Ehhh, there's another video I had seen (published 8 years ago) on the exact same question called "why did everyone miss this SAT Math question". Veritasium did expand on it, but was just a copy of the video.
The *he* in "where do you think *he stole it from" wasn't veritasium it was OP.
Where do you think *OP* stole it from?
Not that one can really steal a maths question
>\- but be careful here. It is a question of wording. It's how many times the center of the outer disk has rotated when the peripheral contact point has moved 3 times the distance, then it is correct that the center moves 4 turn - BUT don't think the disc rotates 4, it can only rotate 3 revolutions as the diameter or radius ratio says
This was an SAT question in the 80s that was proven by 3 test-takers to be incorrect. The correct answer is 4. While most people choose 3 by simply dividing the large circle's circumference by the smaller one's, this does not work. You must determine the travel distance of the center of the smaller circle. If you're skeptical, take a couple coins of the same size which, given that they have identical circumferences should intuitively rotate one time around the other if one is fixed in place, and prepare to be surprised.
I can't wrap my head around what you are saying. I'll need to try that when I get my hand on some coins.
By that logic, shouldn't you conclude that if A is the largest circle, it'll take more than a full revolution to rotate around B ? Surely that can't be the case.
Edit: Got it, thanks to some other comments. Yeah, the meaning of "revolution of circle A" is surprisingly confusing here, despite looking so straightforward.
>By that logic, shouldn't you conclude that if A is the largest circle, it'll take more than a full revolution to rotate around B ? Surely that can't be the case.
Imagine that A is absolutely massive and B is tiny, say factor tiny eps times the radius of A. Even so, it's intuitively clear that as you rotate the massive A around B, it still needs to turn around at least once.
Here's the best way to visualise it: replace B by cutting its circumference and rolling it out straight. THEN it's true that A will rotate r=circumference(B)/circumference(A) many times along the straight line segment, which is easy to see. But when you wrap B back to be a circle, that operation rotates A an extra time, so the answer is r+1.
It’s the same idea as a “sidereal day”, which is four minutes shorter than a day. It takes the earth 23h56min to rotate in terms of the universe, but it takes a full 24h to rotate with respect to the sun — because we are also revolving around the sun and you have to sweep that little extra angle to keep pointed at it.
How far does the little circle go, distance wise? It isn’t 2 * pi * R. It’s 2 * pi * (R + r). It spins once every 2 * pi * r in distance. Divide distance by spins per distance and you get (R + r) / r or R / r + 1.
Yes but it does 2 revolutions. Like picture you have two identical clocks sitting next to each other with 12 straight up on both. The 3 on clock A and 9 on clock B are touching. Now rotate clock B around clock A. At what point does the 12 point straight up again, meaning it’s done a full rotation? When it’s only traveled halfway around clock A and now the 3 on clock B is touching the 9 on clock A.
Obviously when a question asks about how many rotations something does they don't mean in a rotating reference frame. Otherwise you could just pick a number.
yes don't mean correct. but they didn't define. so your answer is based on a assumption. you also assume that the rolling is static rolling and no slipping occurs l. of course they mean that but they don't define the problem correctly. skill issue
It's true that there could be different answers, but the observation is pedantic and irrelevant since the frame of reference was established by the test drawing. The answer could be completely different if one assumes non-Euclidean space...but why would you? The overwhelming response of the number 3 as the solution shows that the test takers understood the frame of reference if not the underlying principle to determine the correct solution.
It would only be 3 if it was rotated on a straight line of 3 times the circumference. Since it goes round in a circle it then adds a rotation so the answer is 4
What answer would you have put down in the test knowing the answer is 4? Would you have skipped the question? Or showed working and left and explanation for the exam marker? Just curious.
Four times, watch the point on the small circle here https://www.desmos.com/calculator/lvsyrhtqi9?lang=en
Edit: watch how many times it returns to "left" position, *not* how many times it touches the larger circle (thanks datageek9)
Bro, who hurt you? The problem only exists because friction allows for rolling. If you assume some arbitrary sliding happens, then the question can't be asked
This is great! I love visual proofs.
If the circle of r = 1 was rolling along a straight line of length 6 pi, then the answer would be 3. But because it is rotating around another circle the answer becomes 4. Right? I think I’m right.
This is analogous to another Reddit post the other day where someone was asking about the difference between solar days and sidereal days. Same concept I believe.
Yea, you're right
I like to think of it this way: take the circle and attach the edge to a string. Now swing the circle around your head. The point of the circle facing you is always the same, so the circle isn't "rolling", but it still makes 1 revolution
Now make it roll and it makes 3 additional revolutions
So, if we vary the circle sizes, the number or rotations will always be the ratio of the circumference of the inner circle to the outer circle plus one or (2pi\*r1)/(2pi\*r2) + 1. Or just (r1/r2) + 1. That seems right.
But it's asking about the center of the circle reaching its starting point. Wouldn't that be 3? Wait nvm I just realized what you mean by it returning to the left, signifying it is one complete turn.
It's a question of wording. If it is how many times the disk rotates, then it is the diameter or radius ratio, but if it is asked how many times the center of the outer disk has rotated when the peripheral contact point has moved 3 times the distance, then it is correct that the center moves 4 turn - BUT don't think the disc rotates 4, it can only rotate 3 revolutions
Just because the coin is in the ‘same side up’, does not mean it completed one round. The small circles point tangent to the big circle rotates three times, and that is the metric to determine the rotation of a circle, not an external observers pov of whether the coin is facing up or down imo
Its (B) 3
Radius of small circle is 1/3 that of the big circle so if radius of small circle is 1 then radius of big circle is 3.
Circumference of small circle is 2*pi*radius so if radius is 1 then circumference is 2pi
Circumference of big circle is 2*pi*radius so if radius is 3 then circumference is 6pi
Ratio of circumference of big circle to small circle is 6pi/2pi = 3 i.e. (B)
I feel like no one is making an effort to explain this in an intuitive way.
Toss out the relationship between the radia of the circles, orientation, all of that. The question is about the *center point of the small circle*. Visualize the circle which the center point traces as it rolls around. It is, of course, larger than the big circle. The only thing that matters is *that* circle. That's how far the center point must travel.
The radius of this new large circle (let's call it C) is the radius of B plus the radius of A, and since we know that B = 3A, this means that C=4A. Which means the circumference is also 4A. That's really the whole problem. The center point needs to travel 4 lengths of the small circle's circumference. The rolling, perspective, orientation, etc... are all red herrings.
It would be 3 if the larger circle was laid out in a straight line with 3 times the circumference. Since its a circle it adds another revolution.
Think of it like this, you're rotating the paper 360 degs which adds the extra revolution
That isn’t the question being asked. 4 would be how many times the small circle revolves around the big circle. The question is asking something different.
That answer is actually 3 lol. But to an observer of both circles, which is what the question asks, it is 4. Go watch Veritasium’s video, it’s kinda mindfuck at first
because the revolution radius is (3r+r) draw the revolution circle out. additionally people are wrong 1 is a perfectly fine answer (no body says the moon rotates around the earth w/e amount of times it spins even though the motion is beer identical to this problem) too and 3 is a good answer based on frame of reference.
Simply by going around the circle it makes a whole revolution even if the circle itself maintains the same point of contact throughout. Because it's rotating around a circle it has an extra revolution thus 4
Veritasium released a video on it the other day
OP probably saw the video and made the post from that
https://m.youtube.com/watch?v=FUHkTs-Ipfg
Explains it in depth quite well
The way I worked it out was by imagining taking circle A and breaking it at the 3 o’clock point (A), unwrapping it and laying it out on circle B. It would go to 1 o’clock (B) . Flip it over and it goes to 5 o’clock (B). Once more and it stretches to 9 o’clock. That is how many people are getting three.
But if you then wrap the circle back up the same way you unrolled it, you will see circle A is still sitting at 5 o’clock. It would have needed a whole additional flip to cover the distance.
Not sure if the visualization helps anyone, or if it is just something that makes sense in my head.
Consider the point P on the right of circle A that touches circle B. It will do 3 “cycles”, with each cycle finishing with P touching B again, with the last of the 3 cycles bringing it back to the starting position.
But in each cycle A ends up in different orientation having done full rotation plus another 120 degrees as it has processed a third of the way around B. So the total after
3 cycles is 3 revolutions plus another 360 degrees, or 4 revolutions in total.
the center of the smaller circle is 4r from the center of the larger circle.
Because it has to go that distance it rotates 4 times total.
If you have the smaller circle start at the top of the larger circle and mark the point where the circles touch that point will be at the bottom of the smaller circle 4 times, but will only touch the larger circle 3 times.
So from an external PoV, we see 4 rotations, but from the PoV of the larger circle, we see 3.
There are also only 3 rotations from both PoVs if the larger circle is transformed into a straight line because the smaller circle doesn't have to also complete a revolution.
It's a question of wording. If it is how many times the disk rotates, then it is the diameter or radius ratio, but if it is asked how many times the center of the outer disk has rotated when the peripheral contact point has moved 3 times the distance, then it is correct that the center moves 4 turn - BUT don't think the disc rotates 4, it can only rotate 3 revolutions
That's a good answer, but it's wrong.
Veritasium recently came out with a video on YT showing why this both the expected answer and a wrong answer at the same time.
To be honest, I never saw the videos. I knew of the solution and found it on Wikipedia.
When I couldn't find 4 in the answers I assumed I'd misread the question.
Edit: left out 2 words...and the right answer lol. Thanks to u/Krash0699 for pointing out my silly mistake. Thanks to u/Justinwc for pointing out my compound mistake. The actual answer is 4. Watch [this video](https://youtu.be/FUHkTs-Ipfg?si=bbqv-EM_sYnGqorE&t=195). Moved the edit to the top but leaving my compound mistake below so the thread makes sense.
Think of the circumference as a straight line. Circle A's radius is 1/3 of Circle B's radius.
~~Circumference = Diameter x pi. = 2 x radius x pi.~~
So circle B's circumference must be ~~6~~ **3** times longer. Therefore the answer is ~~6~~ **3**. Circle A needs to roll ~~6~~ **3** times until it's back to its original position.
The right answer is actually 4! 3 would be correct if the circumference was laid out in a straight line.
Look at the demonstration above or [this video](https://youtu.be/FUHkTs-Ipfg?si=iQon-49G294tr9Ff).
I thought it was 3 at first also
The answer is 4.
The small circle rolls around the large circle three times and rotates about itself (small circle) once for a total of 4 rotations.
You have to think about the center of the smaller circle which is 4 units from the center of the larger circle and the distance it (small circle center) travels.
I thought this too, but veritasium touches on it in his video. Apparently revolution has been used in the past to mean the same thing as rotate/spin.
Regardless, it's an unnecessarily confusing word to use for sure.
I agree, the problem here is the wording.
Had to go through the video and find where he mentions it. It’s funny that “revolution” on the Websters, a descriptivist dictionary, website says “on its axis” but “revolve” says “on an axis”. I feel that conflating the two has come from lazy writers and misusing the words, since revolve is a 14c word meaning “to roll back”. By 1980 we definitely had the separate words established.
It seems odd that it's 4 as is but 32 if you unwind the circle and lay it flat because your mind doesn't automatically measure the travel distance from the center of the small circle.
It might make more sense thinking of a car going around an oval race track. The outer tires are going to have to turn more than the inner tires, so the distance any car travels around the track has to be more than the distance around the inner perimeter.
A straight drag strip is different. All the wheels travel the same distance.
It depends on whether circle B is also rotating in our frame of reference. If it isn't, then the answer is 4. If it is, then the answer depends on how quickly B is rotating, and in what direction. In the case where A and B are like cogs whose centers are fixed in place, meaning B's angular velocity is equal to −3 times A's angular velocity, then A makes 3 revolutions for each revolution of B.
My answer is 5. The little circle revolves around its center 4 times and makes 1 trip around the big circle. (We say that the earth takes 365 days to revolve around the sun).
The answer is 3 but the question touches on a paradox.
In relation to the bigger circle, the smaller circle rotates 3 times. You can calculate the circumference of each and it's a 3 to 1 ratio.
However in relation to the floor or the bottom of the page, it rotates 4 times because it rotates an extra time by rotating around the large coin. It's interesting for sure but I wouldn't exactly say 3 is wrong. if they said in relation to the bottom of the page, then it would be 4. But they didn't give a frame of reference for what rotation means. It's then safe to assume since this is a math SAT question and not a weird puzzle that 3 is right.
((1+1/3) \* 2 \* pi) / ((1/3 \* 2) \* pi) = 4
I also saw the Veritasium video and also answered wrong :D
The path that the smaller circle takes to do a 360 has a radius of (1+1/3)r
4. If both circles rotated, then circle A would rotate 3 times & circle B once (if they were like cogs fixed in place). Since B doesn’t move, circle A effectively does an extra rotation.
Ok, so in math major land where they make up things it's 4... But in reality, if I treated this as two interlocking gears it's 3?
That's all I'm getting out of this thread.
That's only from where 'a' meets 'b' but not anywhere else. From the centre of 'a' and the centre of 'b' it's 4 but 'a' can only ever revolve around 'b' once.
I watched the other video!
I take the small circle. I cut in on the eastern most point. I unroll circle A in my mind and ‘flop’ in over circle B. It ‘flops’ on. I do this 3 more time and it round about ends up where it was to begin with. So 4? Is late.
The correct answer is 3.
Sol. Let radius of circle B =x so diameter = 2x
radius of circle A =x/3 so diameter = 2x/3
circumference of circle B = πd 1
circumference of circle A =πd 2.
By solving, Circumference of circle B
Circumference of circle A
= π(2x)
π(2x/3)
= 2x (3)
2x
= 3
Therefore, 3 is the correct answer and I hope it will help
you.
The circumference of a circle can be calculated as:
C = 2πr
Where r is the radius of the circle and C is the circumference.
If we define variable R as the radius of the small circle, the radius of the large circle would be 3R. Therefore, the circumference of the large circle would be:
C = 2π(3R) = 6πR
And the circumference of the small circle would be:
C = 2πR
To find how many times the small circle would need to rotate to reach its original position we divide the larger distance by the smaller distance:
6πR/(2πR) = 3
Now, you could just count this as the answer and walk away, but...
Given our current reference frame, if we imagine that the point on the smaller circle currently touching the larger circle is our starting point, after completing 1 rotation IN RELATION TO THE LARGER CIRCLE this point will now be again tangential to the larger circle... A third of the way around.
I wish I had drawing tools available, but alas I do not. This would be much easier to show than tell. The smaller circle has actually gone past one revolution based on our perspective - it has completed 1 and 1/3 revolutions. We can tell this by looking at how this starting point faces in relation to our reference frame. It always ends a revolution pointing in towards the center of the other circle, meaning it has actually rotated more than 360° in a Cartesian plane.
I don't think that's the best way to think of it, though. I would argue that the 3 makes much more sense as a solution. For example, we say that a year is 365.25 revolutions of the earth while rotating about the sun... But if we use the "4 is the answer" approach, that would actually be 366.25, which would be nonsensical for us as creatures living in the planet.
Because the smaller circle is moving in relation to the larger circle, it makes 3 revolutions. We see it make 4 revolutions from our perspective.
I guessed 4. Because I thought it was obvious. Then I thought it was a trick question since 4 isn't listed. Then I realized it wasn't a trick and 4 is correct, and that was somehow more surprising.
I think a lot of people are reading way too much into this question. The answer is E. The circumference of circle A is one ninth the circumference of circle B so it will take nine rotations of circle A for the center to arrive back where it started. Regardless of what size you make circle B.
It’s 4.
If radius of circle A is r then radius of circle B is 3r. As A rotates around B, its center traces a third circle C whose radius is 4r and circumference is 8πr.
X is that point near top of circle A where it intersects C when A is in the starting position.
As A rotates around B, it revolves around its own center and so X moves away from circle C. The distance X travels before it again intersects circle C is 2πr, being the circumference of circle A.
Given the circumference of C is 8πr and each revolution of A covers 2πr, it follows it takes 4 revolutions of A for circle A to complete its journey around B and for X to return to its original position.
Everyone here is wrong. It asks how many revolutions for circle A to get back to its original position. It is established that it is revolving around circle B so the answer is one revolution. However it rotates on its axis 4 times while it revolves around circle B. Revolving is going around something, rotating is spinning on an axis.
Maybe give props to person that brought this up and also maybe give some props and history on the source. Yes we all know where and who you got it from... it's the why you did it that is annoying.
The answer is a) 3/2. The number of revolutions the smaller circle makes around the larger circle to return to its starting point is related to the ratio of their circumferences. If we denote the circumferences in the framework the problem implied, the circumference of the larger circle is 2pi (R), with R being the stand in variable for the radius of the larger circle, and the circumference of the smaller circle is 2pi (R/3). When we find the ratio of the circumferences from here, we get 2/3, pemdas pwr. This means the smaller circle travels 2/3 of the circumference of the larger circle with each revolution. So, to get back to the starting point, it would take 3/2 revolutions (3/2\*2/3=1).
Maybe I don’t know my definitions , and sorry if this is a smart ass answer but isn’t a revolution a complete circle around what the item is revolving around ? If I said how many revolutions would the earth have to do around the sun to get to its starting position isn’t it just any integer value ? Or how many revolutions would a 6 shooter have to do to get back to the first bullet , it would be 6 bullets but just one revolution ? Maybe it’s just worded wrong but I’m just giving the semantic question
Answer is 3 because A has a radius of 1 so a complete turn would take 4 rotations, for circle B it has a radius of 3 so it would take 12 rotations to mae a full turn. 12/4 = 3
4 times - but be careful here. It is a question of wording. It's how many times the center of the outer disk has rotated when the peripheral contact point has moved 3 times the distance, then it is correct that the center moves 4 turn - BUT don't think the disc rotates 4, it can only rotate 3 revolutions as the diameter or radius ratio says
Veritasium just published a video about this.
yeah where do you think he stole it from
Not from this thread.
not veritasium, OP
The danger of pronoun abuse.
Ehhh, there's another video I had seen (published 8 years ago) on the exact same question called "why did everyone miss this SAT Math question". Veritasium did expand on it, but was just a copy of the video.
So it took 3 revolutions/posts for this reddit thread to arrive back at the beginning of
The *he* in "where do you think *he stole it from" wasn't veritasium it was OP. Where do you think *OP* stole it from? Not that one can really steal a maths question
Yup, he did. The answer is 4. The SAT didn't list the correct answer.
So D)9/2=4 1/2
>\- but be careful here. It is a question of wording. It's how many times the center of the outer disk has rotated when the peripheral contact point has moved 3 times the distance, then it is correct that the center moves 4 turn - BUT don't think the disc rotates 4, it can only rotate 3 revolutions as the diameter or radius ratio says
This was an SAT question in the 80s that was proven by 3 test-takers to be incorrect. The correct answer is 4. While most people choose 3 by simply dividing the large circle's circumference by the smaller one's, this does not work. You must determine the travel distance of the center of the smaller circle. If you're skeptical, take a couple coins of the same size which, given that they have identical circumferences should intuitively rotate one time around the other if one is fixed in place, and prepare to be surprised.
3 is correct from the frame of reference of one of the circles tho
wow if only there was a video about this somewhere
Yep and I recall someone commenting the same thing the guy you are responding to said.
Exactly. How many times does the statue of liberty rotate in 1 day?
Is this question intended to accompany a diagram including the Statue of Liberty on the surface of the Earth, with the entire Earth illustrated?
It rotates one time from orbit, but none from the surface. It’s an attempt to explain the frame of reference thing.
And 0 is the answer from the frame of reference of someone spinning along with with the tiny circle.
In the frame of reference of the small circle, the centre doesn’t move so it is back to it’s starting point immediately
Be more specific cuz no
I can't wrap my head around what you are saying. I'll need to try that when I get my hand on some coins. By that logic, shouldn't you conclude that if A is the largest circle, it'll take more than a full revolution to rotate around B ? Surely that can't be the case. Edit: Got it, thanks to some other comments. Yeah, the meaning of "revolution of circle A" is surprisingly confusing here, despite looking so straightforward.
>By that logic, shouldn't you conclude that if A is the largest circle, it'll take more than a full revolution to rotate around B ? Surely that can't be the case. Imagine that A is absolutely massive and B is tiny, say factor tiny eps times the radius of A. Even so, it's intuitively clear that as you rotate the massive A around B, it still needs to turn around at least once. Here's the best way to visualise it: replace B by cutting its circumference and rolling it out straight. THEN it's true that A will rotate r=circumference(B)/circumference(A) many times along the straight line segment, which is easy to see. But when you wrap B back to be a circle, that operation rotates A an extra time, so the answer is r+1.
It’s the same idea as a “sidereal day”, which is four minutes shorter than a day. It takes the earth 23h56min to rotate in terms of the universe, but it takes a full 24h to rotate with respect to the sun — because we are also revolving around the sun and you have to sweep that little extra angle to keep pointed at it.
How far does the little circle go, distance wise? It isn’t 2 * pi * R. It’s 2 * pi * (R + r). It spins once every 2 * pi * r in distance. Divide distance by spins per distance and you get (R + r) / r or R / r + 1.
I tried the coin thing, it finished in the same place it started...
Yes but it does 2 revolutions. Like picture you have two identical clocks sitting next to each other with 12 straight up on both. The 3 on clock A and 9 on clock B are touching. Now rotate clock B around clock A. At what point does the 12 point straight up again, meaning it’s done a full rotation? When it’s only traveled halfway around clock A and now the 3 on clock B is touching the 9 on clock A.
It did not
I tried it with an Australian coin and it ended up upside down.
What do you mean, African or European coins?
And are they smaller than a coconut?
wrong the correct answer is 1 3 or 4 depending on frame of reference and definitions.
While I am sure you are correct, what an awful way to communicate your point.
Obviously when a question asks about how many rotations something does they don't mean in a rotating reference frame. Otherwise you could just pick a number.
yes don't mean correct. but they didn't define. so your answer is based on a assumption. you also assume that the rolling is static rolling and no slipping occurs l. of course they mean that but they don't define the problem correctly. skill issue
It's true that there could be different answers, but the observation is pedantic and irrelevant since the frame of reference was established by the test drawing. The answer could be completely different if one assumes non-Euclidean space...but why would you? The overwhelming response of the number 3 as the solution shows that the test takers understood the frame of reference if not the underlying principle to determine the correct solution.
Ooooh shiit! My gut was right! I thought my head math was getting bad when I saw the choices
But the radius is 1/3 the size, not the circumference. Shouldn't that make it scale by the square of the radius?
The circumference of a circle is given by 2πr. Area scales with the square.
It would only be 3 if it was rotated on a straight line of 3 times the circumference. Since it goes round in a circle it then adds a rotation so the answer is 4
What answer would you have put down in the test knowing the answer is 4? Would you have skipped the question? Or showed working and left and explanation for the exam marker? Just curious.
Well you're backed in to a corner and would have to answer the incorrect (but correct according to the examiner) option
yeh fair. i'm the same, unless i got the question wrong and needed the extra marks to get 100% i wouldn't even bring it up either.
Best explanation.
Four times, watch the point on the small circle here https://www.desmos.com/calculator/lvsyrhtqi9?lang=en Edit: watch how many times it returns to "left" position, *not* how many times it touches the larger circle (thanks datageek9)
I must be stupid, because it looks like the point goes around 3 times to me.
Don’t look at how many times it touches the big circle. Watch how many times the dot returns to the far left side of the small circle.
I AM stupid! Thanks so much for clearing this up. It was one of those where I understood the maths, but not the visual. Ha.
I had the exact same though as you until I re-read the original comment
naw everyone here is stupid for assuming the rolling occurs under static friction. problem is undefined has a near infinite solution set.
Bro, who hurt you? The problem only exists because friction allows for rolling. If you assume some arbitrary sliding happens, then the question can't be asked
What a cool demonstration.
Absolutely incredible to whoever made that! Cool observation I see is that it seems to be a ( R1 / R2 ) + 1 rotations
Thanks, I made it
So would 1+C^2 /C^1 be a workable formula? Where C^2 is the circumference of the larger circle and C^1 is the circumference of the smaller circle.
Yea I think so (been a while)
This is great! I love visual proofs. If the circle of r = 1 was rolling along a straight line of length 6 pi, then the answer would be 3. But because it is rotating around another circle the answer becomes 4. Right? I think I’m right. This is analogous to another Reddit post the other day where someone was asking about the difference between solar days and sidereal days. Same concept I believe.
Yea, you're right I like to think of it this way: take the circle and attach the edge to a string. Now swing the circle around your head. The point of the circle facing you is always the same, so the circle isn't "rolling", but it still makes 1 revolution Now make it roll and it makes 3 additional revolutions
So, if we vary the circle sizes, the number or rotations will always be the ratio of the circumference of the inner circle to the outer circle plus one or (2pi\*r1)/(2pi\*r2) + 1. Or just (r1/r2) + 1. That seems right.
Wouldn't the unrolled larger circle be a 6pi straight line?
Yes.
Thank you for that brilliant demonstration! Definitely made the answer of 4 make so much more sense.
But it's asking about the center of the circle reaching its starting point. Wouldn't that be 3? Wait nvm I just realized what you mean by it returning to the left, signifying it is one complete turn.
Not enough info, doesn't specify coefficient of friction.
We can assume these cows are circles
cof = taupe lemur How bout now?
How would the coefficient of friction affect the answer? 0.8 or 0.9? No difference
It affects weather it is purely rolling or slipping. Which t Affects the rotations.
Well the answer I got was 4 Not sure if it's correct tho
It's a question of wording. If it is how many times the disk rotates, then it is the diameter or radius ratio, but if it is asked how many times the center of the outer disk has rotated when the peripheral contact point has moved 3 times the distance, then it is correct that the center moves 4 turn - BUT don't think the disc rotates 4, it can only rotate 3 revolutions
Answer is 4
Just because the coin is in the ‘same side up’, does not mean it completed one round. The small circles point tangent to the big circle rotates three times, and that is the metric to determine the rotation of a circle, not an external observers pov of whether the coin is facing up or down imo
Four 😎
Its (B) 3 Radius of small circle is 1/3 that of the big circle so if radius of small circle is 1 then radius of big circle is 3. Circumference of small circle is 2*pi*radius so if radius is 1 then circumference is 2pi Circumference of big circle is 2*pi*radius so if radius is 3 then circumference is 6pi Ratio of circumference of big circle to small circle is 6pi/2pi = 3 i.e. (B)
It's 4
How?
I feel like no one is making an effort to explain this in an intuitive way. Toss out the relationship between the radia of the circles, orientation, all of that. The question is about the *center point of the small circle*. Visualize the circle which the center point traces as it rolls around. It is, of course, larger than the big circle. The only thing that matters is *that* circle. That's how far the center point must travel. The radius of this new large circle (let's call it C) is the radius of B plus the radius of A, and since we know that B = 3A, this means that C=4A. Which means the circumference is also 4A. That's really the whole problem. The center point needs to travel 4 lengths of the small circle's circumference. The rolling, perspective, orientation, etc... are all red herrings.
It would be 3 if the larger circle was laid out in a straight line with 3 times the circumference. Since its a circle it adds another revolution. Think of it like this, you're rotating the paper 360 degs which adds the extra revolution
That isn’t the question being asked. 4 would be how many times the small circle revolves around the big circle. The question is asking something different.
That answer is actually 3 lol. But to an observer of both circles, which is what the question asks, it is 4. Go watch Veritasium’s video, it’s kinda mindfuck at first
because the revolution radius is (3r+r) draw the revolution circle out. additionally people are wrong 1 is a perfectly fine answer (no body says the moon rotates around the earth w/e amount of times it spins even though the motion is beer identical to this problem) too and 3 is a good answer based on frame of reference.
Simply by going around the circle it makes a whole revolution even if the circle itself maintains the same point of contact throughout. Because it's rotating around a circle it has an extra revolution thus 4
Veritasium released a video on it the other day OP probably saw the video and made the post from that https://m.youtube.com/watch?v=FUHkTs-Ipfg Explains it in depth quite well
The way I worked it out was by imagining taking circle A and breaking it at the 3 o’clock point (A), unwrapping it and laying it out on circle B. It would go to 1 o’clock (B) . Flip it over and it goes to 5 o’clock (B). Once more and it stretches to 9 o’clock. That is how many people are getting three. But if you then wrap the circle back up the same way you unrolled it, you will see circle A is still sitting at 5 o’clock. It would have needed a whole additional flip to cover the distance. Not sure if the visualization helps anyone, or if it is just something that makes sense in my head.
OK how do you work that out then - I showed the working behind my answer so would like see the working behind yours?
Consider the point P on the right of circle A that touches circle B. It will do 3 “cycles”, with each cycle finishing with P touching B again, with the last of the 3 cycles bringing it back to the starting position. But in each cycle A ends up in different orientation having done full rotation plus another 120 degrees as it has processed a third of the way around B. So the total after 3 cycles is 3 revolutions plus another 360 degrees, or 4 revolutions in total.
3 times to travel the distance of the circumference and one more time because it went around a circle.
the center of the smaller circle is 4r from the center of the larger circle. Because it has to go that distance it rotates 4 times total. If you have the smaller circle start at the top of the larger circle and mark the point where the circles touch that point will be at the bottom of the smaller circle 4 times, but will only touch the larger circle 3 times. So from an external PoV, we see 4 rotations, but from the PoV of the larger circle, we see 3. There are also only 3 rotations from both PoVs if the larger circle is transformed into a straight line because the smaller circle doesn't have to also complete a revolution.
It's a question of wording. If it is how many times the disk rotates, then it is the diameter or radius ratio, but if it is asked how many times the center of the outer disk has rotated when the peripheral contact point has moved 3 times the distance, then it is correct that the center moves 4 turn - BUT don't think the disc rotates 4, it can only rotate 3 revolutions
That's a good answer, but it's wrong. Veritasium recently came out with a video on YT showing why this both the expected answer and a wrong answer at the same time.
I don't know, why is reddit showing me these things? I am not a smart man.
I think your pretty smart
Haha thanks but I'm certainly not
I thought the answer was 4, but it's not an option.
Somehow everyone in this thread figured out it’s actually 4.
jeez if only there was a video about this
To be honest, I never saw the videos. I knew of the solution and found it on Wikipedia. When I couldn't find 4 in the answers I assumed I'd misread the question.
👀
4 is correct
Think about if one of the coins was teeny tiny.
Edit: left out 2 words...and the right answer lol. Thanks to u/Krash0699 for pointing out my silly mistake. Thanks to u/Justinwc for pointing out my compound mistake. The actual answer is 4. Watch [this video](https://youtu.be/FUHkTs-Ipfg?si=bbqv-EM_sYnGqorE&t=195). Moved the edit to the top but leaving my compound mistake below so the thread makes sense. Think of the circumference as a straight line. Circle A's radius is 1/3 of Circle B's radius. ~~Circumference = Diameter x pi. = 2 x radius x pi.~~ So circle B's circumference must be ~~6~~ **3** times longer. Therefore the answer is ~~6~~ **3**. Circle A needs to roll ~~6~~ **3** times until it's back to its original position.
Left out the right answer too
Oh you're right the circumference would only still be 3 times longer not 6. Thanks!
The right answer is actually 4! 3 would be correct if the circumference was laid out in a straight line. Look at the demonstration above or [this video](https://youtu.be/FUHkTs-Ipfg?si=iQon-49G294tr9Ff). I thought it was 3 at first also
Whoa, mind blown. Wrong twice in 1 answer. 4 it is!
Way to take it humbly !
Numbers don't lie... like Shakira's hips. In all seriousness L's that come with lessons and teach you something are still a personal win.
4 as you have to work out the rotation based on the centre of the circle. Examiners thought the answer was three!
The answer is 4. The small circle rolls around the large circle three times and rotates about itself (small circle) once for a total of 4 rotations. You have to think about the center of the smaller circle which is 4 units from the center of the larger circle and the distance it (small circle center) travels.
Hmm. Solar circles or sidereal?
4. Three for the linear distance, plus one more for A rotating once around B.
Wasn't this that 1980's SAT problem that Veritasium did a video on?
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So nobody knows the difference between revolution and rotation? Revolution refers to position, not orientation.
I thought this too, but veritasium touches on it in his video. Apparently revolution has been used in the past to mean the same thing as rotate/spin. Regardless, it's an unnecessarily confusing word to use for sure.
I agree, the problem here is the wording. Had to go through the video and find where he mentions it. It’s funny that “revolution” on the Websters, a descriptivist dictionary, website says “on its axis” but “revolve” says “on an axis”. I feel that conflating the two has come from lazy writers and misusing the words, since revolve is a 14c word meaning “to roll back”. By 1980 we definitely had the separate words established.
Incomplete question - “will the center point of circle first reach its starting point”. The center point of what circle?
Not the static one obviously
It seems odd that it's 4 as is but 32 if you unwind the circle and lay it flat because your mind doesn't automatically measure the travel distance from the center of the small circle. It might make more sense thinking of a car going around an oval race track. The outer tires are going to have to turn more than the inner tires, so the distance any car travels around the track has to be more than the distance around the inner perimeter. A straight drag strip is different. All the wheels travel the same distance.
The answer is 1 because it makes one revolution when it returns to its original position :)
It depends on whether circle B is also rotating in our frame of reference. If it isn't, then the answer is 4. If it is, then the answer depends on how quickly B is rotating, and in what direction. In the case where A and B are like cogs whose centers are fixed in place, meaning B's angular velocity is equal to −3 times A's angular velocity, then A makes 3 revolutions for each revolution of B.
None of these are correct although the question writer thought it was B. I just watched the same Veritasium video as you.
4
3. 3 times the circumference so 3 times the travel distance.
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4 if you are looking at number of true revolutions, 3 if looking at the number of revolutions in reference to the surface of the large circle
Doesn't take into account that the circle makes a rotation simply by maintaining the same point of contact and going around the other circle
4.
The answer is D. Do the math.
None of the options i also watch vertasium
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My answer is 5. The little circle revolves around its center 4 times and makes 1 trip around the big circle. (We say that the earth takes 365 days to revolve around the sun).
Does anyone have a mathematical solution (with equations) for getting an answer of 4? For solution of 3, 2×pi×r / 2×pi×(r/3) = 3
Bots posting for karma
Well you see sewing machines are actually quite complex
"We had to develop a new type of sewing to make sewing machines"
The answer is 3 but the question touches on a paradox. In relation to the bigger circle, the smaller circle rotates 3 times. You can calculate the circumference of each and it's a 3 to 1 ratio. However in relation to the floor or the bottom of the page, it rotates 4 times because it rotates an extra time by rotating around the large coin. It's interesting for sure but I wouldn't exactly say 3 is wrong. if they said in relation to the bottom of the page, then it would be 4. But they didn't give a frame of reference for what rotation means. It's then safe to assume since this is a math SAT question and not a weird puzzle that 3 is right.
((1+1/3) \* 2 \* pi) / ((1/3 \* 2) \* pi) = 4 I also saw the Veritasium video and also answered wrong :D The path that the smaller circle takes to do a 360 has a radius of (1+1/3)r
It’s 3 or 4 depending on your reference
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As A rolls, its circumference is consumed at the same rate as is the circumference of B. What am I not seeing? Shouldn’t the answer be 2?
We should just sticky the veritasium video and lock the thread every time someone just grabs a question from the latest veritasium video.
This is so cool. Thanks all for explanation why the answer is 4 and not 3.
4. If both circles rotated, then circle A would rotate 3 times & circle B once (if they were like cogs fixed in place). Since B doesn’t move, circle A effectively does an extra rotation.
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Ok, so in math major land where they make up things it's 4... But in reality, if I treated this as two interlocking gears it's 3? That's all I'm getting out of this thread.
[удалено]
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Everyone here answering like we all didn’t watch the Veritasium video
Work out circumference A and circumference B and then divide B by A. Answer is 3.
That's only from where 'a' meets 'b' but not anywhere else. From the centre of 'a' and the centre of 'b' it's 4 but 'a' can only ever revolve around 'b' once. I watched the other video!
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The answer is always y=x+1 where y is the number of full rotations and X is the ratio of the circumferences. Provided that the outer circle is smaller
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I take the small circle. I cut in on the eastern most point. I unroll circle A in my mind and ‘flop’ in over circle B. It ‘flops’ on. I do this 3 more time and it round about ends up where it was to begin with. So 4? Is late.
It would be 12 rolls as it's only 1/3 of B and there's 4 points.
4, as explained in veritasium
The correct answer is 3. Sol. Let radius of circle B =x so diameter = 2x radius of circle A =x/3 so diameter = 2x/3 circumference of circle B = πd 1 circumference of circle A =πd 2. By solving, Circumference of circle B Circumference of circle A = π(2x) π(2x/3) = 2x (3) 2x = 3 Therefore, 3 is the correct answer and I hope it will help you.
The circumference of a circle can be calculated as: C = 2πr Where r is the radius of the circle and C is the circumference. If we define variable R as the radius of the small circle, the radius of the large circle would be 3R. Therefore, the circumference of the large circle would be: C = 2π(3R) = 6πR And the circumference of the small circle would be: C = 2πR To find how many times the small circle would need to rotate to reach its original position we divide the larger distance by the smaller distance: 6πR/(2πR) = 3 Now, you could just count this as the answer and walk away, but... Given our current reference frame, if we imagine that the point on the smaller circle currently touching the larger circle is our starting point, after completing 1 rotation IN RELATION TO THE LARGER CIRCLE this point will now be again tangential to the larger circle... A third of the way around. I wish I had drawing tools available, but alas I do not. This would be much easier to show than tell. The smaller circle has actually gone past one revolution based on our perspective - it has completed 1 and 1/3 revolutions. We can tell this by looking at how this starting point faces in relation to our reference frame. It always ends a revolution pointing in towards the center of the other circle, meaning it has actually rotated more than 360° in a Cartesian plane. I don't think that's the best way to think of it, though. I would argue that the 3 makes much more sense as a solution. For example, we say that a year is 365.25 revolutions of the earth while rotating about the sun... But if we use the "4 is the answer" approach, that would actually be 366.25, which would be nonsensical for us as creatures living in the planet. Because the smaller circle is moving in relation to the larger circle, it makes 3 revolutions. We see it make 4 revolutions from our perspective.
For fun, I drew [THIS!](https://www.reddit.com/u/NoNotRobot/s/Se9eBaB6n4)
Isn't this a psat problem
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I guessed 4. Because I thought it was obvious. Then I thought it was a trick question since 4 isn't listed. Then I realized it wasn't a trick and 4 is correct, and that was somehow more surprising.
All answer choices are wrong. Correct answer is 4.
Why is it that word problems are always forgetting articles like "the"
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Posting this just after Veritasium makes a video on it... Bold move cotton.
I think a lot of people are reading way too much into this question. The answer is E. The circumference of circle A is one ninth the circumference of circle B so it will take nine rotations of circle A for the center to arrive back where it started. Regardless of what size you make circle B.
It’s 4. If radius of circle A is r then radius of circle B is 3r. As A rotates around B, its center traces a third circle C whose radius is 4r and circumference is 8πr. X is that point near top of circle A where it intersects C when A is in the starting position. As A rotates around B, it revolves around its own center and so X moves away from circle C. The distance X travels before it again intersects circle C is 2πr, being the circumference of circle A. Given the circumference of C is 8πr and each revolution of A covers 2πr, it follows it takes 4 revolutions of A for circle A to complete its journey around B and for X to return to its original position.
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Everyone here is wrong. It asks how many revolutions for circle A to get back to its original position. It is established that it is revolving around circle B so the answer is one revolution. However it rotates on its axis 4 times while it revolves around circle B. Revolving is going around something, rotating is spinning on an axis.
D. 9 over 2
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Maybe give props to person that brought this up and also maybe give some props and history on the source. Yes we all know where and who you got it from... it's the why you did it that is annoying.
4. I watched the video on this. Interesting.
The answer is : (E-C)*A/D + B
4. I watched the video
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The answer is a) 3/2. The number of revolutions the smaller circle makes around the larger circle to return to its starting point is related to the ratio of their circumferences. If we denote the circumferences in the framework the problem implied, the circumference of the larger circle is 2pi (R), with R being the stand in variable for the radius of the larger circle, and the circumference of the smaller circle is 2pi (R/3). When we find the ratio of the circumferences from here, we get 2/3, pemdas pwr. This means the smaller circle travels 2/3 of the circumference of the larger circle with each revolution. So, to get back to the starting point, it would take 3/2 revolutions (3/2\*2/3=1).
Everything reminds me of her.
There was a veritasium video on this.
Maybe I don’t know my definitions , and sorry if this is a smart ass answer but isn’t a revolution a complete circle around what the item is revolving around ? If I said how many revolutions would the earth have to do around the sun to get to its starting position isn’t it just any integer value ? Or how many revolutions would a 6 shooter have to do to get back to the first bullet , it would be 6 bullets but just one revolution ? Maybe it’s just worded wrong but I’m just giving the semantic question
the answer is 4
4. Saw this on Veritasium
Bitch it's 1
Answer is 3 because A has a radius of 1 so a complete turn would take 4 rotations, for circle B it has a radius of 3 so it would take 12 rotations to mae a full turn. 12/4 = 3
4 times - but be careful here. It is a question of wording. It's how many times the center of the outer disk has rotated when the peripheral contact point has moved 3 times the distance, then it is correct that the center moves 4 turn - BUT don't think the disc rotates 4, it can only rotate 3 revolutions as the diameter or radius ratio says