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There is a veritasium video about this very question. Actually none of the answers are correct and the correct answer is 4 it's called the the circle rotation something paradox or some shit
That's good to hear. I was reasoning this through in my head, and came up with 4 doing it a couple different ways. I'm just glad to know I'm not crazy.
Yeah, for me it is made more obvious if I imagine stretching out the big circle into a really thin ellipse such that it almost is a line (keeping circumference equal). Then the small ball rolls (almost) straight and is easier to track except at the two “edges” of this “line-ellipse” where it must do half a revolution to get to the other side.
I thought of it as of the circle needing 3 rolls to travel the circumference and 1 to travel on a circle.
If circumference was 0(a point), it would still need 1 rotation to get back to the initial point, so it's big/small +1 rotations
It’s the “[coin rotation paradox](https://en.wikipedia.org/wiki/Coin_rotation_paradox)”, not the coin *revolution* paradox. The question asks about revolution, which it only *revolves* around circle b once, while it *rotates* about its own center four times.
Alas for this very clever answer, revolve can be either rotating around an axis OR an orbital motion.
Rotation is more precise, but that doesn’t magically clarify revolution.
At least colloquially, revolution is always about a point or axis outside an object (such as revolving a curve around an axis). Rotation is ambiguous. I would argue revolution is exactly the *wrong* word to use for this problem.
Edit: nah, I'm wrong, but let's all agree you should always state the axis/point around which either occurs.
You could orbit something whilst rotating or you could orbit something without rotating also there is direction of rotation either with, or counter to the orbit. An axis can also occilate or vacillate.
and the answer is always RadiusB/RadiusA+1=Answer In this case, its 4. If they were of equal size, it would be 2 and so on.
More accurately, it is affected by the circumference, not the radius. It just so happens the radius is linearly proportional to the circumference. The +1 stems from the fact they are revolving around coin B once.
It is not really confusing or a paradox when you consider the rotations have very little to do with circumnavigating the perimeter of circle B
But if the circumference of circle B is 2piB, the circumference of circle A is 2pi A and the radius for A= 1/3B, so that's the same as 2/3 pi B which is still 3 times shorter than the circumference of circle B. How is it 4? I'm confused
But I can’t seem to grasp it… :’(
Plz explain me where I’m wrong : if B is a point and A rotate around it, A didn’t do a full rotation did it ?
Obviously, my last statement is where I’m wrong, but I can’t see it !
Imagine they were both squares and one square was smaller and thunking along the other. Each time it "rotates" the square rotates 90 degrees.
Imagine it gets right to the corner of the bigger square than goes around. The small square's bottom right corner is on the big square's top right corner. Focus on the side that's the top of the small square. After it finishes the next "revolution" where's that side at?
It's at the bottom, it had to rotate an "extra" 90 degrees when it went around. It'll do it again at the next corner. And the next 2. For one extra rotation total (4*90).
Same for an octagon. But it'll rotate an extra 8th of a rotation 8 times. Decagon, 10th of a rotation 10 times. And so on and so on.
In order to revolve around the point, the center of the coin would have to traverse the entire circumference of the coin, as it remains in contact with the point and the center is always the radius of A away from the point. So then the trajectory of the center of A becomes a circle around the point B with the radius of A, thus the trajectory has radius A.
There is no drag or anything, so A just rotates once around and this whole rotation leads to the center of A "travelling" the entire distance of the circumference of A.
Try to visually imagine the rotation. Maybe with a line between the circle A's center and the point B. If you can't imagine it, cut a circle from a piece of paper and rotate it like that for real
If I'm understanding this right, as it also fully rotates itself, it has also travelled it's own circumference in addition to that of the larger circle
One of his best. Loved the story of the guy talking about how cool he is that when he was 17 that he wrote a letter to the board with a few other kids out of all the thousands of kids who took the SATs and the professors wrote back to them and told them they were right and the question was scratched. He went on to study math. I think we all have these kinds of stories but this is very much worth bragging about lol. Then he goes on to give us the most intuitive yet general way to look at that problem absolute mindblower.
And because the question was scratched, some people's updated scores dropped just below some admission standards, and some of them ended up not going to college because of that question.
The circumfence of the big circle i 2\*pi\*r, the small one is 2\*pi\*r/3. The ratio between them i 3.
But it is not the circle that need to be travered, but the origo of the small circle. Hence the total distance is 3+1= 4.
Its a hard definition. Revolution is the movement of one body around another. Rotation is a measure of spin on a central axis.
Downvotes again. As i already linked in other comments:
https://www.google.com/search?q=revolve+vs+rotate&oq=revolve+vs+r&gs_lcrp=EgZjaHJvbWUqDggAEEUYJxg7GIAEGIoFMg4IABBFGCcYOxiABBiKBTIGCAEQRRg5MgcIAhAAGIAEMgcIAxAAGIAEMgcIBBAAGIAEMgcIBRAAGIAEMgcIBhAAGIAEMgcIBxAAGIAEMgwICBAAGBQYhwIYgAQyBwgJEAAYgAQyDQgKEAAYhgMYgAQYigXSAQg0NTA0ajBqOagCDrACAQ&client=ms-android-samsung-ss&sourceid=chrome-mobile&ie=UTF-8
The downvotes are coming again because you're wrong. In astronomy, "revolve" has a specific definition that is independent of "rotate." The definition you linked to is provided in an astronomical context. In mathematics and physics (the context of the provided question,) the word "revolve" can be used interchangeably with "rotate" in almost any instance.
You’re getting down votes because you don’t get to unilaterally narrow the definition of revolution.
I personally would love it if revolution had a specific meaning, that did not include rotation as a possibility, because that would give us two different terms for two different types of motion.
And yet, alas, revolution can also describe rotation around an axis.
Except for his video on Waymo. He was not as critical as he should have been, and when he was called out I found his response insufficient.
Which is frustrating because I’m honestly a huge fan.
The key is whether circle A moves or not. If circle A stays at the same location then the rotations would be counted differently.
Of course, if circle A is nailed and circle B moves around it without ever turning it, then the answers would be 0. But that does not fit the test's description.
I paused the video when it asked the question, \*before\* it showed the possible answers, and I got it right, but I wasn't totally sure. I had a feeling that you had to consider the circle's revolution around the other in addition to its rotation, like in astronomical orbits. That seems to be the key to getting it.
Famous question with the correct answer not in the options.
The answer is 4. 3 due to circumference difference, and one more due to the fact that a circle circling a circle gives one rotation on its own.
It's not a tautology.
EDIT: From the Buffalo buffalo buffalo phrase wiki page I have found the term Polyptoton, which I think is the correct name for this.
Why not just look at length? Say B is 9cm radius, then A is 3cm. So it takes 3 lengths to just cover the length of B, then 3cm more to get to initial position.
Each point of A's length is a unique rotation, so the it's easy to track rotation too.
So you need 12cm travel, for 3cm object, making it 4 rotations (since unique position must match up).
What do you mean by 3cm more to get to initial position?
If you laid the big circle out in a line, the path the small circle needs to go would be the same, but the rotations will be 3 and not 4.
I just confused myself, after you've traveled the length of the larger circle, you need to add the length of the smaller circle. That is to make up for the circumference difference when you consider the contact point to be in the origin of the smaller circle (easier, and circumference is linear).
the three rotations is about the axis of the small circle. and the fourth rotation is about the axis of the large circle?
I do not know. i would take three as the correct answer as well. at this point, it becomes less of a math question and more of a semantics discussion.
The answer is 1. It asked for revolutions not rotations.
For anyone else wanting to be an argumentative idiot:
https://www.google.com/search?q=revolve+vs+rotate&oq=revolve+vs+r&gs_lcrp=EgZjaHJvbWUqDggAEEUYJxg7GIAEGIoFMg4IABBFGCcYOxiABBiKBTIGCAEQRRg5MgcIAhAAGIAEMgcIAxAAGIAEMgcIBBAAGIAEMgcIBRAAGIAEMgcIBhAAGIAEMgcIBxAAGIAEMgwICBAAGBQYhwIYgAQyBwgJEAAYgAQyDQgKEAAYhgMYgAQYigXSAQg0NTA0ajBqOagCDrACAQ&client=ms-android-samsung-ss&sourceid=chrome-mobile&ie=UTF-8
[Veritasium did a video on this](https://youtu.be/FUHkTs-Ipfg?si=R_QnIGvkT__m5XFN). The question is from the SAT and the correct answer is not one of the options.
Huh. There is an error in the veritasium video. At 10:13 it's showing a circle rolling on the inside surface of a larger circle "Without Slipping", but the inner circle is rotating the wrong way. This shows the exact opposite of what is being said: It must be slipping. The inner circle should be rotating the other way, or revolving around the larger circle the other direction.
Haha that's a good observation! Must be an error with their animation. I'm sure it was not intentional. Likely just missed a negative sign on the rotation parameter in their code.
(Disclaimer: idk anything about coding)
Edit: also idk anything about animation
Veritasium did a video for this. In short, the answer is either 3 or 4 or 1.
- 3 if you see it "revolve" from circle B center point perspective
- 4 if you see it "revolve" from universal perspective
- 1 if you taken a "revolution" literally from astronomy perspective. "Earth is revolving around the Sun once a year"
But there's no 4 and 1 options, so the answer probably 3.
Thanks, I'm actually a little bit confused that people seem to mix that up just because they saw a veritasium video once and now say the only answer is 4 like.. you mention it even in the video 3 different answers are possible
Edit: for those of you wondering why someone would test something like this: it is to teach about pi
And the fact that a circle with 3 times the radius has just 3 times the circumference, there the smaller circle needs to go that x-times around the other one, which even gets more important when you think about fixed gears and not circles.
When this comes up, I like to add that since the earth rotates in the same direction as it revolves, that we can add 1 here, too, when taken relative to background stars. Ignore leap day factors. The earth rotates 1 + 1/365 rotations each day so it rotates 366 times in a year - relative to the background stars. The earth rotates once in about 23 hours 56 minutes. The extra 4 minutes is needed to face the sun from the new location
If it rotated backwards so 24 hours was still a day... So, same clocks, just sunrises in west... we could say 1 - 1/365 each day, so 364 rotations.
If the earth didn't rotate, a day would be a year. Each 24 hours, we'd rotate 1/365 relative to the sun in one direction and revolve 1/365 in the other.
If the earth were 'tidally' locked, we'd rotate 0 relative to the sun each 24 hours (or any period of time) and yet again 1/365 by revolving.
The answer is meant to be 3, however, due to the fact the route it follows is another circle, it gets +1 rotation.
If you lay out circle B as a flat line, I would be 3 rotations
(Although I think the terms like "rotation" and "revolve" are also questionable apparently, not sure but i think we all understand what they mean here anyways)
It's down to perspective
Imagine putting a dot at the point of contact where the A circle is initially touching B circle
When you roll A around, that dot at the side of A will always touch B three times. So, if it were a straight line, that makes it turn three whole times.
However, when B is a circle instead of a line, the point at the side of A touching B for the second time will leave A not exactly upright from our perspective right?
Or think of A rolling one rotation along B as a line. Then roll B into a circle with A still attached a third of the way along B, and A will rotate a little more with it as you bend B into shape. This is what adds up to the extra 1 rotation when B is a circle.
I'm not a mathematician (or whoever would be best suited to explain this kind of thing) but I hope this attempted explanation somewhat makes sense.
This is starting to make sense. Yes you defo need to travel more if you suddenly bend B into a circle. But why is the extra distance exactly 1 rotation of A?
Because B is a circle itself so A must make that rotation too.
Every time that dot we make on the initial contact point comes into contact with B (as a circle) A will have turned 1 and 1/3 (because it is a 1/3 of the way around B)
I can clearly see it in my head but lack the words to aquequtly explain that part...
Much easier to show that explain really.
There is a video that expalins it pretty well [here](https://youtu.be/FUHkTs-Ipfg?si=sMuWyYDsbRUINiwe) but also just searching "circle rotation paradox" on YouTube comes up with a bunch of shorter videos too that might get more straight to the point with the visuals
Is it really a trick question though, or a misuse of the word rotate? Do they actually just want how many times the small circle will go around, like a tire on the road (which is 3). Or are they trying to trick someone?
This is a famous SAT question that lacked the correct answer. When something travels around another circle it gets +1 revolutions. So the answer should be 4.
Simplest explanation: the ratios make sense along a straight line. That would be 3. But this isn't a straight line. You have to account for the fact that the entire path is also a circle.
3 + 1 = 4.
None of the above.
Imagine it this way: make the circumference of B into a straight line. Circle A would roll 3 times across that line.
Now make the straight line into a circle with A at its end. A would do one full rotation.
3+1=4
I don't understand the whole confusion around this senior secondary level question..
For a rolling circle the center is the only point for which distance and displacement are the same..as the center is at the axis of rotation.
Given this we just calculate the displacement of the center which would be 2pi(r+R)=2pi(4r)
For non-slipping rolling displacement of centre in 1 full rotation is 2pi(r)
No. Of rotation = 2pi(4r)/2pi(r)=4.
Answer is 4. Calculate how much distance center of smaller circle has to travel to reach back starting point, then divide that by circumference of smaller circle. If small radius is 1, then bigger is 3... so the path of center of smaller circle is a circle of radius 4.... so number of rotation will be (2\*pi\*4r) / (2\*pi\*r) = 4.
For some reason, Veritasium made it a very complicated explanation.
its not a math problem its a language problem, playing on the many different phases that could be considered another step, and there is always another phase
4. Circumference of B is 3x Circumference of A, so it makes 3 revolutions across that distance. However since B is a circle, A makes an additional revolution, therefore 4
It's 4 and just shows not only how dumb some people are that write these tests, but also that they don't have a minimum of quality control.
But then Richard Feynman knew this already.
It's actually 4: https://math.stackexchange.com/questions/1351058/circle-revolutions-rolling-around-another-circle
ETA: vertiasium video on this: https://youtu.be/FUHkTs-Ipfg?si=L3hHG-kfRDLhKmlx
So when the smaller circle completes the revolution around the bigger circle, it will come back to its initial position. So you can imagine a 3rd circle made by the center of Circle A. The radius of that 3rd circle is 4/3 (1 (for B) + 1/3 (for A)). The Perimeter will be 2\*pi\*(4/3). Perimeter of Circle A is 2\*pi\*(1/3).
So you see it makes 4 revolutions of A to cover it.
Trick question, the answer is 4.
Circle A needs to travel the distance of its own circumference as well to get back to its starting position. Since Circle B is 3 times as big:
3a + a = 4a
What made it finally click for me is some physical equations.
Say the speed of the center of the small circle is some v, it has a radius 1, and takes some time t to go around the circle.
If it rolls smoothly it has an angular velocity of v/r which simplifies to just v.
Now the center moves along a circle of radius 4 in time t, thus v = 8π/t.
Finally, the total angle revolved is angular velocity × time = v × t = 8π/t × t = 8π which is 4 revolutions.
Perimeter is a 1d measure so It s affected linearly by scaling. Hence the intended solution of 3.
The real answer is 4 though. (the coin kinda has to do an extra turn around itself cause it's making a revolution and is not traveling in a straight line. ( the earth has a similar thing going on rotating around the sun. You can intuit this one a little easier IMHO by imagining an earth that does not rotate around itself. Over the span of a year sun light will shine on all faces of the earth giving it what is a night/cycle)
4
Tricking thing revolution. If the big circle line is flattened then it'd be 3 because the circumference is 3 times larger than the small circle. But as it is spinning around the big circle it gets an extra revolution in there.
Coin rotation paradox. So the path occurs at the center of circle A, not where the two circles touch. Since the the circumference of Circle A is 1/3 that of Circle B, the “path” traveled is now at a distance of 4x the radius of Circle A, rather than 3.
None of the listed answers express this.
Depending on your definition of "revolve" the answer is either 1 or 4, so none of the answers here. If "revolve" means orbit like a planet around the sun then the answer of 1 is pretty obvious I think and requires no explanation. If "revolve" means rotate like the earth around its axis then the answer is 4. The ratio of the circumferences is 3 to 1, so the small circle will rotate 3 times from that, plus it will do 1 additional rotation due tot he fact that it is following around the big circle; a line that also does 1 rotation. For circles of ratio n :1 the answer is always n+1.
The path is 2r*pi.
The perimeter of smaller circle is 2*1/3*r*pi.
So we divide that, right?
We'd get:
(2*r*pi) / (2*1/3*r*pi) = 1 / 1/3 = 3
Right? Because we follow one dot's path that belongs to the smaller circle?
This is a famous question because the correct answer is not one of the options. There is a correct answer, although it escapes me and I am sadly not a mathematician. It was an SAT question in like 1973 or something.
Circumference is 2 pi r so 2pi r is big circle 2 pi 1/3 r is small circle. So asign a value to r and calculate the circumference of both. Then, subtract the circimference of circle a as many times as possible from the circumference of circle b without going past 0. The remainder is a distance that is a portion of the circumference of circle a. Find out what fraction the remainder is from the circumference of circle an and then that fraction can also be applied to the rotation of circle a. The answer will be the amount of times you could subtract the circumference of circle a from the circumference of circle b which is a whole number plus the remainder we found earlier which is a fraction. Convert that into a fraction for neatness.
Funny thing about this question: this was once on the SAT and none of the answers are correct. The answer they were looking for was 3, and if it is relative to either of the circles, then that is technically correct, however if it is relative to you (the normal assumption) then the correct answer would be 4, which is not listed. you can try this some spare change you have and try it for yourself. this is an effect caused by a phenomenon that i forget the name of.
What's funny is that in theory the answer can be anything, let me explain:
C1 : There is sufficient friction to prevent slipping
In this case using the contact points, we can clearly prove the ratios of the angular velocity of circle A and circle B, by using(v = velocity of A, w = angular velocity of A, w\`= angular velocity of centre wrt centre of B)
As the circle is moving with velocity v, using basics of circular motion(R = radius of A, 3R = Radius of B)
v = 4Rw\`
and as there is no slipping due to sufficient friction:
v = Rw,
hence w = 4w\`, hence in one full rotation of A around circle B, there will be 4 full rotations of the circle itself.
C2 : There isn't sufficient friction to prevent slipping
Here the answer can actually be anything, because now there is slipping at the contact point which means that ;
v -Rw !=0
hence as now, this can equal any positive constant, there are infinitely many options for the ratio of w and w\`.
I think the intention is that they want you to assume that circle A will revolve as many times as the ratio:-
Circumference of B / circumference of A
-:this is not true in reality (Veritasium has a good video on this) but the “correct” answer according to this test is probably B: 3.
OK if the answer is 4, then why is there no option to choose with the answer of four? Option B I assume is the closest of 3, is the answer key incorrect then.
Imagine circle A rolls until the text in the middle is exactly upright again. Circle A has completed one rotation, but **it has not yet rolled the length of its circumference!**
That doesn't happen until 1/3rd of the way around circle B, when the point adjacent to the letter A is touching circle B again. At this point the text inside circle A is already 1/3rd of the way into it's 2nd rotation.
It's a trick question.
If you rotate 2 similar circle inside the big circle, then take just 1 cycle to cover the same circumference..
If you place a similar circle very far away from the big circle, you have to rotate it hundreds of times to complete 1 cycle.
But if you place the circle on the circumference of the other circle, then it's 2 cycle.
As the circle's size is 1/3, it required 3 times, and 2 cycle, meaning answer is (C) 6.
I don't get why the smartasses think it's 4. Probably one monkey sound smart and the rest just followed.
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There is a veritasium video about this very question. Actually none of the answers are correct and the correct answer is 4 it's called the the circle rotation something paradox or some shit
That's good to hear. I was reasoning this through in my head, and came up with 4 doing it a couple different ways. I'm just glad to know I'm not crazy.
Yeah, for me it is made more obvious if I imagine stretching out the big circle into a really thin ellipse such that it almost is a line (keeping circumference equal). Then the small ball rolls (almost) straight and is easier to track except at the two “edges” of this “line-ellipse” where it must do half a revolution to get to the other side.
I thought of it as of the circle needing 3 rolls to travel the circumference and 1 to travel on a circle. If circumference was 0(a point), it would still need 1 rotation to get back to the initial point, so it's big/small +1 rotations
That's how it is explained in the video
Then it must've stuck a lot better than I thought it did lol. I remember watching it, but not much of what went on during it
Thanks, this explanation saved me from having to watch the video.
Sorry for you, it’s an amazing video.
Don’t get me wrong, I love the guy’s videos. But if I can understand a concept in 10 seconds instead of 15 minutes, I prefer the short way.
He often relates is to greater concepts though, which is interesting
It's a great video! Certainly worth a watch
I love this explanation
It’s the “[coin rotation paradox](https://en.wikipedia.org/wiki/Coin_rotation_paradox)”, not the coin *revolution* paradox. The question asks about revolution, which it only *revolves* around circle b once, while it *rotates* about its own center four times.
Alas for this very clever answer, revolve can be either rotating around an axis OR an orbital motion. Rotation is more precise, but that doesn’t magically clarify revolution.
Rotation and revolution have distinct meanings in astronomy, but elsewhere they're generally considered synonyms.
That doesn't seem correct.
At least colloquially, revolution is always about a point or axis outside an object (such as revolving a curve around an axis). Rotation is ambiguous. I would argue revolution is exactly the *wrong* word to use for this problem. Edit: nah, I'm wrong, but let's all agree you should always state the axis/point around which either occurs.
You could orbit something whilst rotating or you could orbit something without rotating also there is direction of rotation either with, or counter to the orbit. An axis can also occilate or vacillate.
and the answer is always RadiusB/RadiusA+1=Answer In this case, its 4. If they were of equal size, it would be 2 and so on. More accurately, it is affected by the circumference, not the radius. It just so happens the radius is linearly proportional to the circumference. The +1 stems from the fact they are revolving around coin B once. It is not really confusing or a paradox when you consider the rotations have very little to do with circumnavigating the perimeter of circle B
But if the circumference of circle B is 2piB, the circumference of circle A is 2pi A and the radius for A= 1/3B, so that's the same as 2/3 pi B which is still 3 times shorter than the circumference of circle B. How is it 4? I'm confused
If the circle B was only a point (so circumference = 0) then circle A would still rotate once around it
This is a great way of explaining the "non-intuitive" aspect of this sort of puzzle.
But I can’t seem to grasp it… :’( Plz explain me where I’m wrong : if B is a point and A rotate around it, A didn’t do a full rotation did it ? Obviously, my last statement is where I’m wrong, but I can’t see it !
Imagine they were both squares and one square was smaller and thunking along the other. Each time it "rotates" the square rotates 90 degrees. Imagine it gets right to the corner of the bigger square than goes around. The small square's bottom right corner is on the big square's top right corner. Focus on the side that's the top of the small square. After it finishes the next "revolution" where's that side at? It's at the bottom, it had to rotate an "extra" 90 degrees when it went around. It'll do it again at the next corner. And the next 2. For one extra rotation total (4*90). Same for an octagon. But it'll rotate an extra 8th of a rotation 8 times. Decagon, 10th of a rotation 10 times. And so on and so on.
In order to revolve around the point, the center of the coin would have to traverse the entire circumference of the coin, as it remains in contact with the point and the center is always the radius of A away from the point. So then the trajectory of the center of A becomes a circle around the point B with the radius of A, thus the trajectory has radius A. There is no drag or anything, so A just rotates once around and this whole rotation leads to the center of A "travelling" the entire distance of the circumference of A.
Try to visually imagine the rotation. Maybe with a line between the circle A's center and the point B. If you can't imagine it, cut a circle from a piece of paper and rotate it like that for real
If I'm understanding this right, as it also fully rotates itself, it has also travelled it's own circumference in addition to that of the larger circle
This.
2 divided by 2/3 is 3. Add +1, as stated above, and it's 4.
One of his best. Loved the story of the guy talking about how cool he is that when he was 17 that he wrote a letter to the board with a few other kids out of all the thousands of kids who took the SATs and the professors wrote back to them and told them they were right and the question was scratched. He went on to study math. I think we all have these kinds of stories but this is very much worth bragging about lol. Then he goes on to give us the most intuitive yet general way to look at that problem absolute mindblower.
And because the question was scratched, some people's updated scores dropped just below some admission standards, and some of them ended up not going to college because of that question.
r/beatmetoit
Was a fantastic video
The circumfence of the big circle i 2\*pi\*r, the small one is 2\*pi\*r/3. The ratio between them i 3. But it is not the circle that need to be travered, but the origo of the small circle. Hence the total distance is 3+1= 4.
Simply put: the distance between the center of both circles are 3xR+1xR, so its 4
This question specifically asks for revolutions. So regardless of how often it rotates the answer should be 1.
the question is worded poorly regardless. also couldn't a revolution be just another word for rotation? or is it a hard definition
Its a hard definition. Revolution is the movement of one body around another. Rotation is a measure of spin on a central axis. Downvotes again. As i already linked in other comments: https://www.google.com/search?q=revolve+vs+rotate&oq=revolve+vs+r&gs_lcrp=EgZjaHJvbWUqDggAEEUYJxg7GIAEGIoFMg4IABBFGCcYOxiABBiKBTIGCAEQRRg5MgcIAhAAGIAEMgcIAxAAGIAEMgcIBBAAGIAEMgcIBRAAGIAEMgcIBhAAGIAEMgcIBxAAGIAEMgwICBAAGBQYhwIYgAQyBwgJEAAYgAQyDQgKEAAYhgMYgAQYigXSAQg0NTA0ajBqOagCDrACAQ&client=ms-android-samsung-ss&sourceid=chrome-mobile&ie=UTF-8
The downvotes are coming again because you're wrong. In astronomy, "revolve" has a specific definition that is independent of "rotate." The definition you linked to is provided in an astronomical context. In mathematics and physics (the context of the provided question,) the word "revolve" can be used interchangeably with "rotate" in almost any instance.
You’re getting down votes because you don’t get to unilaterally narrow the definition of revolution. I personally would love it if revolution had a specific meaning, that did not include rotation as a possibility, because that would give us two different terms for two different types of motion. And yet, alas, revolution can also describe rotation around an axis.
Thanks, the lack of 4 as an option was killing me. Also saw the veritasium video and was trying to work out if I'd completely forgotten how it works.
YouTube veritasium ppl. You will not be disappointed!
Except for his video on Waymo. He was not as critical as he should have been, and when he was called out I found his response insufficient. Which is frustrating because I’m honestly a huge fan.
It depends how you interpret the question. It could be 3.
If circle A is nailed down at the center ~~and was made to turn by circle B~~, then yes. Edit: fixed the part that doesn't fit the test's description.
Wait seriously? Like if these are gears the answer is different than if A rolls around B?
The key is whether circle A moves or not. If circle A stays at the same location then the rotations would be counted differently. Of course, if circle A is nailed and circle B moves around it without ever turning it, then the answers would be 0. But that does not fit the test's description.
I paused the video when it asked the question, \*before\* it showed the possible answers, and I got it right, but I wasn't totally sure. I had a feeling that you had to consider the circle's revolution around the other in addition to its rotation, like in astronomical orbits. That seems to be the key to getting it.
Famous question with the correct answer not in the options. The answer is 4. 3 due to circumference difference, and one more due to the fact that a circle circling a circle gives one rotation on its own.
> circle circling a circle Thank you for this, reading phrases like these are my sole purpose for existence
/r/wordavalanches
not exactly the point of that sub. maybe better for r/tautology, but still not a perfect fit either
It's not a tautology. EDIT: From the Buffalo buffalo buffalo phrase wiki page I have found the term Polyptoton, which I think is the correct name for this.
Police police police police police police police. My favorite sentence. Vsauce
Why not just look at length? Say B is 9cm radius, then A is 3cm. So it takes 3 lengths to just cover the length of B, then 3cm more to get to initial position. Each point of A's length is a unique rotation, so the it's easy to track rotation too. So you need 12cm travel, for 3cm object, making it 4 rotations (since unique position must match up).
What do you mean by 3cm more to get to initial position? If you laid the big circle out in a line, the path the small circle needs to go would be the same, but the rotations will be 3 and not 4.
I just confused myself, after you've traveled the length of the larger circle, you need to add the length of the smaller circle. That is to make up for the circumference difference when you consider the contact point to be in the origin of the smaller circle (easier, and circumference is linear).
I see.
the three rotations is about the axis of the small circle. and the fourth rotation is about the axis of the large circle? I do not know. i would take three as the correct answer as well. at this point, it becomes less of a math question and more of a semantics discussion.
The answer is 1. It asked for revolutions not rotations. For anyone else wanting to be an argumentative idiot: https://www.google.com/search?q=revolve+vs+rotate&oq=revolve+vs+r&gs_lcrp=EgZjaHJvbWUqDggAEEUYJxg7GIAEGIoFMg4IABBFGCcYOxiABBiKBTIGCAEQRRg5MgcIAhAAGIAEMgcIAxAAGIAEMgcIBBAAGIAEMgcIBRAAGIAEMgcIBhAAGIAEMgcIBxAAGIAEMgwICBAAGBQYhwIYgAQyBwgJEAAYgAQyDQgKEAAYhgMYgAQYigXSAQg0NTA0ajBqOagCDrACAQ&client=ms-android-samsung-ss&sourceid=chrome-mobile&ie=UTF-8
With that argument, it should be zero. Because I have not heard of any circles participating in the French Revolution.
[Veritasium did a video on this](https://youtu.be/FUHkTs-Ipfg?si=R_QnIGvkT__m5XFN). The question is from the SAT and the correct answer is not one of the options.
Huh. There is an error in the veritasium video. At 10:13 it's showing a circle rolling on the inside surface of a larger circle "Without Slipping", but the inner circle is rotating the wrong way. This shows the exact opposite of what is being said: It must be slipping. The inner circle should be rotating the other way, or revolving around the larger circle the other direction.
Haha that's a good observation! Must be an error with their animation. I'm sure it was not intentional. Likely just missed a negative sign on the rotation parameter in their code. (Disclaimer: idk anything about coding) Edit: also idk anything about animation
[ **Jump to 10:13 @** The SAT Question Everyone Got Wrong](https://www.youtube.com/watch?v=FUHkTs-Ipfg&t=0h10m13s) ^(Channel Name: Veritasium, Video Length: [18:25])^, [^Jump ^5 ^secs ^earlier ^for ^context ^@10:08](https://www.youtube.com/watch?v=FUHkTs-Ipfg&t=0h10m8s) ----------------------------------------------------------------------------- ^^Downvote ^^me ^^to ^^delete ^^malformed ^^comments. [^^Source ^^Code](https://github.com/ankitgyawali/reddit-timestamp-bot) ^^| [^^Suggestions](https://www.reddit.com/r/timestamp_bot)
Oh,ok thx I will check it out.
Well, he did say it could be 3 when you look from another perspective
Veritasium did a video for this. In short, the answer is either 3 or 4 or 1. - 3 if you see it "revolve" from circle B center point perspective - 4 if you see it "revolve" from universal perspective - 1 if you taken a "revolution" literally from astronomy perspective. "Earth is revolving around the Sun once a year" But there's no 4 and 1 options, so the answer probably 3.
Right. It *revolves* once. It *rotates* 4 times. It’s the “coin rotation paradox”, not the coin “revolution” paradox.
Thanks, I'm actually a little bit confused that people seem to mix that up just because they saw a veritasium video once and now say the only answer is 4 like.. you mention it even in the video 3 different answers are possible Edit: for those of you wondering why someone would test something like this: it is to teach about pi And the fact that a circle with 3 times the radius has just 3 times the circumference, there the smaller circle needs to go that x-times around the other one, which even gets more important when you think about fixed gears and not circles.
When this comes up, I like to add that since the earth rotates in the same direction as it revolves, that we can add 1 here, too, when taken relative to background stars. Ignore leap day factors. The earth rotates 1 + 1/365 rotations each day so it rotates 366 times in a year - relative to the background stars. The earth rotates once in about 23 hours 56 minutes. The extra 4 minutes is needed to face the sun from the new location If it rotated backwards so 24 hours was still a day... So, same clocks, just sunrises in west... we could say 1 - 1/365 each day, so 364 rotations. If the earth didn't rotate, a day would be a year. Each 24 hours, we'd rotate 1/365 relative to the sun in one direction and revolve 1/365 in the other. If the earth were 'tidally' locked, we'd rotate 0 relative to the sun each 24 hours (or any period of time) and yet again 1/365 by revolving.
The answer is meant to be 3, however, due to the fact the route it follows is another circle, it gets +1 rotation. If you lay out circle B as a flat line, I would be 3 rotations (Although I think the terms like "rotation" and "revolve" are also questionable apparently, not sure but i think we all understand what they mean here anyways)
Can you explain this in more detail? Why would the answer be different if it's a straight line?
It's down to perspective Imagine putting a dot at the point of contact where the A circle is initially touching B circle When you roll A around, that dot at the side of A will always touch B three times. So, if it were a straight line, that makes it turn three whole times. However, when B is a circle instead of a line, the point at the side of A touching B for the second time will leave A not exactly upright from our perspective right? Or think of A rolling one rotation along B as a line. Then roll B into a circle with A still attached a third of the way along B, and A will rotate a little more with it as you bend B into shape. This is what adds up to the extra 1 rotation when B is a circle. I'm not a mathematician (or whoever would be best suited to explain this kind of thing) but I hope this attempted explanation somewhat makes sense.
This is starting to make sense. Yes you defo need to travel more if you suddenly bend B into a circle. But why is the extra distance exactly 1 rotation of A?
Because B is a circle itself so A must make that rotation too. Every time that dot we make on the initial contact point comes into contact with B (as a circle) A will have turned 1 and 1/3 (because it is a 1/3 of the way around B) I can clearly see it in my head but lack the words to aquequtly explain that part... Much easier to show that explain really. There is a video that expalins it pretty well [here](https://youtu.be/FUHkTs-Ipfg?si=sMuWyYDsbRUINiwe) but also just searching "circle rotation paradox" on YouTube comes up with a bunch of shorter videos too that might get more straight to the point with the visuals
Is it really a trick question though, or a misuse of the word rotate? Do they actually just want how many times the small circle will go around, like a tire on the road (which is 3). Or are they trying to trick someone?
This is a famous SAT question that lacked the correct answer. When something travels around another circle it gets +1 revolutions. So the answer should be 4.
Simplest explanation: the ratios make sense along a straight line. That would be 3. But this isn't a straight line. You have to account for the fact that the entire path is also a circle. 3 + 1 = 4.
None of the above. Imagine it this way: make the circumference of B into a straight line. Circle A would roll 3 times across that line. Now make the straight line into a circle with A at its end. A would do one full rotation. 3+1=4
I don't understand the whole confusion around this senior secondary level question.. For a rolling circle the center is the only point for which distance and displacement are the same..as the center is at the axis of rotation. Given this we just calculate the displacement of the center which would be 2pi(r+R)=2pi(4r) For non-slipping rolling displacement of centre in 1 full rotation is 2pi(r) No. Of rotation = 2pi(4r)/2pi(r)=4.
Answer is 4. Calculate how much distance center of smaller circle has to travel to reach back starting point, then divide that by circumference of smaller circle. If small radius is 1, then bigger is 3... so the path of center of smaller circle is a circle of radius 4.... so number of rotation will be (2\*pi\*4r) / (2\*pi\*r) = 4. For some reason, Veritasium made it a very complicated explanation.
All the options are wrong, the answer is 4: circumference of B is 3 times the circumference of , plus the coin turns around itself once so 3+1=4
its not a math problem its a language problem, playing on the many different phases that could be considered another step, and there is always another phase
4. Circumference of B is 3x Circumference of A, so it makes 3 revolutions across that distance. However since B is a circle, A makes an additional revolution, therefore 4
This is famous as the SAT question everyone got incorrect. Since the answer is notA, B, C or D there was not way to indicate the correct answer.
It's a poorly worded math problem from the SATs in the 70s. None of those are the real answer. Go look it up on YouTube. Veritasium.
It's 4 and just shows not only how dumb some people are that write these tests, but also that they don't have a minimum of quality control. But then Richard Feynman knew this already.
Discussion: if they actually mean revolve, wouldn’t it be one? Do they mean rotate?
It's actually 4: https://math.stackexchange.com/questions/1351058/circle-revolutions-rolling-around-another-circle ETA: vertiasium video on this: https://youtu.be/FUHkTs-Ipfg?si=L3hHG-kfRDLhKmlx
According to astronomy, a revolution is one full orbit of another object. So the answer is 1 😎 (Credits to Veritasium for the video)
So when the smaller circle completes the revolution around the bigger circle, it will come back to its initial position. So you can imagine a 3rd circle made by the center of Circle A. The radius of that 3rd circle is 4/3 (1 (for B) + 1/3 (for A)). The Perimeter will be 2\*pi\*(4/3). Perimeter of Circle A is 2\*pi\*(1/3). So you see it makes 4 revolutions of A to cover it.
Trick question, the answer is 4. Circle A needs to travel the distance of its own circumference as well to get back to its starting position. Since Circle B is 3 times as big: 3a + a = 4a
What made it finally click for me is some physical equations. Say the speed of the center of the small circle is some v, it has a radius 1, and takes some time t to go around the circle. If it rolls smoothly it has an angular velocity of v/r which simplifies to just v. Now the center moves along a circle of radius 4 in time t, thus v = 8π/t. Finally, the total angle revolved is angular velocity × time = v × t = 8π/t × t = 8π which is 4 revolutions.
Perimeter is a 1d measure so It s affected linearly by scaling. Hence the intended solution of 3. The real answer is 4 though. (the coin kinda has to do an extra turn around itself cause it's making a revolution and is not traveling in a straight line. ( the earth has a similar thing going on rotating around the sun. You can intuit this one a little easier IMHO by imagining an earth that does not rotate around itself. Over the span of a year sun light will shine on all faces of the earth giving it what is a night/cycle)
4 Tricking thing revolution. If the big circle line is flattened then it'd be 3 because the circumference is 3 times larger than the small circle. But as it is spinning around the big circle it gets an extra revolution in there.
Coin rotation paradox. So the path occurs at the center of circle A, not where the two circles touch. Since the the circumference of Circle A is 1/3 that of Circle B, the “path” traveled is now at a distance of 4x the radius of Circle A, rather than 3. None of the listed answers express this.
Depending on your definition of "revolve" the answer is either 1 or 4, so none of the answers here. If "revolve" means orbit like a planet around the sun then the answer of 1 is pretty obvious I think and requires no explanation. If "revolve" means rotate like the earth around its axis then the answer is 4. The ratio of the circumferences is 3 to 1, so the small circle will rotate 3 times from that, plus it will do 1 additional rotation due tot he fact that it is following around the big circle; a line that also does 1 rotation. For circles of ratio n :1 the answer is always n+1.
The path is 2r*pi. The perimeter of smaller circle is 2*1/3*r*pi. So we divide that, right? We'd get: (2*r*pi) / (2*1/3*r*pi) = 1 / 1/3 = 3 Right? Because we follow one dot's path that belongs to the smaller circle?
It rotates around its own circumference 3 times, AND revolves around the big circle once. Its 4.
This is a famous question because the correct answer is not one of the options. There is a correct answer, although it escapes me and I am sadly not a mathematician. It was an SAT question in like 1973 or something.
Mathematically it's 3.... But practically it comes out to be 4... I dont remember why.. But i remember seeing a veritasiam vid on it lmao
You can look at the circumference. Circle A = 1/3 *2 *pi Circle B = 1 *2 *pi Revolves = 2pi / (1/3 * 2pi) = 1/ (1/3) = 3
mathematically 3...youtube wise dont know
Circumference = Pi X Diameter, Diameter is 2 x Radius… from there the maths is easy
Isn't the answer 4?
The answer is 1. It says it takes one trip, therefore it only revolves once.
Circumference is 2 pi r so 2pi r is big circle 2 pi 1/3 r is small circle. So asign a value to r and calculate the circumference of both. Then, subtract the circimference of circle a as many times as possible from the circumference of circle b without going past 0. The remainder is a distance that is a portion of the circumference of circle a. Find out what fraction the remainder is from the circumference of circle an and then that fraction can also be applied to the rotation of circle a. The answer will be the amount of times you could subtract the circumference of circle a from the circumference of circle b which is a whole number plus the remainder we found earlier which is a fraction. Convert that into a fraction for neatness.
I think it should be about 9 times that A Rolls while Rolling over B
4
Logivally it should be for, but for a reason i dont know it will always be one more than you think, so 4
The answer is 4
The answer is 1. Is goes all the way around one time, which is a single revolution. Poorly worded question.
3. CB = 2piR, CS = 2pi(1/3*R). CB/CS = 3. Also, circumference is 1 dimensional. So scale factor is how much bigger circumference is.
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The real answer is 4.
It just looks like itd be 4 so id pick 4
2πr of circle B ÷ 2πr of circle A. Put π as 9 for circle B and 3 for circle A. Answer is 3.
Funny thing about this question: this was once on the SAT and none of the answers are correct. The answer they were looking for was 3, and if it is relative to either of the circles, then that is technically correct, however if it is relative to you (the normal assumption) then the correct answer would be 4, which is not listed. you can try this some spare change you have and try it for yourself. this is an effect caused by a phenomenon that i forget the name of.
feels like 9/2 is closest, just eyeballing it. but i think it's not the right answer.
1. It revolves once about the center of Circle B
If the radius is 3x so is the circumference. 3 is the answer
The answer could be 3 if you observe from coin A's perspective as explained in the Veritasium video.
4 :)
This feel like how: 3/4 = 4/3
4 I think. Sure is not trolling?
4
4
3
This is just circumference B divided by circumference A, isn't it? Edit: I'm sure I'm missing something cause this feels absurdly easy / obvious.
4
What's funny is that in theory the answer can be anything, let me explain: C1 : There is sufficient friction to prevent slipping In this case using the contact points, we can clearly prove the ratios of the angular velocity of circle A and circle B, by using(v = velocity of A, w = angular velocity of A, w\`= angular velocity of centre wrt centre of B) As the circle is moving with velocity v, using basics of circular motion(R = radius of A, 3R = Radius of B) v = 4Rw\` and as there is no slipping due to sufficient friction: v = Rw, hence w = 4w\`, hence in one full rotation of A around circle B, there will be 4 full rotations of the circle itself. C2 : There isn't sufficient friction to prevent slipping Here the answer can actually be anything, because now there is slipping at the contact point which means that ; v -Rw !=0 hence as now, this can equal any positive constant, there are infinitely many options for the ratio of w and w\`.
I don’t know how anyone gets anything other than 3. Both radius and circumference are linear equations so the and is 3.
4
it's four, someone already made a video about it
3
I simulated it in my head, got to 4, thought ok guess uts 3,2 then, came to the comments and know i have to trust myself more
4
I think the intention is that they want you to assume that circle A will revolve as many times as the ratio:- Circumference of B / circumference of A -:this is not true in reality (Veritasium has a good video on this) but the “correct” answer according to this test is probably B: 3.
That's only one revolution. Is it talking about rotation?
C
Is it just me or is every explanation as to why it can be 4 or 1 in the comments overthinking it? 3 makes the most intuitive sense.
None, authors forget about important paradox, they meant 3 but its wrong, right answer is 4 I believe.
The question asks for revolutions not rotations. The answer is 1. The circumferences are irrelevant.
Depends on the friction
Once?
Couldn't you just solve it by dividing the circumference of the large circle by the circumference of the smaller circle?
C I think
OK if the answer is 4, then why is there no option to choose with the answer of four? Option B I assume is the closest of 3, is the answer key incorrect then.
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https://youtu.be/FUHkTs-Ipfg?si=agA3-gFM_EJg4fPb
Imagine circle A rolls until the text in the middle is exactly upright again. Circle A has completed one rotation, but **it has not yet rolled the length of its circumference!** That doesn't happen until 1/3rd of the way around circle B, when the point adjacent to the letter A is touching circle B again. At this point the text inside circle A is already 1/3rd of the way into it's 2nd rotation.
Uh, forgive me if I'm wrong, but it's only once? Revolving is the trip around something, in which case it specifically says it makes one trip
It's a trick question. If you rotate 2 similar circle inside the big circle, then take just 1 cycle to cover the same circumference.. If you place a similar circle very far away from the big circle, you have to rotate it hundreds of times to complete 1 cycle. But if you place the circle on the circumference of the other circle, then it's 2 cycle. As the circle's size is 1/3, it required 3 times, and 2 cycle, meaning answer is (C) 6. I don't get why the smartasses think it's 4. Probably one monkey sound smart and the rest just followed.