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Showing content with the highest reputation on 12/04/2018 in all areas

  1. 1 point
    Or, in terms of the original statement of the paradox, it is impossible to have: X2 - X1 = R * (T2 - T1) x2 - x1 = r * (t1 - t1) X2 - X1 = x2 - x1 T2 - T1 = t2 - t1 and R > r where X2 - X1 and x2 - x1 are the distances traveled by the big and small wheels, respectively, T2 - T1 and t2 - t1 are the angles (theta) that both wheels rotate (e.g. 2pi radians) and R and r are the radii. That is a mathematical statement of Aristotle's paradox. Darrell
  2. 1 point
    Hi Max, Maybe "resolve" (my word) or "solve" isn't the right term. Of course, Aristotle's paradox cannot be "resolved" if all of the conditions are enforced. It is impossible for two wheels that are rigidly attached to each other to turn without slipping on two different tracks if the radii are different. I was simply trying to point out that mathematically, there must be slippage somewhere in the system. Formally, it is impossible to have: V = RW v = rw V = v W = w and R > r where V, v = tangential velocities of the big and small wheels respectively, W, w = their respective angular velocities, and R, r = their respective radii. That is what Aristotle's paradox demands. Darrell
  3. 1 point
    False, the crux of the paradox is that both wheels cannot do a "true roll" without slippage. Reality is that the smaller wheel is slipping, reality is not what you're imagining. Aristotle brought that second track in, your suggestion that that is some newfangled invention of ours is disingenuous, we just keep to the original formulation! Further, nobody claims that that wheel behaves differently when this track is "brought in", it only is a reference that makes clear that the smaller wheel is not rolling out its circumference, but makes another movement that we call slipping.
  4. 1 point
    But you haven't identified any paradox yet. What was thought to be paradoxical in the first place? Specifically what do you think that you're dispensing with by saying the above? You get so easily lost in your own circular reasoning. You're telling us that you've answered the question. We respond that you haven't, and that you don't even know what the question was, and you don't answer. The second track was not "brought in," but was an original element of the "paradox." It is essential to the "paradox." Without it, there is no apparent contradiction or logically unacceptable conclusion to bother Aristotle (or whomeverthefuck). How many times do we have to remind you? No matter how many times, you keep reverting to your previous mistaken premise. Write the shit down, Tony. Make a checklist since you can't hold all of the information in your mind at one time. Stop going back to false arguments that have been soundly refuted already. J
  5. 1 point
    I can't remember, but you stated your position again so explicitly, that I was wondering why it should be so important. As you'll have seen, there are many different definitions of a paradox, and also many different kinds of paradoxes. No problem for me, I don't believe so much in the "one and only" correct definition à la Rand (e.g. her definition of altruism). My viewpoint is, that so many of those classic "paradoxes" are known as "paradoxes", that I see no reason not to use that term for that kind of "paradoxes", genuine or not. For myself I use the definition: an apparent contradiction in an argument caused by a more or less hidden error in the argument or in the premises. In general it isn't difficult to move the error from a false premise to an error in the argument, and an error in the argument can always be thought of as the result of an implicit false premise, there is no sharp distinction between the two options. Changing the formulation a bit can change the formal expression of a paradox, without really changing its essence. Therefore I think my definition isn't that much different from yours, only less restricting, while I also admit false premises. But as I said, I find definitions not that important (the only correct one!) as long as you state them clearly.
  6. 1 point
    Ron Paul is right about Bush, Sr. and his involvement in drug smuggling. The late Rodney Stich, a former Federal Aviation Administration investigator, in his book Defrauding America provides evidence that Bush Sr. helped smuggle cocaine from Mexico when he was in the CIA.
  7. 1 point
    Abusing his host, showing his ugly leftist ass all over an Ayn Rand site, for years, how many years, now? to satisfy his superiority complex trolling and belittling the Randroids. You are a pathetic loser, Billy. Cheer up, tho, I’m sure impeachment is right around the corner, #FuckingMoron
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  9. 1 point
    Darrell, That's just cruel. My post is way back on Page 43 of posts. We are now on Page 55. Who's gonna read it due to your mentioning it? So here is the link to my post of Nov. 22nd. Ah... That's better... (btw - I'm glad you liked it. The world was swimming through my brain at the time... I would like to restrict that to the past tense, too, but alas... ) Michael
  10. 1 point
    Hi Tony, After reading MSK's post from Nov. 22nd --- I'll catch up eventually --- I realized that there are two ways to resolve the paradox. Perhaps the second way is easier for you. Let R, W, and V be the radius, angular velocity and tangential velocity of the big wheel. Then V = RW. Define r, w, and v similarly for the small wheel so that v = rw. Then, if R > r either V > v or w > W. Either the tangential velocity of the big wheel is larger or the angular velocity of the small wheel is larger. So, another way of resolving the paradox is to say that the wheels are actually separate wheels that turn at different rates. If that is easier for you to visualize, that works too. Darrell
  11. 1 point
    That is quite disingenuous. Of course - the small wheel rolls its own circumference. It, too, revolves - once. BUT, the ~distance travelled~ is greater than its own circumference, and you know what I meant, in my brief way of stating that. So, this fails: "that is by definition slipping"... No, it is by definition - "rolling". Repeat: it does not "get past its own circumference". It ~traverses a distance~ greater than its own circumference. Geez. Poor attempt.
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  13. 1 point
    https://mobile.twitter.com/GmanFan45 @Potus steamrolls Europe! Anyone that thinks #Trump fingerprints are not all over the revolution in France and elsewhere is not paying attention. Bannon was never fired, this was all part of the show. He is doing what he did for Trump in 2016 for Europe! 9:29 AM · Dec 3, 2018
  14. 1 point
    Applying the same argument to the large circle: every cycloid of the large circle (cycloid length = 8*R) is greater than 2*pi*R . Therefore, the large circle rolls farther than it would by pure rolling? You've created a new paradox! At least I'm not the one who is drowning.
  15. 1 point
    Maybe you are not sure about this one. Let’s do another one. The small wheel rolls it’s road without slip, and what is the large wheel doing?
  16. 1 point
    Here the small wheel rolls it’s road without slip. Watch it once to confirm this fact. Then watch it again, this time focusing on the large wheel and the lower road - does it look like the large wheel is rolling that road without slip?
  17. 1 point
    Here is what happens when the wheel rolls without any slip on the road. Why isn’t the small wheel staying in point-to-point contact? Is Jon performing a trick, or is there something real, something about all of this that is, in reality, keeping the small wheel from performing honest roll?
  18. 1 point
    Here is what honest, point-to-point rolling without skid or slip looks like. Point-to-point is easily confirmed; each successive tooth is falling into the next pocket in the chain.
  19. 1 point
    Max, That is an even better way of saying it than I did. We assume the diagram is showing a wheel that is rolling and not slipping. But there's no reason to assume that. As to your second point, I agree. I was addressing where the misunderstanding mostly arises. (Apropos, in cases like this, I use posts and discussion to think through the issue, not teach others what I know.) Thus, since most people are arguing as if at least one of the wheels is not slipping, I took that as the default. I should have qualified my thoughts rather than presumed this was clear. In fact, presumption in lieu of qualification where something seems off is the same epistemological error that the diagram induces people to do. In further fact: This is exactly what I was saying. Except I was presuming (to use just one example) that the larger wheel's circumference, if unrolled on the ground like a roll of toilet paper, would be the same as the length of the road. In that case, the point on the smaller circle represents that distance of rotation in relation to the actual road length (i.e., the rotation circumference of the larger circle), not the rotation length of its own circumference. This is because it is not a separate circle, but part of an assembly of circles where the larger circle rules, so to speak. (Remember, this only applies to the case where the large circle does not slip.) I know that sounds a bit convoluted, but conceptually, I know it's correct. As you say, the point represents rotation. I get that. But rotation can be represented by a straight line length after rolling. If the road line measured is not the same length as the circumference of the rotation, that means the wheel slipped. And there's nothing in the diagram that says the rotation represents a non-slipping wheel. Either wheel. And there's nothing that says it represents a slipping wheel, either. Those are merely presumptions. I make no apology. I can get simple later. Convoluted is exactly what working through a thorny idea looks like. I mean, why be simple when complicated also works? Michael
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  21. 1 point
    Here is a wheel with a small wheel firmly attached...
  22. 1 point
    The diagram is in so far incorrect, that it doesn't represent a wheel that is rolling without slipping (which was the supposition in the description of the paradox): the distance traveled after one revolution is smaller than the circumference of the large circle. But apart from that, it is a completely valid diagram, it's perfectly possible that both wheels are slipping. No, that line doesn't represent the movement around the circumference of just one of the circles, it just marks two points on those circumferences, thereby forming a mark for the amount of rotation. You could very well paint such a line on a real wheel, and it would rotate exactly the same way. Those two intersection points rotate completely synchronously. However, a different thing is that at least one of those points is also slipping along its tangent line. You can see that also in slow motion.
  23. 1 point
    I found a great solution to the paradox yesterday--at least from an optical illusion perspective--and have not had time to write about it. I found it in my sleep right before waking up. I'll try to present it graphically a little later. Ah, hell. Let me present the same GIF as before. Where I got this GIF from is an egghead site called Wolfram Mathworld (see here). They said: And of course I didn't understand jack when I first read it. But then something in the second paragraph stuck in the back of my mind: "one-to-one correspondence of points." Later my eureka moment happened while coming out of dozing. The line going through the small circle and ending on the rim of the large circle has nothing to do with both circumferences. Nothing at all. It exists only to represent the circumference of only one of the circles. And, it doesn't matter which. That will depend on which circumference corresponds to the straight line in reality (the stretch of road, so to speak). The line intersecting the rim of the other circle is a projection inward or outward, sort of like a perspective view in a painting. If that "perspective view" aspect is eliminated, and the same stretch of road remains constant, one wheel has to slip if the other does not. I'm not a graphic artist, so I am loathe to draw a perspective view of a person near the viewer and a same size person farther away, but to get the effect of distance, the nearer person will have to be drawn larger than the one farther away. However, if we put a drawing of a tree next to the near person (the larger-drawn person) and a same size tree next to the farther away person (the smaller drawn person), we either destroy the effect of distance, or destroy the effect of the two people being the same size. If we keep one, we destroy the other, it doesn't matter which. This corresponds to what happens with the slippage of one wheel when the other doesn't slip.. The two circles have the same line intersecting their rims and rotating with them, not two different lines. That's because they are connected. If they were separate with each running on their own, they could have two center-to-rim lines. But when there is only one center-to-rim line, that will mean there is only one end-point that counts in relation to the road, not two end-points like it seems (one end point for each circle). Note that the same size road line was used for both circles in the diagram, sort of like the same-size tree was for both people. That's why, when there are two roads for real against two wheels for real, and the wheels are connected, there is slippage in one wheel. Michael
  24. 1 point
    Here, once again, is what the Wikipedia page had on the "paradox" prior to Merlin's fucking with it: 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.
  25. 1 point
    Oh, now you reply to me! Too funny. Look Tony, whenever I go too fast on my motorcycle, the inner circles go slower than the rest of the wheels. When I slow down, they catch up. Reality. Try it and you’ll see.
  26. 1 point
    Whenever I roll a wine bottle it shatters instantly. This is because the neck is trying to rotate at a different rate, and does. Are you saying that when you roll a wine bottle it doesn’t shatter? I’ll have to see a video of that.
  27. 1 point
    Michael, I really think we need to start at the beginning. The Paradox setup requires an abstraction. The paradox is: A wheel rolls a road and it appears lesser diameters within the wheel roll the same road length, which is impossible due to their reduced circumference. The Paradox doesn’t exist until we abstract the lesser diameters as wheels in themselves, rolling on a drawn road. Until we do that, you are right, an inner circle is not a wheel and it is moving through air, not in contact with any road. There is just a rolling wheel, and there is no paradox. There is only a paradox after we abstract the inner circle as a wheel in itself rolling down its drawn road. NOW we have a paradox: Inner diameters which, when thought of as wheels rolling on their road, can and do roll distances in excess of their circumferences, which is impossible. That’s a paradox. That’s Aristotle’s Wheel Paradox.
  28. 1 point
    Ellen wrote: Every inner circle travels the same distance the wheel travels, and that is correct. end quote My parents bought me and my older brother Schwinn’s one Christmas and mine was too big for me. I needed a wooden block attached to each pedal to raise its height but I finally mastered the big bike. I found that I could out travel kids with smaller bike tires and I pedaled less hard. I remember switching bikes with another kid and his little wheeled bike required that I work harder and harder to go the same distance as the bigger Schwinn. He wanted to trade. Now if the inner and outer portions of the tire were not attached and instead made up two separate bikes one with big wheels and one with small wheels the smaller wheeled bike would initially out sprint me but after X amount of revolutions my bigger Schwinn would win the race. The other point I want to make is that “girls’ bikes” make more sense for a boy’s anatomy because the boy’s genitals are in less danger of being squished. Peter