Because of this explanation I finally understand this weird paradox which isnt weird at all! The thing is that the distance a point on the coin moves is still only one time the circumference, but because the coin travels on the surface of one complete circle, the coin rotates one "extra" circle. If we would simply take an arc instead of a circle and the arc has the length of the circumference of the circle, we can better understand this principle!! Because in case of an arc we understand that the coin also traverses over a curved surface and hence that adds to the rotation of the coin, but, and a very important but: the distance traversed by a single point on the circle is still the length of the arc. It is so weird that with a circle I did not grab this concept but with an arc I did, so the more "arcy" the path to traverse is, the more the coin will rotate extra 🤗
@@mathaha2922 btw, this also made me realize that when instead of traveling the outer surface of a circle we travel the inner surface of a circle, the coin rotates ZERO times if they both have the same diameter! Same logic, just substracting 1 instead of adding and for any arc this just means substracting the arcness, which for a circle is the value 1. Going even further we could calculate the arcness of any spline and calculate the number of rotations a coin would travel along the spline. Interesting problem perhaps? Calculating the arcness of a spline?
What's interesting to me is that this is a perspective issue: it makes more sense if the coins stay still and you just rotate both coins. When the moving coin is above the stationary coin, the top of the moving coin is pointing away from the stationary coin. When the moving coin is below the stationary coin, the top of the moving coin is pointing at the stationary coin. This is the same, rotationally speaking, as if you just rotated the moving coin in place half a full rotation. With this in mind, it is easy to see that if you fix your perspective such that the coins only rotate, you see that the path is really just both coins rotating once at the same time.
I read an answer on quora and my mind blasted could not process it even after reading it twice, and after just 4 minutes of explanation this dude perfectly explained it. Thank you 🙏
@@howieduin915 He had a better Imagination And Nailed it that way too!! Yes table thing could have helped people understand better But still he did his best .
Hi Brett! I showed my brother this video today. He enjoyed it just as much as I did when I first watched it. Thank you! My brother added some of hos own ideas/explanations. I told him to share them with you himself because I don't think I can repeat them correctly. Your videos are great! Keep 'em coming!
I agree. This guy doesn't provide the best example, a much bigger coin than the one rotating. If you rolled a quarter around a coin the size of the earth it would be obvious that more than one rotation would be needed to reach 360 degrees, or to "return home," as this guy says. Understanding the paradox is easier when testing with a much bigger diameter coin, not a speck. A quarter coin revolving around its own center reaches home after one rotation. If you increase the size of the center point to a quarter it takes two rotations... a car tire, many rotations... an earth size coin, countless rotations! Rotations have reference points called 'centers' and you need to enlarge the radius of the center to understand the change in number of rotations needed to return home.
You need to put a mark at the bottom of the smaller circle where it touches the larger circle. THEN rotate the small coin around the larger one counting how many times the mark touches the larger coin.
As the rolling coin goes around the stationary coin their surfaces match just like a zipper with teeth ("no slippage") or 2 gears in a machine. If the rolling coin goes around a coin 20x its own circumference then it's 20 rolls. In this case it's 1:1, so 1 roll. If the coin goes around a dot in space (not a coin, at all), think of the coin going around a thumbtack in a corkboard..then the rolling coin rotated 1x. (As the host says, it's number of rolls PLUS ONE for its own rotation!)
Thanks to your video I think I get it. I would visualize it in a slightly different way. Start rolling the coin along the wire and stop half way through. The image on the coin is now upsidedown. Now rotate the wire along the second staitionary coin to show how it's position changes to right side up without covering any distance on the wire.
Wow. Similar to the question “does a planet in tidal lock rotate?” Would seem the answer is No. But it does rotate 1 time per revolution about its sun.
What? No... there is a leap year simply because one exact orbit of the sun does not land exactly on an integer amount of day rotations. The action of revolving while rotating does not inherently mean there would be a non integer solution. This wouldnt even apply to an orbit anyway as the whole premise comes from an object rotating without slipping due to friction. The added rotation required had no connection with why a leap year is needed.
The non-stationary coin (coin 1) doesn't travel the circumstance of the stationary coin (coin 2). Coin one travels on its canter of mass, to find the circumstance in which it travels you use (R+r)Pi. To simplify everything cancels out to make it (R+r)/r. In this case twice coin 1s R.
Really be honest and think about this: Would you approach this most difficult mind bending question, an anamoly of nature that it is if the question was put this way: How many times would this quarter (A) rotate around itself, that is around it's own center, after it revolves around the stationary quarter (B) just one time by means of rolling around it.
It's a trick. When you rotate the turning coin to the bottom, you haven't actually gone one full rotation. You've only gone 1/2 rotation, while turning the 2-coin assembly upside down. Try doing it again, while making a small mark somewhere on the rotating coin. And put that mark so that it is touching the other coin. Now turn that coin. When you get to the bottom (half-way) point, it looks like a full rotation but that's only because as you turned the one coin, you've also turned the whole thing upside down. Look where the mark is at that halfway point. Is it back where it started touching the stationary coin? Now, It is now on the side OPPOSITE the edge that touches the other coin. That's only a HALF rotation. Now do this again while holding both coins stationary. The top one stays on top, and the bottom one stays on bottom. You are doing the same thing, you're just not letting the whole thing turn upside down. But what happens? BOTH COINS HAVE TO TURN. That's what's happening. When you count the first rotation, you're counting BOTH COINS' ROTATIONS. So when you are counting the rotations when you held the bottom coin stationary and moved the top one around it was in fact 1/2 turn (as you can see with the mark) of one coin, and 1/2 turn OF THE OTHER COIN. 1/2 + 1/2 = 1. So it looks like you have one rotation, when in fact you don't, which you can see by observing the mark.
I must say that this is not a Paradox at all . Its just somewhat intuitive to me and i was able to find the answer when I first saw this problem and it was a good one.......................just used the CONCEPTS OF ROTATIONAL BODY DYNAMICS AND PURE ROLLING i read all the comments and some people were talking that it will be very interesting and tough as well to solve the same question in generalized manner SUCH AS CONSIDERING RADII AS R1 FOR INNER CIRCLE AND R2 FOR OUTER ONE ......................I HAVE CALCULATED ANSWER FOR THAT PART ALSO The answer is (R1+R2)/R2
I think of it as two gears of equal sizes. If one gear turns once so does the other gear, however in the case of the two coins one is stationary so the one in motion steals the stationary coins single rotation. 1 to1 gear rotation.
Great explanation! Since I am the 366th like I just have to mention that this must mean that the Earth rotates 366 times in a year and 367 in a leap year 🤔
take a cut-out circle - no image or lettering - place it on a straight line that's equal in length to the circumference of the circle - mark the point that is touching (touch point) - roll the circle until the touch point is touching the circle again - which will be at the other end of the circle make the straight line a slight downward curve - roll the circle until it touches again - the touch point touches again at the end of the curved line make the straight line a full circle (center circle) - start the rolling circle at the top of the center circle - with the touch point touching the top of the center circle - roll the rolling circle about the now center circle - the touch point will be touching when the original circle is atop the center circle - the starting point in all of these experiments - the rolling circle rolled just once - and reached the other end of the line --- back to the straight line - place the rolling circle at the start of the line with the touch point touching the start of the line - add an image indicating up (the "up" point will be directly across of the touch point) - roll the circle across the straight line - it will reach the end of the line with the touch point is touching - and the up orientation will be directly upwards - the image will have rotated on it's axis 1 time - along with the touch point make the straight line a slight downward curve - start the rolling circle with the touch point touching - and the orientation directly upwards - roll the rolling circle along the curved line - the rolling circle will end with the touch point touching - but the orientation will not be straight up - it will have gone past being vertical make the straight line a full circle (center circle) - start the rolling circle at the top of the center circle - with the touch point touching the top of the center circle and the orientation directly up - roll the rolling circle around the center circle - the touch point will be touching when the original circle is atop the center circle - but the orientation has gone past exactly 2 times - in other words - it has "rotated on its axis" 2 times -- note that in the latter case - when the rolling circle is at the bottom of the center circle - the touch point is on the opposite side of the center circle - the top of the rolling circle is now touching the circle - this is the opposite of how the rolling circle started out - imagine straightening out the center circle's edge into a straight line - the rolling circle would be upside down on the straight line - exactly halfway to the end of the straight line what's happened? we see imaged or lettered rolling circles as oriented on the edge of the paper or screen - but the rolling circle is actually oriented to the line on which it is rolling - we are seeing an illusion - which is why the number of image rotations changes - while the touch point counts remain the same
@@mathaha2922 - i reread my post and found i made some errors - i corrected them - but it really needs to be setup visually like yours and the many others
This puzzle is counterintuitive. The outer coin travels the inner coin (linear distance) and also travels around itself (circular distance). When there is no inner coin it only travels around itself but not the linear distance. When there is a linear path it only travels linear distance but not around itself. This puzzle has circular and linear motion but we miss the circular part at first and it seems a paradox. If the inner coin is 100 times or 50 times the linear distance changes but circular distance of outer coin is always the same.
Just consider the center of gravity, the center of the circle, of the piece which is in rotation. If its radius is r and turn around a circle of radius R, then the circle described by the center of the circle in rotation is not 2 pi R but 2 pi (r+R). So, it takes that length as equivalent length to travel. Divide by 2 pi r to get the number of turns, you obtain (r + R) / r or 1+ R/r turns.
YOUR REFERENCING IS WAY OFF BASE Sorry, but you couldn't possibly be more mistaken. This problem, as it is presented, must of course be seen as being in 2D space that surrounds the coins which are NOT strung out from each other but are instead continuously in contact with each other and, THEREFORE, it is the BEGINNING and ENDING CONTINUOUSLY MOVING MUTUAL CONTACT POINT of the outer edge circumference of both coins that MUST, therefore, be the only thing considered RELATIVISTICALLY TO EACH OTHER in order to rightfully have considered either coin to ACTUALLY have completed that singular(((ONE TIME ONLY))) Rotation/Revolution TRAVEL of that mutual and constantly CHANGING POINT OF CONTACT between those two De Facto INTERLOCKING GEARS of the same size and shape. The next time you try that identical coin experiment, simply concentrate on the ONLY thing that you SHOULD have been considering from the beginning, which of course should only have been that CONSTANTLY CHANGING POINT OF A DE FACTO GEAR CONTACT POINT and avoid all those false and IRRELEVANT parameters. ONE ROTATION/REVOLUTION TRAVEL OF THE MUTUALLY MOVING POINT OF CONTACT BETWEEN TWO IDENTICAL GEARS WILL ALWAYS ENCOMPASS JUST ONE AND (((ONLY))) ONE CIRCUMFERENCE LENGTH.
The COM of the coin goes around a distance of 4πr and not 2πr and since it is pure rolling so that means that if the COM travels 4πr distance then the coin should rotate twice.
Solution The circumference formula is 2*pi*r, coin 1 rotates around the circumference of coin 2 to find the distance that coin 1 will travel. You must find the circumference of a circle that's center is coin 2's center but also intersects coin 1's center. To do this, all you need to do is add the radius of coin 1 to the radius of coin 2, then you will have the distance between coin 1's center and coin 2's center, then you plug in the distance between the two centers as your radius for the circumference of the circle that coin 1 will be following. To find the number of rotations it will take to land back where you started, all you need to do is divide the circumference of the circle that coin 1 will be following with the circumference of coin 1 and you will be left with the exact number of rotations needed to return to your start point. Paradox The reason this feels like a paradox is because most people think of finding the circumference of the center coin and the circumference of the rotating coin and then dividing to get the number of rotations needed. But this doesn't work because that equation would only work if the center of the rotating coin is on the circumference or meets with the end of the radius of the center coin. (Basically, your thinking the distance that the coin will rotate is the circumference of the center coin, but that couldn't be the case because it never follows the circumference of the center coin. The rotating coin follows the circumference of an invisible circle that has a radius of the sum of the 2 coins radii.)
I don’t get it. But in trying to get it, something tells me that if they’re both free to rotate such that both centres do not move, then two coins of the same diameter just turn the distances of their circumference (as though rolling on a flat surface expects). This is all that feels right after not understanding the problem or the solution.
If i didn't already understand this paradox I don't think i would have been able to understand from this video but that is maybe because i am a visual learner and good graphic animation or at least a whiteboard is needed for my learning style.
Alice Ant and Bob Ant are together on the rims of their respective equally sized coins where they touch. If Bob's coin rolls round Alice's how many tours of their respective coins must each make to stay with the other?
Nope Nope, Nopey McNope Nuh Uh. The coin has to make the same number of rotations for the straight line as for the circle. You have to look @1.40 and compare that to @1:53 to see the error mathAHA is making. He correctly says that the eagle is in the standing position (head pointing skyward) @1:40 and again @1:53, and he incorrectly assumes that this means the coin has made one complete rotation (he actually means revolution but we will deal with definitions later) Even though the eagles head is facing up at both times noted, upon close observation, you will see that @1:40 the eagles feet are touching the second coin and @1:53 the eagles head is touching the other coin - this means there has only been 1/2 revolution. If you keep going until the eagle's feet once again touch the coin then that will be one complete revolution. ROTATION - Spinning on it's own axis. REVOLUTION - one object moving around another object
All good..but why hold all the pieces up vertically?....why not place both coins on a flat table and show the same thing?....would think it shows better that way without your fingers in the way.
I'm a mechanical engineer, not that it should count for anything here. My answer is 3. The examiner was correct. That was why Einstein postulated the relativity theory. It's all about where the observer is. When you observe the rotation of the small circle from a distance, you assume that it rotated in the first quarter... But it did not. It was due to the curve track of the second circle that the smaller circle appears to have completed a full rotation. If you were the small circle or even on it, you would experience only 3 rotations around the bigger circle and not 4. Imagine you are on earth and roll a ball or circle around it. Because up is always away from the centre of the earth, the observer on earth would always get 3, not 4. Mathematically it also makes sense -the circumferences are at a ratio of 1 to 3. To proof what I stated above; the examiner did not say that the small circle must be rolled around the big one, nor did the examiner state that the bigger circle is stationary. Please repeat your experiment but now, rotate the bigger circle one full rotation. You will see that the smaller one will only rotate 3 times and not 4.
Your point scenario is not quite right I don't think. Imagine a clock hand turning by the time you get to three O'clock it is at a 90 degree orientation to its starting point. Two coins of the same size one fixed and the other rotating around it is like the clock hand with added rotation. Half the orientation is a product of its angle to the circle and the other half is that it will have travelled half it's circumference. Despite having turned a full rotation the two coins have only touched along half their circumference. On a straight line half a coins circumference will leave it upside down. The curve of the semi circle of the static coin provides half the orientation.
This is relative to a system of mathematics that I developed that has no zero. The Alpha theorem is A + 1 divided by 2 times A equal X. A is any whole # into eternity as far as you dare. X is A and every whole # less than A added together. Unlimited numbers into eternity added in milliseconds.. A long with five other theorems I can add, subtract, multiply, and divided any combination of whole numbers into eternity instantly. 6 theorems. On the 7 day HE rested
Adding one to it is a bad reason. The simple and correct reason is the outer coin rotates around its own axis. Period. This draws the circumference that is the denominator. The numerator is the circumference of the outer coin. Divide and that’s your answer.
This is just blowing my mind so much I'm having such a hard time wrapping my head around it! I'm thinking like this, if you had two equal size wheels each with 360 equally spaced and interlocking teeth, then as you rotated one wheel around the other, then shouldn't the teeth always interlock with a ratio of 1:1? (Or any specific number of teeth; I'm just using 360 because it relates to 360 degrees of a circle.) Or I guess there's some kind of flaw in the way that I'm thinking of it??? EDIT: While your wonderful video is from 7 years ago, I found a video from less than a year ago that uses gears: ua-cam.com/video/lGA6Ivvf2Pw/v-deo.htmlfeature=shared
the math answer is : Number of Rotations = 1 + (R/r) R = Radius of Coin that is NOT rotating, r = Radius of Coin doing the ROTATION.. Ex_ if R = r then N rotations = 1+ (R/R) = 1+1 = 2 ....... What if r = 0 ? then N = 1 + R/0 = 1 + INFINITY....= Infinity... go figure
It is not a paradox. The answer is not 1 more. The observer thinks the coin rotates completely when it reaches the bottom of the stationary coin when, in fact, it rotated itself just half way, using up just half the string if it were being unspooled from the rotating coin. Because of the reference point of the observer (you), you are duped merely because the rotating coin appears again "upright" at the moment it traveled just halfway around the stationary coin. If you were an observer tracking the rotating coin from the center of the stationary coin, it becomes obvious that the rotating coin has, in fact, rotated just half way. Using two clocks of equal size, the rotating clock would be tangental to the stationary clock at the rotating clock's 6 O'clock point at the start. At the point it appears to have rotated once, it is now tangental to the stationary clock at the rotating clock's 12 O'clock position. It still needs to travel 6 more hours before completing it's first full rotation.
So all this is saying is that the total number of revolutions the coin makes is equal to the number of revolutions about its center plus the number of revolutions about the other coin.
That's a fair comment. This video is more about removing our cloudy understanding of what we mean when we say "one coin rotates around another" and less about the mathematical rigor one should also certainly pursue as regards this question. Thanks for the comment!
Wrong. At the start, the contact point of the rotating coin is it's bottom. When you start rotating it and it looks upright again, the contact point is the TOP of the coin. The coin it upright, but it has done only half a revolution. When the coin arrives at its starting point, the contact point is the bottom point again. To solve this, you simply need to draw dots on both coins at the starting point. The problem is a fantastic illusion. You confuse the UPRIGHT state of the coin with the true CONTACT point.
It’s about the angle of rotation. Similar to Aristotle’s Paradox. Drop the small circle to the bottom and it will not track the same distance. It’s the difference in angle to the surface of the inner circle.
It is not a paradox... Paint the center coin... stop when the rotating coin is fully painted. Rotation of a circle is always measured from the circle's center and a central theta angle... You are measure in terms of an outside, Euclidian reference frame... You are using the wrong coordinate system. Stop fooling children.
Is not fooling children. It's a very interesting "problem" that leads to a better analysis and understanding of rotation in circumferences. Instead of just strongly denying something, we could take some minutes to analyze what the other person is trying to communicate and trying to make us see.
I wonder how this relates to the quantum spin-1/2 Fermion statistics and Pauli exclusion principle. Also recommend to take a look at ua-cam.com/video/1VPfZ_XzisU/v-deo.html
Do you know what paradox is? I don't think so. What you got here is word/lingo game. It is worded to mislead then call it paradox. Don't whisper and don't skip words: You do it unknowingly and the people who came up with it, I am more than certain that they did it deliberately. I don't see anyone saying it this way and if you did or the ones who came up with it said it this way, it would not have become some big crap to figure out. READ CAREFULLY HERE AND THEN WATCH HOW YOU NEVER SAID IT THIS WAY: [[ This quarater rotates around the center quarter one time. In doing so, the question is how many times it will rotate around IT'S OWN CENTER OF AXIS]]
This type of paradox is known as a "veridical" paradox. In such paradoxes there is no contradiction, just a _seeming_ contradiction. Thanks for your comment.
@@mathaha2922 I am sure you have seen that question about a small circle 1/3 diameter of the big circle, making a one trip around it. How many times the small circle will revolve. Notice, the simple seeming straight forward word, "revolve". I answered 1 time. That is right the small circle revolves around the big circle one time if it makes one trip. Anyone tries to argue with me will instantly show what they nicely skipped. And the ones who which are the vast majority answering 3 times, those vast poor souls ( which are not poor souls, they play the civility role, the respectibility role, while I am the abrasive one), these folks took on the brunt of it as they interpreted it points of contact just like your quarter deal which of course those clever wise know hows take full advantage of it. They know how people think and use use words just enough to leave the sheep to the point of assumption. And even the bright ones... will swear up and down it revolves 3 times and the quarter revolves one time. They all drink from the sewege of point of contact. I come around and tell the mainupulators who are also self-minupulating just the same and tell them to go back to school and learn how to state your question in the most clearly manner as possible so no one mis interpret it other than what you are really getting at. Your quarter revolves around the center quarter one time. Period. Even a child can tell you that. Your moving quarter revolves around its own center 2 times. In fact if this is the true intention, why the hell bring in the other stationary quarter in the picture in the first place? Think about that. It is to mislead, it is no different than magic trick. Get them to pay attention to something else, the center quarter is used as a decoy, slide of hand, distraction. And it should be told. And I am the abrassive asshole.
Because of this explanation I finally understand this weird paradox which isnt weird at all! The thing is that the distance a point on the coin moves is still only one time the circumference, but because the coin travels on the surface of one complete circle, the coin rotates one "extra" circle.
If we would simply take an arc instead of a circle and the arc has the length of the circumference of the circle, we can better understand this principle!! Because in case of an arc we understand that the coin also traverses over a curved surface and hence that adds to the rotation of the coin, but, and a very important but: the distance traversed by a single point on the circle is still the length of the arc. It is so weird that with a circle I did not grab this concept but with an arc I did, so the more "arcy" the path to traverse is, the more the coin will rotate extra 🤗
Thanks for your comment! It was interesting to read about your own way of understanding the problem.
@@mathaha2922 btw, this also made me realize that when instead of traveling the outer surface of a circle we travel the inner surface of a circle, the coin rotates ZERO times if they both have the same diameter! Same logic, just substracting 1 instead of adding and for any arc this just means substracting the arcness, which for a circle is the value 1.
Going even further we could calculate the arcness of any spline and calculate the number of rotations a coin would travel along the spline. Interesting problem perhaps? Calculating the arcness of a spline?
I made a very long explanation video, maybe you want to see it?
ua-cam.com/video/lGA6Ivvf2Pw/v-deo.html
THIS IS THE BEST EXPLANATION THAT ONE CAN GET ON COIN PARADOX.
Thank you for the kind comment!
Superior explanation! Especially the one with the circle with zero diameter!
Thanks for the comment! Am glad to hear it!
Yes - The ‘zero’ radius/ diameter coin clarifies the matter.
@@richardgreenlees7088 Glad to hear it!
Yes!! The zero-circumference explanation is what made me finally understand this phenomenon!
What's interesting to me is that this is a perspective issue: it makes more sense if the coins stay still and you just rotate both coins. When the moving coin is above the stationary coin, the top of the moving coin is pointing away from the stationary coin. When the moving coin is below the stationary coin, the top of the moving coin is pointing at the stationary coin. This is the same, rotationally speaking, as if you just rotated the moving coin in place half a full rotation. With this in mind, it is easy to see that if you fix your perspective such that the coins only rotate, you see that the path is really just both coins rotating once at the same time.
Excellent point!
I read an answer on quora and my mind blasted could not process it even after reading it twice, and after just 4 minutes of explanation this dude perfectly explained it.
Thank you 🙏
Same 🤣🤣
Glad the video was helpful!
Very clear explanation. Too bad he didn't have a table to demonstrate on. The fact that he was doing all that in his hands was driving me nuts lol.
@@howieduin915 He had a better Imagination And Nailed it that way too!!
Yes table thing could have helped people understand better But still he did his best .
Hi Brett! I showed my brother this video today. He enjoyed it just as much as I did when I first watched it. Thank you! My brother added some of hos own ideas/explanations. I told him to share them with you himself because I don't think I can repeat them correctly. Your videos are great! Keep 'em coming!
Hi Mareike! Great to hear! Thank you for the very kind comment and say hi to your brother.
You sir, saved my day! I was going crazy about this until I found this explanation
Glad the video could be of help!
the explanation about the rotation of the coin around the microscopic point is super !!
Glad to hear it!
If you use the point of tangency as the indication of a full revolution instead of a vertical reference the revolutions will be one less.
Best explanation yet. I’m a visual learner and you showed the reason perfectly. 💡
Great teacher - Great communicator - Great statesman!
Thanks!
I watched this to try understand why it was a paradox in the first place.
I hope the video could at least help with that question.
This is a really good explanation. I like your thinking and practical explanation.
Thanks for the kind comment!
Crazy explanation. Loved the effort! It'll really be a good exercise to extend this to circles of different radii.
Glad it was helpful!
I agree. This guy doesn't provide the best example, a much bigger coin than the one rotating. If you rolled a quarter around a coin the size of the earth it would be obvious that more than one rotation would be needed to reach 360 degrees, or to "return home," as this guy says. Understanding the paradox is easier when testing with a much bigger diameter coin, not a speck. A quarter coin revolving around its own center reaches home after one rotation. If you increase the size of the center point to a quarter it takes two rotations... a car tire, many rotations... an earth size coin, countless rotations! Rotations have reference points called 'centers' and you need to enlarge the radius of the center to understand the change in number of rotations needed to return home.
This man can explain Rocket Science too!
Thanks for the comment!
Seriously! I was plucking out my hair, thanks a ton! ❤
You need to put a mark at the bottom of the smaller circle where it touches the larger circle.
THEN rotate the small coin around the larger one counting how many times the mark touches the larger coin.
Why?
Exactly
You nailed it! It is relativity to the center of the coin.
Glad to hear it!
So, when can we expect your solution to the Riemann Hypothesis or the Goldbach Conjecture? 😊
As the rolling coin goes around the stationary coin their surfaces match just like a zipper with teeth ("no slippage") or 2 gears in a machine. If the rolling coin goes around a coin 20x its own circumference then it's 20 rolls. In this case it's 1:1, so 1 roll. If the coin goes around a dot in space (not a coin, at all), think of the coin going around a thumbtack in a corkboard..then the rolling coin rotated 1x. (As the host says, it's number of rolls PLUS ONE for its own rotation!)
Thanks to your video I think I get it. I would visualize it in a slightly different way. Start rolling the coin along the wire and stop half way through. The image on the coin is now upsidedown. Now rotate the wire along the second staitionary coin to show how it's position changes to right side up without covering any distance on the wire.
Good idea. Thanks for the comment!
Even when viewing it-my brain tells me you’re only going halfway around when you’re saying its 1
I know what you mean. This question can make your head spin!
Super good job explaining.
Thanks!
finally a good explanation!
glad to hear it!
Noooo... I don't think it is right.
Wow. Similar to the question “does a planet in tidal lock rotate?” Would seem the answer is No. But it does rotate 1 time per revolution about its sun.
Revolution vs rotation...hmmm. . I think you just explained why we have a leap year every 1461 days. 🤔 Keep the videos coming.
Will do! :)
What? No... there is a leap year simply because one exact orbit of the sun does not land exactly on an integer amount of day rotations.
The action of revolving while rotating does not inherently mean there would be a non integer solution. This wouldnt even apply to an orbit anyway as the whole premise comes from an object rotating without slipping due to friction. The added rotation required had no connection with why a leap year is needed.
no nothing like that
The non-stationary coin (coin 1) doesn't travel the circumstance of the stationary coin (coin 2). Coin one travels on its canter of mass, to find the circumstance in which it travels you use (R+r)Pi. To simplify everything cancels out to make it (R+r)/r. In this case twice coin 1s R.
Really be honest and think about this:
Would you approach this most difficult mind bending question, an anamoly of nature that it is if the question was put this way:
How many times would this quarter (A) rotate around itself, that is around it's own center, after it revolves around the stationary quarter (B) just one time by means of rolling around it.
Great job! Still making me smarter!
Thank you! Although if this is the Christopher Salinas I know, it is certainly you that always made me smarter!
Very interesting! Nice videos.
Thanks! Glad you like the videos -- that makes it worth it to make them!
Thank you so much for this excellent explanation!
You are welcome!
It's a trick. When you rotate the turning coin to the bottom, you haven't actually gone one full rotation. You've only gone 1/2 rotation, while turning the 2-coin assembly upside down.
Try doing it again, while making a small mark somewhere on the rotating coin. And put that mark so that it is touching the other coin. Now turn that coin. When you get to the bottom (half-way) point, it looks like a full rotation but that's only because as you turned the one coin, you've also turned the whole thing upside down. Look where the mark is at that halfway point. Is it back where it started touching the stationary coin? Now, It is now on the side OPPOSITE the edge that touches the other coin. That's only a HALF rotation. Now do this again while holding both coins stationary. The top one stays on top, and the bottom one stays on bottom. You are doing the same thing, you're just not letting the whole thing turn upside down. But what happens? BOTH COINS HAVE TO TURN. That's what's happening. When you count the first rotation, you're counting BOTH COINS' ROTATIONS.
So when you are counting the rotations when you held the bottom coin stationary and moved the top one around it was in fact 1/2 turn (as you can see with the mark) of one coin, and 1/2 turn OF THE OTHER COIN. 1/2 + 1/2 = 1. So it looks like you have one rotation, when in fact you don't, which you can see by observing the mark.
100%
I must say that this is not a Paradox at all . Its just somewhat intuitive to me and i was able to find the answer when I first saw this problem and it was a good one.......................just used the CONCEPTS OF ROTATIONAL BODY DYNAMICS AND PURE ROLLING
i read all the comments and some people were talking that it will be very interesting and tough as well to solve the same question in generalized manner
SUCH AS CONSIDERING RADII AS R1 FOR INNER CIRCLE AND R2 FOR OUTER ONE ......................I HAVE CALCULATED ANSWER FOR THAT PART ALSO
The answer is (R1+R2)/R2
U CAN ALSO USE CONCEPT OF IAOR (INSTANTANEOUS AXIS OF ROTATION )
Thanks! The zero radius drove it home for me.
You are welcome!
I think of it as two gears of equal sizes. If one gear turns once so does the other gear, however in the case of the two coins one is stationary so the one in motion steals the stationary coins single rotation. 1 to1 gear rotation.
An enlightening way of looking at it! Thanks!
Clear explanation. Thank you!
You are welcome!
I know it's 2 but if you go by the contact points or the point of view of the stationary coin, it's 1.
VERY good video!
Thank you very much for the kind comment!
Great explanation! Since I am the 366th like I just have to mention that this must mean that the Earth rotates 366 times in a year and 367 in a leap year 🤔
Thanks for your comment!
Which coin rotates around which???
take a cut-out circle - no image or lettering - place it on a straight line that's equal in length to the circumference of the circle - mark the point that is touching (touch point) - roll the circle until the touch point is touching the circle again - which will be at the other end of the circle
make the straight line a slight downward curve - roll the circle until it touches again - the touch point touches again at the end of the curved line
make the straight line a full circle (center circle) - start the rolling circle at the top of the center circle - with the touch point touching the top of the center circle - roll the rolling circle about the now center circle - the touch point will be touching when the original circle is atop the center circle - the starting point
in all of these experiments - the rolling circle rolled just once - and reached the other end of the line
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back to the straight line - place the rolling circle at the start of the line with the touch point touching the start of the line - add an image indicating up (the "up" point will be directly across of the touch point) - roll the circle across the straight line - it will reach the end of the line with the touch point is touching - and the up orientation will be directly upwards - the image will have rotated on it's axis 1 time - along with the touch point
make the straight line a slight downward curve - start the rolling circle with the touch point touching - and the orientation directly upwards - roll the rolling circle along the curved line - the rolling circle will end with the touch point touching - but the orientation will not be straight up - it will have gone past being vertical
make the straight line a full circle (center circle) - start the rolling circle at the top of the center circle - with the touch point touching the top of the center circle and the orientation directly up - roll the rolling circle around the center circle - the touch point will be touching when the original circle is atop the center circle - but the orientation has gone past exactly 2 times - in other words - it has "rotated on its axis" 2 times
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note that in the latter case - when the rolling circle is at the bottom of the center circle - the touch point is on the opposite side of the center circle - the top of the rolling circle is now touching the circle - this is the opposite of how the rolling circle started out - imagine straightening out the center circle's edge into a straight line - the rolling circle would be upside down on the straight line - exactly halfway to the end of the straight line
what's happened?
we see imaged or lettered rolling circles as oriented on the edge of the paper or screen - but the rolling circle is actually oriented to the line on which it is rolling - we are seeing an illusion - which is why the number of image rotations changes - while the touch point counts remain the same
Thanks for the detailed explanation!
@@mathaha2922 - i reread my post and found i made some errors - i corrected them - but it really needs to be setup visually like yours and the many others
This puzzle is counterintuitive.
The outer coin travels the inner coin (linear distance) and also travels around itself (circular distance).
When there is no inner coin it only travels around itself but not the linear distance.
When there is a linear path it only travels linear distance but not around itself.
This puzzle has circular and linear motion but we miss the circular part at first and it seems a paradox.
If the inner coin is 100 times or 50 times the linear distance changes but circular distance of outer coin is always the same.
I was trying to figure out what's
perplexing and what even is the problem,
This answered that.
Thank you 😎
P.S My mom took me out at 3rd grade😕
No home school☹️
You are welcome!
Just consider the center of gravity, the center of the circle, of the piece which is in rotation. If its radius is r and turn around a circle of radius R, then the circle described by the center of the circle in rotation is not 2 pi R but 2 pi (r+R). So, it takes that length as equivalent length to travel. Divide by 2 pi r to get the number of turns, you obtain (r + R) / r or 1+ R/r turns.
That's it! You nailed it.
This. Frames of reference kids.....
YOUR REFERENCING IS WAY OFF BASE
Sorry, but you couldn't possibly be more mistaken. This problem, as it is presented, must of course be seen as being in 2D space that surrounds the coins which are NOT strung out from each other but are instead continuously in contact with each other and, THEREFORE, it is the BEGINNING and ENDING CONTINUOUSLY MOVING MUTUAL CONTACT POINT of the outer edge circumference of both coins that MUST, therefore, be the only thing considered RELATIVISTICALLY TO EACH OTHER in order to rightfully have considered either coin to ACTUALLY have completed that singular(((ONE TIME ONLY))) Rotation/Revolution TRAVEL of that mutual and constantly CHANGING POINT OF CONTACT between those two De Facto INTERLOCKING GEARS of the same size and shape.
The next time you try that identical coin experiment, simply concentrate on the ONLY thing that you SHOULD have been considering from the beginning, which of course should only have been that CONSTANTLY CHANGING POINT OF A DE FACTO GEAR CONTACT POINT and avoid all those false and IRRELEVANT parameters.
ONE ROTATION/REVOLUTION TRAVEL OF THE MUTUALLY MOVING POINT OF CONTACT BETWEEN TWO IDENTICAL GEARS WILL ALWAYS ENCOMPASS JUST ONE AND (((ONLY))) ONE CIRCUMFERENCE LENGTH.
The COM of the coin goes around a distance of 4πr and not 2πr and since it is pure rolling so that means that if the COM travels 4πr distance then the coin should rotate twice.
Thanks for the comment.
He replied.
Now this is epic.
@@mathaha2922 Hats off Sir your work is amazing keep it up.
@@prakharpandey2392 Thanks again!
Solution
The circumference formula is 2*pi*r, coin 1 rotates around the circumference of coin 2 to find the distance that coin 1 will travel. You must find the circumference of a circle that's center is coin 2's center but also intersects coin 1's center. To do this, all you need to do is add the radius of coin 1 to the radius of coin 2, then you will have the distance between coin 1's center and coin 2's center, then you plug in the distance between the two centers as your radius for the circumference of the circle that coin 1 will be following. To find the number of rotations it will take to land back where you started, all you need to do is divide the circumference of the circle that coin 1 will be following with the circumference of coin 1 and you will be left with the exact number of rotations needed to return to your start point.
Paradox
The reason this feels like a paradox is because most people think of finding the circumference of the center coin and the circumference of the rotating coin and then dividing to get the number of rotations needed. But this doesn't work because that equation would only work if the center of the rotating coin is on the circumference or meets with the end of the radius of the center coin. (Basically, your thinking the distance that the coin will rotate is the circumference of the center coin, but that couldn't be the case because it never follows the circumference of the center coin. The rotating coin follows the circumference of an invisible circle that has a radius of the sum of the 2 coins radii.)
I don’t get it. But in trying to get it, something tells me that if they’re both free to rotate such that both centres do not move, then two coins of the same diameter just turn the distances of their circumference (as though rolling on a flat surface expects). This is all that feels right after not understanding the problem or the solution.
Rolling along a flat surface will get you a different answer...
Small coin rotates around large coin. Formula: (R + r)/r. R = large radius. r = small radius.
Yes, I suppose that is another way of saying R/r + 1. Thanks for the comment!
Great, thank you. However it would have been a lot easier to demontrate on a flat horizontal surface..
Thanks for the feedback. You are probably right!
Not really, no.
If i didn't already understand this paradox I don't think i would have been able to understand from this video but that is maybe because i am a visual learner and good graphic animation or at least a whiteboard is needed for my learning style.
Thanks for watching all the same!
@@mathaha2922 not detracting from you at all, i know that it takes a lot of tech savvy, time and effort to make a video of the sort I am referring to.
You are confusing coin orientation with 360 degrees of rotation.
Exactly.
It's all about perspective 😮
That is true. If you stood at the center of the moving coin, you would have a different answer. Thanks for your comment!
Alice Ant and Bob Ant are together on the rims of their respective equally sized coins where they touch. If Bob's coin rolls round Alice's how many tours of their respective coins must each make to stay with the other?
Nope Nope, Nopey McNope Nuh Uh. The coin has to make the same number of rotations for the straight line as for the circle.
You have to look @1.40 and compare that to @1:53 to see the error mathAHA is making.
He correctly says that the eagle is in the standing position (head pointing skyward) @1:40 and again @1:53, and he incorrectly assumes that this means the coin has made one complete rotation (he actually means revolution but we will deal with definitions later)
Even though the eagles head is facing up at both times noted, upon close observation, you will see that @1:40 the eagles feet are touching the second coin and @1:53 the eagles head is touching the other coin - this means there has only been 1/2 revolution. If you keep going until the eagle's feet once again touch the coin then that will be one complete revolution.
ROTATION - Spinning on it's own axis.
REVOLUTION - one object moving around another object
All good..but why hold all the pieces up vertically?....why not place both coins on a flat table and show the same thing?....would think it shows better that way without your fingers in the way.
Thanks for the sensible suggestion!
I'm a mechanical engineer, not that it should count for anything here. My answer is 3. The examiner was correct. That was why Einstein postulated the relativity theory. It's all about where the observer is. When you observe the rotation of the small circle from a distance, you assume that it rotated in the first quarter... But it did not. It was due to the curve track of the second circle that the smaller circle appears to have completed a full rotation. If you were the small circle or even on it, you would experience only 3 rotations around the bigger circle and not 4. Imagine you are on earth and roll a ball or circle around it. Because up is always away from the centre of the earth, the observer on earth would always get 3, not 4. Mathematically it also makes sense -the circumferences are at a ratio of 1 to 3. To proof what I stated above; the examiner did not say that the small circle must be rolled around the big one, nor did the examiner state that the bigger circle is stationary. Please repeat your experiment but now, rotate the bigger circle one full rotation. You will see that the smaller one will only rotate 3 times and not 4.
awesome explanation.....
glad to hear it.....
Your point scenario is not quite right I don't think. Imagine a clock hand turning by the time you get to three O'clock it is at a 90 degree orientation to its starting point. Two coins of the same size one fixed and the other rotating around it is like the clock hand with added rotation. Half the orientation is a product of its angle to the circle and the other half is that it will have travelled half it's circumference. Despite having turned a full rotation the two coins have only touched along half their circumference. On a straight line half a coins circumference will leave it upside down. The curve of the semi circle of the static coin provides half the orientation.
Good explanation but what an incredibly difficult way to video it. Put the coins on a table and film from above!!!
Thanks for the good suggestion!
This is relative to a system of mathematics that I developed that has no zero. The Alpha theorem is A + 1 divided by 2 times A equal X. A is any whole # into eternity as far as you dare. X is A and every whole # less than A added together. Unlimited numbers into eternity added in milliseconds.. A long with five other theorems I can add, subtract, multiply, and divided any combination of whole numbers into eternity instantly. 6 theorems. On the 7 day HE rested
Adding one to it is a bad reason. The simple and correct reason is the outer coin rotates around its own axis. Period. This draws the circumference that is the denominator. The numerator is the circumference of the outer coin. Divide and that’s your answer.
Thanks for your comment!
This is why there are 365.2422 solar days in a year but one more, 366.2422, sidereal days.
This is just blowing my mind so much I'm having such a hard time wrapping my head around it!
I'm thinking like this, if you had two equal size wheels each with 360 equally spaced and interlocking teeth, then as you rotated one wheel around the other, then shouldn't the teeth always interlock with a ratio of 1:1? (Or any specific number of teeth; I'm just using 360 because it relates to 360 degrees of a circle.) Or I guess there's some kind of flaw in the way that I'm thinking of it???
EDIT: While your wonderful video is from 7 years ago, I found a video from less than a year ago that uses gears: ua-cam.com/video/lGA6Ivvf2Pw/v-deo.htmlfeature=shared
Thanks for the comment and the link!
hmmm is it the same with ellipses?
Good question!
the math answer is : Number of Rotations = 1 + (R/r) R = Radius of Coin that is NOT rotating, r = Radius of Coin doing the ROTATION..
Ex_ if R = r then N rotations = 1+ (R/R) = 1+1 = 2 ....... What if r = 0 ? then N = 1 + R/0 = 1 + INFINITY....= Infinity... go figure
Ouch! My head hurts!! Never trust math formulas again. It is what it is unless it's not.
It is not a paradox. The answer is not 1 more. The observer thinks the coin rotates completely when it reaches the bottom of the stationary coin when, in fact, it rotated itself just half way, using up just half the string if it were being unspooled from the rotating coin. Because of the reference point of the observer (you), you are duped merely because the rotating coin appears again "upright" at the moment it traveled just halfway around the stationary coin. If you were an observer tracking the rotating coin from the center of the stationary coin, it becomes obvious that the rotating coin has, in fact, rotated just half way.
Using two clocks of equal size, the rotating clock would be tangental to the stationary clock at the rotating clock's 6 O'clock point at the start. At the point it appears to have rotated once, it is now tangental to the stationary clock at the rotating clock's 12 O'clock position. It still needs to travel 6 more hours before completing it's first full rotation.
Thanks for your insight!
So all this is saying is that the total number of revolutions the coin makes is equal to the number of revolutions about its center plus the number of revolutions about the other coin.
Thanks for your comment!
The eagle was standing but one rotation was not completed mr.
Are you perhaps referring to this question: When has the coin completed a rotation? Relative to the viewer or relative to the other coin?
still,I m not satisfied with the lack of mathematical rigour*
That's a fair comment. This video is more about removing our cloudy understanding of what we mean when we say "one coin rotates around another" and less about the mathematical rigor one should also certainly pursue as regards this question. Thanks for the comment!
Wrong. At the start, the contact point of the rotating coin is it's bottom. When you start rotating it and it looks upright again, the contact point is the TOP of the coin. The coin it upright, but it has done only half a revolution. When the coin arrives at its starting point, the contact point is the bottom point again. To solve this, you simply need to draw dots on both coins at the starting point. The problem is a fantastic illusion. You confuse the UPRIGHT state of the coin with the true CONTACT point.
Thanks for noticing what I noticed right away too. The fact this guy looks so smug while getting it so wrong is hilarious.
Sorry but I am not going to listen to your explanation because I want to solve it own my own no matter how much time does it take
That sounds like a good plan. Go for it!
@@mathaha2922 I will solve it tomorrow and will reply how much time it took
Inuse UA-cam comments like my personal diary
@@mastrammeena328 Will look forward to it. :)
It’s about the angle of rotation. Similar to Aristotle’s Paradox. Drop the small circle to the bottom and it will not track the same distance. It’s the difference in angle to the surface of the inner circle.
It is not a paradox... Paint the center coin... stop when the rotating coin is fully painted. Rotation of a circle is always measured from the circle's center and a central theta angle... You are measure in terms of an outside, Euclidian reference frame... You are using the wrong coordinate system. Stop fooling children.
Is not fooling children. It's a very interesting "problem" that leads to a better analysis and understanding of rotation in circumferences.
Instead of just strongly denying something, we could take some minutes to analyze what the other person is trying to communicate and trying to make us see.
Why?!!!
Why?
I wonder how this relates to the quantum spin-1/2 Fermion statistics and Pauli exclusion principle. Also recommend to take a look at ua-cam.com/video/1VPfZ_XzisU/v-deo.html
Do you know what paradox is?
I don't think so.
What you got here is word/lingo game.
It is worded to mislead then call it paradox.
Don't whisper and don't skip words:
You do it unknowingly and the people who came up with it, I am more than certain that they did it deliberately.
I don't see anyone saying it this way and if you did or the ones who came up with it said it this way, it would not have become some big crap to figure out.
READ CAREFULLY HERE AND THEN WATCH HOW YOU NEVER SAID IT THIS WAY:
[[ This quarater rotates around the center quarter one time. In doing so, the question is how many times it will rotate around IT'S OWN CENTER OF AXIS]]
This type of paradox is known as a "veridical" paradox. In such paradoxes there is no contradiction, just a _seeming_ contradiction. Thanks for your comment.
@@mathaha2922 I am sure you have seen that question about a small circle 1/3 diameter of the big circle, making a one trip around it. How many times the small circle will revolve.
Notice, the simple seeming straight forward word, "revolve".
I answered 1 time.
That is right the small circle revolves around the big circle one time if it makes one trip.
Anyone tries to argue with me will instantly show what they nicely skipped.
And the ones who which are the vast majority answering 3 times, those vast poor souls ( which are not poor souls, they play the civility role, the respectibility role, while I am the abrasive one), these folks took on the brunt of it as they interpreted it points of contact just like your quarter deal which of course those clever wise know hows take full advantage of it. They know how people think and use use words just enough to leave the sheep to the point of assumption.
And even the bright ones... will swear up and down it revolves 3 times and the quarter revolves one time.
They all drink from the sewege of point of contact.
I come around and tell the mainupulators who are also self-minupulating just the same and tell them to go back to school and learn how to state your question in the most clearly manner as possible so no one mis interpret it other than what you are really getting at.
Your quarter revolves around the center quarter one time. Period. Even a child can tell you that.
Your moving quarter revolves around its own center 2 times.
In fact if this is the true intention, why the hell bring in the other stationary quarter in the picture in the first place? Think about that.
It is to mislead, it is no different than magic trick. Get them to pay attention to something else, the center quarter is used as a decoy, slide of hand, distraction. And it should be told.
And I am the abrassive asshole.
This guy is awfully smug, but he's wrong.
I am sorry if I come across as smug. It would interest me why you think the explanation is wrong.
@@mathaha2922 The coin makes one full rotation around the other. You're making it appear as if it's two, based on sophistry.
Well explained but really terribly demonstrated
LOL. Thanks for the comment!
USE PAINT... THINK REFERENCE FRAME... there is no paradox!!!! STOP PROPAGATING NONSENSE!!!
Thanks for your comment. Unfortunately, I do not understand it.
Sorry but your explanation doesn't make sense.
No problem. Sorry to hear that!
@@mathaha2922 What I mean is that your explanation is what someone can see in Wikipedia but do you really grasp it?
So, the earth IS flat. I knew it!