! And why the inner one turns 1 : 1 with the outer!  And why everyone has consistently got this wrong! "The rotational speeds are equal..." No, the "tangential" speeds cannot be equal. A wheel would disintegrate if they were so.

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1 hour ago, anthony said:

Forget my occasionally lax terminology. I use a word loosely at times for explication. "Rotational" instead of "tangential".

Geez, the nit-picking here.

Nitpicking? This is as misleading as confusing voltage and current, or force and energy, entropy and enthalpy.

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In your words, you have just conceded the differing 'tangential" velocity (= m/s).

Conceded?!   I used this fact already on February 4  2018 for calculating the translation speed for different points on the wheels:

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Suppose the large wheel/circle rolls without slipping. After 1 period in time T the center of the circle is translated over a distance 2*pi*R, with a uniform translation speed of its center v= 2*pi*R/T. The point at the top of the circle is translated with speed 2*v and the bottom (that touches the line (=support) has translation speed zero. The translation speed of the point at the top of the smaller circle ≡ v2 = 2*pi*(R+r)/T. This can be checked by substituting r=R and r=0. Similarly, the translation speed of the point at the bottom of the smaller circle ≡v3= 2*pi*(R-r)/T > 0 for r < R. So we see that for the smaller circle and its tangent (support) the condition for tracing out the circumference is not met. That the bottom point of the smaller circle has a translation speed > 0 is the mathematical equivalent of saying that the smaller wheel is rotating and slipping.

How do you think these formulas were derived? Where are your calculations? Your proof that my calculation has errors?

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So. What do YOU think is the only cause and driver of the small wheel's and large wheel's equal rotation - i.e. 1: 1 ?

They form a rigid body and rotate about a common center (axis in 3 dimensions).

.

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By trial and error, consider and remove both the other, equal, velocities of a circle and what are you left with?

Right. The varying tangential velocity! Which is, commonsensically, how a wheel can function! And why the inner one turns 1 : 1 with the outer!  And why everyone has consistently got this wrong! "The rotational speeds are equal..." No, the "tangential" speeds cannot be equal. A wheel would disintegrate if they were so.

I cannot make head or tail of this. But one sentence is revealing: "And why everyone has consistently got this wrong!

You'd think this is satire, but I'm afraid it is not...

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Tony  thinks a funnel veers off to one side because the end with smaller circumference is down lower. He thinks that if one precisely raises the surface the small end contacts, then the funnel will roll straight. He doesn’t see the circumference disparity still exists, was thinking of it merely as a “height” issue. Thinks raising the small end solves it the issue.

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Here's a suggestion. Enough mathematizing, go observe a wheel turn.

There cannot be "slippage", and guess why? A wheel is an integrated whole and every point on every different radius within it, is moving at a specific, different, tangential velocity -- As would do an internal wheel, positioned on any radius.

The ONLY way you'll have your slippage, is for the internal velocities to be all equal (oh, but that's what you think).

Then, the only way to try to attain slippage, would be to place a physical track under the inside wheel for it to slip on (oh, but then its different tangential speed will cause drag, and stop both wheels).

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Desperately seeking slippage

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12 minutes ago, anthony said:

Desperately seeking slippage

Only the plate itself rolls without slip.

The large black circle slips, skids, it’s line.

The small black circle skids it’s line even more so than the larger one.

How else can it be, Tony? The black circles lack circumference to roll true like the plate does. Same road length, smaller then smaller again circumferences. You seriously still cannot see this simple fact?

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49 minutes ago, Jon Letendre said:

Tony  thinks a funnel veers off to one side because the end with smaller circumference is down lower. He thinks that if one precisely raises the surface the small end contacts, then the funnel will roll straight. He doesn’t see the circumference disparity still exists, was thinking of it merely as a “height” issue. Thinks raising the small end solves it the issue.

Blah, small potatoes. Tony was experimenting with ideas of Darrell's cones and funnels mimicking a large wheel and small wheel. Weeks ago. Of course the first thing to do is to level the ends.

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3 minutes ago, Jon Letendre said:

Only the plate itself rolls without slip.

The large black circle slips, skids, it’s line.

The small black circle skids it’s line even more so than the larger one.

How else can it be, Tony? The black circles lack circumference to roll true like the plate does. Same road length, smaller then smaller again circumferences. You seriously still cannot see this simple fact?

WHAT? Do me a favor, go look at your bike's wheel and come back and tell me you see slippage - anywhere - at any radius - between tire, wheel, whatever.

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18 minutes ago, Jon Letendre said:

How else can it be, Tony? The black circles lack circumference to roll true like the plate does. Same road length, smaller then smaller again circumferences.

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See, you are looking without seeing. You have a preconception ("lack circumference to roll true...") of how the wheels *should* be turning, and missing what is there.

You cannot see the two points on the circumferences taking a longer/shorter path to return to the bottom point? The outer one moving further and therefore moving faster? In order for both to return simultaneously?

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29 minutes ago, anthony said:

WHAT? Do me a favor, go look at your bike's wheel and come back and tell me you see slippage - anywhere - at any radius - between tire, wheel, whatever.

Of course I see it.

See the tiny little screw in center of plate? It goes a distance much greater then it’s circumference — by skidding. It is rotating, but mostly skidding, it’s road. I see that every time I watch the video. It’s tiny little circumference quite obviously will not get it across the screen in one rotation — it gets across the screen in one rotation mostly by skidding it’s road.

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15 minutes ago, anthony said:

See, you are looking without seeing. You have a preconception ("lack circumference to roll true...") of how the wheels *should* be turning, and missing what is there.

You cannot see the two points on the circumferences taking a longer/shorter path to return to the bottom point? The outer one moving further and therefore moving faster? In order for both to return simultaneously?

They traverse the same road length as the plate, Tony, but they lack the circumference to do so by rolling only. Therefore, they skid. Not should. They do skid their roads/lines. They rotate and skid.

Of course I can see those things. They don’t change the skidding the drawn black wheels of smaller circumference perform on their lines.

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12 hours ago, Jon Letendre said:

Only the plate itself rolls without slip.

You are not even close. I don't have the device itself to measure more accurately, but as best I can tell: The circumference of the disc (calculated from its diameter) is nearly 20% longer than the distance along the wires. The larger circle's circumference (calculated from its diameter) is about 2% shorter than the distance along the wires.  Alternatively, if the larger circle's circumference were about 2% larger, it would equal the distance along the wires.

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12 hours ago, anthony said:

WHAT? Do me a favor, go look at your bike's wheel and come back and tell me you see slippage - anywhere - at any radius - between tire, wheel, whatever.

We already know that you don't see the slippage, Tony. That's what we've been saying. You do not have the visuospatial cognitive abilities to see and comprehend it.

J

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46 minutes ago, merjet said:

You are not even close. I don't have the device itself to measure more accurately, but as best I can tell: The circumference of the disc (calculated from its diameter) is nearly 20% longer than the distance along the wires. The larger circle's circumference (calculated from its diameter) is about 2% shorter than the distance along the wires.  Alternatively, if the larger circle's circumference were about 2% larger, it would equal the distance along the wires.

There are only a few frames of video in which the entire apparatus is shown at once, and even then it is not shown precisely head on, but is shown with perspective distortion caused by the wide lens and the closeness of the camera. Do you know how to compensate for perspective, Merlin? No, you don't. You're not even aware of the relevance of it and the effect that it will have on your measurements and your "best I can tell" guesses.

And let's not forget that you are so visually inept that you initially thought that the disk had a hidden, smaller disk behind it that was rolling on a ledge that you imagined seeing (that's how lacking your knowledge of perspective is: You didn't see the perspective indicators of the area of the apparatus behind the wheel, and thus fooled yourself into believing that it was a vertical section rather than a horizontal continuation of the board on which the disk rolls).

You're guessing and bluffing way beyond your level of knowledge. You don't have a clue how to geometrically account for the perspective in the images.

J

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3 hours ago, Jonathan said:

[More hogwash and lies.]

Both you and Jon are so inept that you asserted the disk rolls without slipping on the bottom of the groove. Its circumference is about 20% longer than the distance along the wires, which proves you are both wrong.

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6 minutes ago, merjet said:

Both you and Jon are so inept that you asserted the disk rolls without slipping on the bottom of the groove. Its circumference is about 20% longer than the distance along the wires, which proves you are both wrong.

Prove it.

Demonstrate that you know how to account for the perspective in the images. Show us the geometry. Plot it out, and show your work. No more unsupported assertions.

Heh.

Um, here's a screen capture of the entire apparatus:

Do you see the vertical red rectangles that I've placed on the left and right sides of the image? They are both the same size. Notice that the one on the left is the same height as the wooden support next to it? See that? And on the other side, the wooden support appears to be shorter. Why is that?!!! Hmmm? Can you figure it out, genius?

Is the post on the right really shorter than the one on the left? If so, do the strings go downhill? When the wheel reaches the right side, do the lines end up lower than the circles to which they are currently tangential? No? They don't? So, what could explain the wooden support on the right appearing to be about 20% smaller than it actually is?

OMG, Merlin, look at this giant dog!!!

His shoulders come up to the deck of the Golden Gate bridge! He's way taller than the north tower of the bridge, but just shorter than the south tower.

Dang, it's a new paradox. How is it that the bridge deck is level when the north tower is so much smaller than the south tower? Is the giant dog a part of the solution?

J

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10 minutes ago, Jonathan said:

Prove it.

Demonstrate that you know how to account for the perspective in the images. Show us the geometry. Plot it out, and show your work. No more unsupported assertions.

Prove that the disk rolls without slipping on the bottom of the groove. Show your work. No more unsupported assertions.

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5 minutes ago, merjet said:

Prove that the disk rolls without slipping on the bottom of the groove. Show your work. No more unsupported assertions.

You're a reality-hating, belligerent idiot.

What would you accept as proof that the disk rolls without slipping? I'd first have to teach you the geometry of perspective in order for you to grasp the proof, and you would be a resistant, belligerent idiot the entire time, denying reality and refusing to see.

J

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12 minutes ago, Jonathan said:

[Blah, blah.]

Pure ad hominem. Entirely lacking in rational argument.

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8 minutes ago, merjet said:

Pure ad hominem. Entirely lacking in rational argument.

Answer the question. What would you accept as proof that the disk rolls without slipping?

J

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On 11/15/2018 at 8:25 PM, Jon Letendre said:

The video Merlin offered is honest in that it depicts the main wheel performing a true roll on the road (which is a couple millimeters below the black deck.) True rolling is evidenced by the equal lengths of the pink arc and pink line.

The slip or skid of the “small wheel” relative to its “road,” which many of us have been referring to from the beginning, is evidenced by the blue arc’s length being less than the blue line’s length.

The blue wheel has traversed a length of blue road equal to the length of the blue line segment, and yet all the while it has made contact with the road with only a short length of wheel (the blue arc,) which is insufficient for all that traversing, thus we can see it has been skidding, slipping on its “road.”

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3 hours ago, merjet said:

Prove that the disk rolls without slipping on the bottom of the groove. Show your work. No more unsupported assertions.

Done, dumb jackass.

Now let’s see if you respond with more rational argument than “hogwash.”

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4 hours ago, merjet said:

Both you and Jon are so inept that you asserted the disk rolls without slipping on the bottom of the groove. Its circumference is about 20% longer than the distance along the wires, which proves you are both wrong.

god you are stupid.