Why did everyone miss this SAT Math question?
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- Опубликовано: 21 окт 2024
- Problem: Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will circle A revolve in total?
1982 SAT Question
In the figure above, the radius of circle A is one third the radius of circle B. Starting from position shown in figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A first reach its starting point?
(a) 3/2
(b) 3
(c) 6
(d) 9/2
(e) 9
Blog post summary: mindyourdecisio...
Sources:
www.donaldsaute...
www.nytimes.com...
Also see the Coin Rotation Paradox
en.wikipedia.o...
Update: some people have made really cool interactive worksheets at Geogebra. You should check these out!
Rolling Circles: tube.geogebra.o...
Rolling coin paradox: tube.geogebra.o...
A question about a rolling circle: tube.geogebra.o...
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Happy 4 year anniversary and 1 million views! Yes the answer really is 4, even the test makers admitted it! But it's a counter-intuitive problem. A lot of people who felt the video was mistaken now see why the video is correct, particularly after reading these links:
Sources
www.donaldsauter.com/rolling-circles.htm
www.nytimes.com/1982/05/25/us/error-found-in-sat-question.html
Coin Rotation Paradox
en.wikipedia.org/wiki/Coin_rotation_paradox
Geogebra demonstrations
tube.geogebra.org/m/787131
tube.geogebra.org/m/112822
tube.geogebra.org/m/107691
The question is revolve not rotate ,so it should be 1 ,
And even if its rotate it should be 3 ,if the sign "circle A" starts by facing circle B then the second rotation should be facing circle B too and again and again , and if we include math ,the radius of circle A is 1/3 of circle B ,
Let's just say B's radius is 360 then A's should be 120 and if we turned radius to distance it'll be the same, 3 circle A is equals to the length of circle B ,a so it means it rotated three times around B but the question is revolve ,so even how many rotation it gets ,it only revolved around B one time.
@@Waddles.st09 The real answer is 4 and he explained it perfectly. If you have doubts, consider checking the links provided.
I think you are wrong!
The answer is still = 3 !
You better mark with red-ink the spot of two-circles kiss^
@@rudychan2003 You're counting the number of rolls that way. If you want to measure rotations/revolutions, you should measure it with respect to X or Y axis.
I think rotation(own axis or center) is the correct word so it would be n+1.Revolution is always 1 just like Earth's revolution around the sun which equates to 1 year. 1 rotation equates to 1 day. Therefore, based on the question, the answer will always be 1 as it asks how many revolution.
It all depends on what you mean by "completing one revolution". If one revolution is completed when the initial point on A touching B once again touches B, which is what I originally thought, then 3 is the correct answer.
I agree with you. But main fault is in language aka 'revolve in total' according to viewer? or according to center of larger circle.
1) On small circle, simply draw a line from center to initial point contracting large circle.
2) Now rotate the model, you can easily realize you didn't complete full revolution.
So simple solution is (big circle circumference) 2 x Pi x 3R / (small circle circumference) 2 x Pi x R = 3
Just imagine you are seeing everything while standing on center of big circle.
Now if you are thinking of full revolve according to viewer, like the wiki explained, its cardioid. Just add 1 to above result. Simple explanation is rolling around a circle gives extra 1 rotation to small circle. If you flatten out the large circle, this extra rotation is gone.
Yes, exactly what I thought. If you were pushing a huge ball around the earth you would see it rotate n times, not n+1.
I would think the answer is 1 which is provided in the question. The small circle revolves around the larger circle one time, but it rotates on its own axis 4 times in the process of revolving around the large circle once.
I agree with you. I think all the incredible confusion about this comes from lack of clarity of what question is being asked. I got 9 first because I used the area formula instead of circumference.
Well "one revolution" obviously means it's returned to its original orientation. Why would you assume it's relative to the tangent of the larger coin? The question is frames as "the large coin is stationary, the small coin is moving" so the frame of reference is clearly the large coin.
The center of the small circle travels around a circle of radius R+r. Not R, as the test writers thought. So the number of rotations is 2pi(R+r)/2pi(r) = R/r + 1.
Excellent explanation!
Plsss, rotation and motion of center of mass are entirely different things, you are clearly mixing two different concepts here and it seems correct to most individuals, see center of mass is moving in a circle of radius R +r and it does not rotate while doing it, the circle does not the center of mass does
@@bhupeshyadav1340 when did he mention center of mass? 🤦🏻🤦🏻🤦🏻
@@macacopaco4664 center of mass of that circle is its center
@@macacopaco4664 I didn't write the reply proper, u could check it again
It rolls 3 times, rotates 4 times, but revolves only once. Hear me out.
A full roll is completed when an object progresses around the a surface of another, without slipping, until the contact point between the object and the surface has gone all the way around the circumference of the object and is touching again; a full rotation is completed when an object turns all the way around and is upright again with respect to the frame of reference (yes, this means that rolls = rotations on a straight surface or where the frame of reference rotates with the normal of the curve); a full revolution is completed when an object travels all the way around another object with respect to that second object (most people are familiar with the Earth rotating on it's own axis vs revolving around the Sun).
In the video example, you can see that:
1) the "bottom" of circle A, identifiable by the orientation of the text, faces the centre of circle B three times as it rolls;
2) the text is upright with respect to the camera 4 times as it rolls; and
3) circle A travels around circle B only 1 time.
Does that make sense to you?
Agree
Totally! Well put.
Yes.
@@user-zr6pl6nb6z Exactly
You are correct - he is not thinking this through
Another way to think of this is that the center of the smaller circle is 4 units away from the center of the larger circle; thus, the center of the smaller circle travels 4 times its original circumference. Since it rolls without slipping, that also means that its edge rolls through 4 times its circumference.
Interestingly, it works out in a similar manner if the smaller circle is INSIDE the larger circle. If we think about the center of the smaller circle, it travels through a circumference of twice its radius, which means the edge travels through twice its circumference. OR we can think of it as rolling around itself three times but it travels BACKWARD one rotation because that's how an inner circle rolls. Either way, and in general, the number of rotations of an inside circle will be n - 1.
What an underrated comment. This is the BEST way to understand it, thanks for explaining it so well!
except that's wrong - the center of the rolling circle is 4 units away - that center is making the circumference you're describing - but the edge of the rolling circle is 3 units away since it is resting on the edge of the fixed circle - the smaller circle's circumference is 1/3 the larger - so it will roll out 3 full turns - the correct answer is the old one - 3 - - put a dot on the rolling circle where it touches the fixed circle - and count the times that dot touches
The phrasing is ambiguous. It doesn't give a frame of reference, from the perspective of the bigger circle it is 3 times. From top down it is 4 times.
***** I know people will disagree about this so I'll give my take. I think the phrasing is precise, the problem is they wrote the wrong word. If you ask how many times the circle ROLLS around, the answer is 3. If you ask how many times it REVOLVES, then we have to count the number of times the circle returns to the same orientation. The question asked used the word REVOLVE and that is why the test makers admitted the error.
MindYourDecisions Honestly with the wording the way it is, I'd argue the correct answer is 1. Circle A only revolved around circle B once! You just told me that after all. :)
MindYourDecisions Buruma is right, it depends on the frame of reference.
If you observe it from the static bigger circle then 4 is the correct answer but if you observe it from the moving smaller circle then 3 is the correct answer. So the error the test prepares made is that they didn't define the frame of reference.
MindYourDecisions isn't that still depending on the reference? There are also two different definitions for the day. We also revolve around the sun but don't use the sidereal day...
I agree with you. it depends on reference point of circle A with respect to B.
Visually mathematically:
Take Circle B's edge. Cut out all the inside circle (so you only have the rim), cut the rim in any place so you can then stretch the rim out flat on the table. This is the perimeter of Circle B. It has a value that we know, its circumference, which is s = 2*pi*r = pi*d, where pi is obviously pi, 2 is 2, r is radius, and d is diameter. Basically, we're just laying out the length of the circumference of B as a straight line on the table and measuring the total length.
Now, take Circle A's edge. Same thing, cut out so you only have a rim, snip the rim, lay the rim out flat. The total length of this circumference of A is also known and can be expressed as s = pi*d.
Now, if we put the line we now have from A next to the line for B, the line for A is 1/3rd the line for B. This means it takes THREE circumferences of A to make ONE circumference of B.
.
Mechanically (e.g. the way you rolled the circles):
The problem with your circle is, at the 1/4th point, Circle A has NOT made a full circle. You're lining it up Cartesian like using the table as a frame of reference to say that Circle A has rotated 360 degrees WITH RESPECT TO THE TABLE. But this is not WITH RESPECT TO CIRCLE B's PERIMETER.
If you notice, the STARTING orientation of Circle A with respect to Circle B's perimeter is where the "bottom" of the circle is tangent to Circle B. Look at how the A's point is pointing AWAY from the center of Circle B. That is, in order to see when Circle A has made "one full rotation with respect to Circle B's perimeter", you must rotate Circle A until the tip of the A is pointing away from the tangent point/center of Circle B. Each time the tip of A points away from Circle B, you have made one full rotation of Circle A with respect to Circle B.
Note that where you stop at the 1/4th point, the tip of the A is NOT pointing AWAY FROM Circle B. It's pointing perpendicular-ish to it.
Timestamp ~ 1:44 you can see what I'm looking at. At 1:44, notice how the tip of the A is (more or less) NOW pointing away from Circle B. So at point 1:44, you have completed one full rotation of Circle A WITH RESPECT TO Circle B.
Notice also that this is at ABOUT the 1/3rd point...
...meaning that mechanically, we've now arrived at the same solution: Circle A makes it 1/3rd of the way around Circle B when it makes one full rotation WITH RESPECT TO Circle B, ergo, it will take Circle A three full rotations WITH RESPECT TO Circle B in order to trace out the full perimeter of Circle B.
.
This is kind of some Relativity problem, so you could argue reference frames here, but the fact remains that Circle A needs to "lay out" its circumference 3 full times around Circle B's outside edge in order to cover the complete circle.
This is true both mathematically and in terms of Circle A's motion WITH RESPECT TO Circle B.
So you can argue that frame of reference can give you different answers (which is true), but 3 is _A_ correct answer, and an entirely valid answer, per Relativity.
.
That is, you can say 3 AND 4 are BOTH correct answers, but you cannot say that 4 is THE correct answer and 3 is wrong. That would make YOU wrong.
Note that I mean this with no animosity, I just dislike people making absolute statements that are not actually dictated by the situation or evidence presented. If you are given a problem that has two answers (say, the square root of 4, which can be 2 or -2), and the list of answers has one of those (say -2) but not the other. Then the correct answer should be the one answer OF THE GIVEN CHOICES that is one of the correct answers.
In this case, they presented both 3 and 4, and 3 is ONE correct answer, therefor, of the options listed, 3 would be THE correct answer, since the other four answers were incorrect.
Now, if they gave you BOTH answers (e.g. for square root of 4 they gave you -2, -1, 0, 1, and 2), then you could argue that BOTH 2 and -2 are correct, so either you pick should be counted as correct.
In which case, going back to the circles, BOTH 3 AND 4 are correct, since it depends on the frame of reference you choose, and the problem did not state a frame of reference.
Agree/disagree? I get that this is a 6 year old video, so you'll probably never see this. :)
I saw my ans was 3 😀
(3*pi)/(1*pi)=3 you can calculate that on head. i mean you see its 3/1 LOL means 3 rotation
I though the answer was 3 without thinking much and was wondering what was making me uncomfortable with his explanation. Thx for pointing out that it was about the reference chosen, I might have ended up thinking about it for a while x)
This illustrates the problem with most, if not all, of these "only a genius" can get the right answer. The problems are usually ambiguous and only a psychic could get the "right" answer.
The question given seems to imply the reference-frame by mentioning that the smaller circle revolves around the larger one. It not only does travel the length of the circumference but it does so, not in a straight line, but revolving around a circle.
I‘m not sure as to which solution is actually correct.
“Rolls around” is actually ambiguous. One rotation could be seen as the point where the same point on the rolling coin retouches the other, in which case the answer is 3.
Yes I agree with this
i don't think i understand your point, when he did the empiric experiment, when the "bottom" point of circle A only touched the "top" point of circle B again, circle A had already rotated 4 times around itself
@@San-lh8us if you were to stand in the center of the circle and watch the smaller circle roll around when one revolution occurs, you should be seeing the smaller circle as it was when it started so you would be able to read it and therfore it only goes around 3 times
More clear than this ...impossible ! The right answer is 3
@@San-lh8us Watch the video again and make a mental "dot" on the bottom of the small circle. When he stops on the "first revolution" the "dot" would not have touched the bigger circle yet, showing that the small circle had not yet completed a full revolution. There are 4 moments where you get the first circle in the same orientation as the starting one, but that does not mean it made a full circle around the bigger circle yet.
However, circle A's "tread" only engaged the surface of circle B exactly 3 times.
From a topological pov, it only made 3 revolutions.
That's the common mistake, it's a circle, it's not about how many times it made contact with circle B in regards to it's circular rotation, but how many times circle A revolves in total, the same question can be applied to any relevant shape, while if it was the former I wouldn't work.
The small circle makes 3 revolutions if your perspective is from the center of the other circle, but 4 revolutions if your perspective is outside the circles. They could have worded the question more clearly to distinguish it
@@Hhhh22222-w you are wrong. You are counting an orbit as a revolution. If this was a gear and the small one was fixed it would rotate only 3 times to get the larger one to rotate 1 time. Perspective is important and unless it specifies in relationship a parallel line at the bottom it only rotates 3 times around the center of the larger circle.
This is similar with the problem where a train 1 mile long enters a tunnel 1 mile long at 1mile/hour and it needs 2 hours to get on the other side
@@mumrah73 excellent point.
Both 3 and 4 are correct answers, depending on semantics. When the small circle has advanced 1/4 of the perimeter of the big circle, it has rotated/revolved one time according to our visual reference, but has not finished a full rotation in terms of where it is tangent to the big circle.
So if the frame of reference is the small circle by itself, the answer is 4.
If the frame of reference is where the two circles are tangent, the answer is 3.
Get it clear.
The small circle does not do 1 rotation in1/4 of the large circle. it takes 1/3!!!
absolutely! thanks for clearing that up! i knew it was some trickery but did not catch that until you pointed it out. only 270 degrees of the small one touched the large one through the first 'rotation' that reached a quarter of the circle. multiply that by 4 and we get 1080 degrees, or three revolutions of circle A. it only looks like 4 revolutions because our minds assigned a top and bottom to circle A without noticing the points of contact around the rotating revolution. it's like the moon kinda, right?
The words revolve and rotate are not synonyms. Look up how those words are used in astronomy for example.
The words revolve and rotate are not synonyms. Look up how those words are used in astronomy for example.
The reason why so many people get it wrong is because they mistake the word, 'revolve', for 'rotate'. The correct answer is 1 because circle A revolves once around circle B. Bahahaha! Case closed.
3 rotations relative to its "track', but since its track is oriented as a circle 4 rotations to our eyes. Flatten the track to a straight line, and it will only be 3 rotations relative to both.
No need to flatten. Orient the small circle on a plane 90* to the plane with the big circle. Now you got your answer.
Thank you. I was so confused because in my head I imagined the line flattened out and easily thought 3 times. I was so confused when he demonstrated 4 because I didn't notice what went differently.
@@gotnoshoes99 what's your point?
Oh, thanks for this explanation. I tried to wrap my head around this "magic", but this helped :D
@@johanness3850 The point was the correct answer was not displayed as an option
The actual correct answer is 1. It rotates 4 times while revolving once around B.😀
No - In this case rotate and revolve mean the same thing. Would the test makers ask you to answer with a number stated in the question?
I also thought the same, there might be some ambiguity while in maths, of course when they make questions they don't want you to think literally.
I think the question is wrong
Math and verbal in one question!
@@me.unpredictable280 well in interviews this Q might make certain discussion as there u need think literally
I got to 3, here is how I did it:
Every rotation of the smaller circle covers a distance equal to it's circumference, therefore, I treated each circle as a straight line, and then compared them.
c= 2pi(r)
With r being the circumference of circle A
C= 2pi(R)
R=3r
C=2pi(3r)
A/B = 3
So, the A circle needs to travel 3 times it's total circumference to make it around B, which means 3 full rotations.
After watching the video and reading the comments:
My answer seems to hold true if the circles are stationary and would spin in place, such as in a gearbox, OR the B distance is a true straight line allowing A to move in a straight line.
However, they aren't stationary, and B is a circle, so that throws a wrench in it.
I'm trying to understand how A rotates 4 times as it travels around B.
My only explanation is that the center point of A is travelling a larger circle than the circumference of B as A rotates around.
This means that the distance A travels in the question is actually longer than the circumference of B.
So the real equation I should use is:
c=2pi(r)
C=2pi(3r+r)=2pi(4r)
A/B=4
Others have pointed out that there is a perspective issue at play.
A rotates 4 times, relative to the perspective of the viewer.
However, when considering the points of contact between A and B, there are only 3 rotations.
ie. the "bottom" of A is only in contact with B at 3 points.
I find this argument compelling, because from the perspective of A, it has only rotated 3 times.
If I was standing on A, looking directly at B from the start, I would only be looking directly at B 3 times during the trip.
So, I suppose it comes down to definition - what does "revolve" mean?
Is it relative to B, or is it relative to the viewer?
Personally, I would argue that the way the question is written, they meant "revolve" to be relative to B, and therefore 3 is correct.
But, it is not explicitly stated which perspective is correct, so it is still vague enough to be open to interpretation.
You explained my thoughts perfectly
There's no relative to B or relative to the viewer. Circle A does rotate 4 times when B is a circle.
You're right when saying that the center of A travels longer when B is a circle, compared to when B is stretched into a straight line.
However consider this, if we forget about the rule r=R/3, and let's say B is so small that its radius is 0 (basically a dot), what would your answer be when asked how many times circle A revolve around circle B ? For me when B is a dot, A revolves 1 time. If you agree with this logic, then the answer to that SAT problem is indeed 4.
Yeah, the gears would be 3, but since the the small one is moving, we were all fooled into thinking it was moving a distance of the circumference of the big one, but that’s false. The little one is actually moving a distance of 2*pi* (R+r) because the small circle is not a singular point, it’s a flat object with length and width, therefore you approximate the small circle with a singular point at its CENTER. So the center of the small one is doing all the translational motion, and it is a distance of (R+r) from the center of the big one, and traces out a circular path whose radius is (R+r). So that’s the total distance neeeded to be covered, not 2*pi*R, but 2*pi*(R+r) by the edge of the small circle.
I'm definitely in the "it rotated 4 times" camp. The question wasn't "how many times does the coin rotate in respect to the center of circle B. It was simply, "how many times does it rotate" and if you were inside of a ring standing at the top of Circle B when it starts rotating, you'd feel yourself go upside down and back upright 4 times. The coin example he used shows this super clearly.
@@nahog99
Instead of rotating a coin around the outside, if we take a pen and move it a distance of 2π around the outside of circle B (where circle B has a circumference of 6π), we'll find it's moved through an angle of 120 degrees. If we only move the pen 90 degrees around circle B's central axis (where circle A will have rotated once), we'll have only gone 1.5π which isn't the same distance as the circumference of circle A.
If we model circle A as the Earth and circle B as the Earth's orbit, and classify a "revolution" as each time the Earth sees noon on its way back to the same axial position relative to the centre of the Sun, and "rotation" as the number of rotations around Earth's central axis that it does in that same time, we arrive at 3 and 4 respectively.
The question is accidentally ambiguous in that revolution can mean how many times does circle A orbit (once), how many times does circle A "see noon" (three times), or how many times does circle A rotate around circle A's central axis (four times) in the same time.
I just figured it out. The actual CIRCUMFERENCE of A touches the CIRCUMFERENCE of B 3 times during its trip around. But the angular position goes through (360*4) degrees because it's a circle going around a circle. So despite the circle ending back upright after 1/4 of the trip around or 90 degrees, the original contact point between circle A and B doesn't come back around and touch B until it's 1/3 the trip around, or 120 degrees.
Yes, so technically, both answers can be correct depending on the frame of reference. If your frame of reference set to have an axis perpendicular to the perimeter of circle B, it would look like how it was drawn with a straight line. If the "simpler" frame of references of the viewer is used, then 4 is the correct answer.
Remember that the frame of reference is probably the most important information in questions regarding motions. If it's not mentioned, many answers are correct.
It's called the coin rotation Paradox.
You are correct! The answer is 3, not n + 1. Circle A begins its rotation contacting circle B from its 6 o’clock position and circle B’s 12 o’clock position. Circle A will have completed one rotation when the 6 o’clock position contacts circle B again, which happens at circle B’s 4 o’clock position or 120 degrees. If you want to duplicate the illustration in the video use tick marks at circle A’s 6 o’clock and circle B’s 12 o’clock positions and you get three rotations for circle A. Math doesn’t lie! You can also illustrate this with lengths of string.
Ohhh, thank you! Of course, because it isn't rolling on a flat surface, it looks like it rotates more... Wow xD
@@altersami9660 hi so you still think the answer can be 4? Can you help me with it? I don't understand the "have an axis perpendicular to the parameter of circle B" part
@@attaboy631 it is a trick question... you know what is right ... but they word it so it is ambiguous....
This Video is incorrect. I wrote this in a reply to somebody, and didn't see anyone else post this.
When he rolls the 'circle A' around 'circle b' there's a total of 3 full rotations by Circle A. A full rotation by circle A is when the initial contact point of circle A comes in contact with circle B. An easy way to visualize this is with his coin example at 2:40. You can notice that the initial contact point is directly below the head on the coin that is being rotated. This initial contact point only makes contact once per full trip around the circumference of the stationary coin.
The visual representation just makes it look like the coin is revolving additional times. The coin\circle just appears to be revolving more because it's being rotated and moved in a circle. It's not actually revolving more.
Actually, the question only concerns how many times circle A revolves, not about circle B, so you do not consider hoe many times the contact point has touched circle B, but the actual spinning of circle A. If you magically cover up circle B (or just use a piece of paper I guess?), you will see that the circle revolves 4 times. Otherwise you are making the same mistake the testmakers did.
Remember that 1 revolution = 360degrees of rotation. Let the moving circle radius be r, the stationary circle radius be R. The moving circle travels around the stationary circle for 360 degrees, so the number of revolutions is (R/r)(a/360)+(a/360) where "a" is the angular distance travelled by the moving circle, in degrees (therefore a/360=1 when the moving circle has returned to its original position the first time). Simplify and you get (a/360)((R/r)+1).
+P Lee it literally states a revolution is when circle A reaches its starting point.
+P Lee "Actually, the question only concerns how many times circle A revolves, not about circle B,"
Actually, that's precisely what it does ask. "around Circle B".
+Missed Lethal "A full rotation by circle A is when the initial contact point of circle A comes in contact with circle B."
pls explain me how you got to that statement
i think a full rotation is, when the circle turns 360°
+MrZauberwuerfel "Circle A rolls around circle B one trip back to its starting point" can be interpreted in two ways, which is causing the confusion. The first way is that circle A rotates around circle B until it gets back to its point on top of circle B. This interpretation will lead to the answer being 4, as it is assumed a revolution is 360°. The second way is that a revolution of circle A is when the initial point of contact to circle B is once again in contact with circle B. This will lead to the answer being 3.
In the context of the test 3 is the correct solution, as it was one of the answers provided. The question is poorly worded and ambiguous, but it is hyperbole to say that everybody got the question wrong.
Can you solve it mathematically?
Not with coins or cardboards.
I can solve it mathematically
I don't know how to mathematically explain the idea of the author that leads to the answer =4.
However, for the answer =3, it's the following:
Let r be the radius of the small circumference. Then the radius of the big circumference is 3*r.
The generic formula for the perimeter of a circumference is P=2*pi*r.
For the small circumference, P_small=2*pi*r
For the big circumference, P_big=2*pi*(3*r)=6*pi*r
The ratio between them is exactly 3.
I think i can
I do it by measuring the distance of the two centers of the circles then dividing it by the radius of the small circle
Yes but 3
Three. You have to consider the starting points. Six o’clock for the smaller circle and 12 o’clock for the larger circle. The circles resume there initial positions after three complete rotations. This mistake is counting a full rotation of the smaller coin when it is oriented upward, not when it has traversed 360* of arc.
Exactly, circle A is revolving relative to circle B, not to you! Draw an outward line on circle A and count a rev every time it touched circle B, you’ll find the answer to be 3
Thank you! I thought I was the only one who noticed this.
Your mistake is also assuming from the circles POV, not from your pov or a person's pov, that's the point of the question and what it's testing, if you assume from a pov your answer will always be wrong
just divide the total possible positions of circle A as it travels around circle B, which is 20 possible position by the possible instances of circle A, which is 5, the hint is that circle A is 1/3 of circle B, in other words you can fit 3 circle A in circle B, then just find the sum of how many times it can fit outside circle B without being in a different angle i.e measure vertically from the starting point, you get 1 (starting) + 1 (adjacent to starting) = 2 + 3 (circle A inside circle B) = 5
20/5=4
@@Hhhh22222-w Sorry, but your explanation is very hard to picture in my mind, and possibly overcomplicated. But, I understand what the video says, which you agree with. However, agreeing with these commenters you're replying to, the issue with this SAT question is that it isn't specific. The truth is both 3 and 4 revolutions are correct depending on the perspective taken of what rotation we are determining. The correct desired perspective was not indicated in the question. If you look at the circle rotating from a birdseye view with you (and North remains North) then it is clear from the video that the smaller circle will rotate 4 times as it goes around - and there have been many good explanations of why this happens on here - I especially like someone's explanation of how the center of the smaller circle will travel a radius of the smaller plus the larger i.e. 3+1=4. If same size coins then 1+1=2, etc. This explains why the n+1 equation.
However, as above, the circumference of the smaller will be 1/3 of the larger, so each point on the smaller coin will touch the larger coin EXACTLY 3 times as it travels around. From this perspective, it rotates 3 times, not 4. and is intuitively what people think the question is asking. They are thinking from within the larger circle and easily see the circumference of the larger is three times as big. They are not thinking of the birdseye view and asking how many times the coin will spin on its axis as it travels around. In fact, that would be a better-stated question, how many times will the smaller coin spin 360 degrees on its axis as it goes around the larger circle once? Then you would have to have learned the n+1 logic the questioner wanted apllied.
@@JohnFeng1 That's a different question, not the one asked.
No, you consider a roll based on surface area touching the circles, and in that case it rolled around 3 times. The act of moving the circle across the 2 dimensional plane distorts your perception of what a full revolution is. The "circle A" won't be straight up when it makes one revolution, it will be tilted to the side somewhat and so on... Think of it as if you drew an arrow from the top of Circle A to the bottom of Circle A. One revolution will be when that arrow is pointing directly at Circle B, not when "Circle A" is right-side up and readable. The correct answer is 3.
Wrong, but you have clearly demonstrated why so many mistakenly think that 3 is correct. You are adding things to the problem that are not there to begin with. You have no basis for choosing "when [the] arrow is pointing directly at Circle B" to be your point of reference for a complete revolution. The problem is presented as two-dimensional objects moving across a two-dimensional plane which we are viewing from above. This is the only valid perspective to take. If the problem asked how many revolutions Circle A made relative to Circle B, then you would be correct, but it did not ask this. From the point of view we are presented, Circle A completes 4 revolutions relative to its starting orientation.
+Vicky Pawar
All that demonstrates is that the ratio of the circumferences is 3:1, but that is not what is being asked. By only looking at the movement along the circumference of the Circle B, you are changing the problem from a two-dimensional problem to a one-dimensional problem. It's like trying to measure the volume of an object by looking at its shadow.
. You did not approach the problem mathematically , the circumference of A is 1/3 that of B , which means that , in order for A to travel a complete revolution it needs 3 x 1/3 x 2 x Radius of B x pi which is 3 whole revolutions .. Good thinking my friend
The answer is three only. If you don’t believe me try doing the coin experiment ( the one in this video) yourself but just mark half of the coin border with a black marker. Now rotate the MARKED coin, it will be 1 revolution/ rotation only ( note that for one full rotation/revolution the black marked part SHOULD come in contact with the other coin)
Jacques Nicolay I know,by my thinking and by my calculations the amswer is 3(Note:I haven't watched the vide yet but i am assuming that his answer is the same)
The problem with the cutout example is that where he stopped at "1 revolution" is not a full revolution. The first revolution occurs when the bottom of circle A touches circle B, not when the letter "A" is vertical again. He was counting 3/4 of a rotation as a full rotation, which is why he got 4 instead of 3.
No, his answer is correct, because rotations are from the perspective of the camera, not from the perspective of circle B. When the bottom of circle A touches circle B like you're saying, circle A will be 1/3 of the way around circle B, but since the bottom isn't pointing down anymore, circle A actually rotated 1+1/3 times. So the total is 4.
Rotation and revolution are not the same thing. Rotation is for orientation. Revolution is for traversing the entire circumference. So, no, his answer is wrong.
If we're using the real definition of revolution, the correct answer is 1, because circle A revolved around circle B only once, regardless of how it rotated. :)
Nope. It doesn't say how many times will it revolve around B. It says how many times will "A" revolve as it circles B. There is a *huge* difference between the two. The language is clear that the revolution to be counted is the number of times the point on A that corresponds to the tangent of A and B moves from a tangential point back to a tangential point, i.e., a rotation. That is what a revolution of A is defined as. That tangential intersection occurs three times as circle A rolls along the circumference of circle B, not 4. [Straightening out the circumference of B into a line (see my first answer below) shows exactly what a rotation of A means. If you mark the spots were the starting point of A hits line B and then roll line B back up into a circle, the circle will have three marks on it. It's that simple.
John Roe
You're absolutely right that A rotates 3 times relative to a perspective centered at B and rotating to always aim at A. But this perspective is not actually a reasonable one, since rotating perspectives are not inertial. From a fixed perspective, e.g. the paper, the test-taker, etc., A rotates 4 times and revolves 1 time--there's no other answers.
What's missing in this question is a specified frame of reference. If the frame of reference is circle B, then the answer is 3. If the frame of reference is the top of the page, then the answer is 4.
That's a good point. But in the absence of any specific designation of a point of reference, then certainly the POR must be that of the viewer who is looking at the problem.
@@Milesco - in my interpretation - the POV is the fixed circle - i envisaged a blank circle - so i didn't visualize the image's orientation at all
I'll admit I missed this one as well, but it sorta reminds me of how the Earth rotates and revolves around the Sun. It only takes 23 hours and 56 minutes for the Earth to rotate 360 degrees, but a day is 24 hours because it takes the Earth an additional 4 minutes to turn another degree or so in order for it to be facing the Sun in the exact same way, because it has advanced about one degree in its orbit.
"Sidereal Day"
The day is actually not 24hrs, we measured it and made it flat. Like we made a year 365 days plus leap year, but it takes 365.25ish days for one ellipse.
there it also helps to look at the reference you use... the 24 hours is about the time it takes for the sun to get back to the highest point (noon - to - noon)... but the 23 hours and 56 minutes is the time it takes for the stars to stand in the exact same position as the previous observation...
you read my mind - and I knew someone else here likely thought of the similarity to astronomy
I was thinking the same thing. The Earth takes about 365.25 days to orbit the sun. But it rotates 366.25 times.
Interesting to see where the extra revolution comes from. If circle B was simply a single point with no circumference, Circle A would still take a single revolution to rotate around that point. That is the extra revolution, regardless of how large or small circle B is.
I like your explanation!
There is no extra "revolution." Take the larger circle and unroll it to a single single line, if the ratio is n then only n revolutions occur. Revolutions confers position as relative to the two circles in this case where the two circles touch (for roughly similar example see the earth revolving around the sun), thus only 3 revolutions occur for the smaller circle in this example. The n+1 is not the amount of revolutions it is the movement relative to an outside observer. Another way you see that the revolution of a smaller circle revolving around a larger circle of n ratio is to put a marker at the point where the smaller circle contacts the larger and you will find that only n revolutions occur.
@@nerofl89 So you're saying that despite the fact that visually observing its trip shows a revolution relative to it's starting position once every 1/4 of the trip, we have to ignore that? If B was a dot, A would essentially be doing a complete revolution around a single point. Even though B has 0 circumference, A still completes a single rotation. Number of rotations relative to starting position is still N+1. I think this is the missing element. Then again, I suck at maths, and I'm sure equations bear you out. It would depend on an interpretation of the question as to the "correct" answer.
@@SheyD78 Here in lies the problem with Presh's reasoning: "Circle A rolls around circle B" that means that perspective wise we are discussing the relation of Circle A to Circle B. For instance let's reword the question to something more familiar: Earth rotates around the sun one trip back to its starting point. How many times will Earth revolve in total? We would never say ~366.25, would we? No, it is ~365.25, why, because we are talking about perspectives of the Earth to the sun. Presh's n+1 argument is a perspective of an external source and is relative solely to Circle A while ignoring the fact that the question makes it quite clear we are dealing with the relations between Circle A and Circle B. Now could the question have been more explicit? Yes, but this video is a clear example of overthinking a question while ignoring the hints it gives.
The mistake is to look at the edges of the circles. Look at what the CENTRE of the smaller circle is doing. It's describing a circle whose radius is the SUM of the radii of the two circles (in this case, 4). Flatten that out as in the illustration, and you get the right answer.
You literally revolved 3 times in your example, you just counted it as 4. Place a mark to where the circle A touches and you will see what I mean. The circumference of the circle * Revolutions= distance traveled. If we let radius of B=1, then the circumB=6.28. Therefore the radius of A=1/3 and the cirumA=2.09. In the original formula, solve for Revolutions and you get Rev.=Distance/Circum. The distance Circle A traveled is equal to the circumB. Therefore 6.28/2.09=3. Not exact because I rounded Pi.
u r right bro
exactly!
No need to round up pi
It gets cancelled
Exactly. You don't need π to work this out. You are working out the difference of two related circumferences. The circumference of a circle is, of course, 2πr. As the radius of B is 3 times that of A, you can work out the answer as a simple calculation [using r as the radius of circle A]: (2π3r)/(2πr). Multiplication is commutative, so this is the same as (3(2πr))/(2πr) which is clearly 3 as the two 2πr terms cancel each other out. The circumference of B is three times that of A, and that was clearly meant to be the desired answer, but as the video states, circle A also does a complete revolution whilst going around the static circle B, so the real answer to the question as stated is definitely 4. That's the great thing about maths - the answer is always correct. It's up to the exam board to ensure the question is correct.
Phil Rodgers-
The reason that your answer is not in the multiple choice is because you are WRONG!
At the beginning, you will note that the writing on the small circle is perpendicular to a line drawn from the centre of the large circle to the point where the two circles touch.
When you rotate the small circle on the circumference of the large one, it will have completed 1 revolution only when the writing is perpendicular to the line again, which is when 1/3 of the large circle has been covered.
You did not do a complete revolution of the small circle, you only brought the writing back to the horizontal.
This means the answer is 3, not 4.
It would seem that the writers of the original question had a definition of 'revolve' that was relative to the center of Circle B. It's like the reckoning for an Earth day; it's slightly more than a full rotation each day (2.737% more) because the measure for a day is relative to the sun, rather than to the 'fixed' stars. This is 'sidereal' versus 'solar,' or 'tropical' reckoning.
Do the experiment again, but before starting, put a mark on Circle A at the contact point. As you spin Circle A around Circle B, count how many times the mark again contacts Circle B as it completes its trip around. The sidereal measure of it is 4, but the tropical measure is 3.
This is a really great explanation. I had this thought process in my head but couldn't get the vocabulary together. Thanks!
I am from the north. We do not 'reckon' up here, we do figurin' instead.
I got 6 because circle will revolve 3 times to get to the radius of circle b=half
Another 3 times =6
Kelli, thank you for explaining that for me. That is exactly what I was thinking when I did the problem, but you helped clear things up for me.
A revolution is with respect to the smaller circle. It did turn around to completely and hence its correct.
The answer is indeed n+1. I dont get what people are tryinf to point by not touching at at the below point because thats not how revolution works in this case. If we were to do this in a straight line then it would have been correct
. but touching of the point at the exact same place is just a method to keep track of revolution and not the definition itself.
To see what i mean for the equal radius condition look at the above and below part, if you were to focus only on the smaller ciecles in the two cases they are indistinguishable hence they DID complete a revolution
I could (if you want) show you mathematically given you know the concept of center of mass and rolling.
If you stretch both circumferences, their lengths are 3 to 1. If you fix the little circle and turn one time the bigger one, the little wiĺl do 3 turns.
When the little circle revolves, it do the 3 turns plus one more because the visual change. The initial contact point isn't the same at 1/4 of the turn, but the little circle center trips.
Maybe I'm misunderstand here... But the question states
the RADIUS of circle A is three times the RADIUS of circle b.
The radius is the measurement from the centre of a circle to
its perimeter. This might make more sense if I give an example.
Let Circle A's radius = 1
Therefore circle b's radius = 3
To work out the circumference the equation is 2Pi x r
Pi is close enough to 3
So the circumference of circle A = (2x3)x1 = 6
The circumference of circle B = (2x3)x3 = 18
If circle A is to roll around circle B it is the same as opening
up both circles to form a line and comparing lengths.
So a line of length 6 will fit onto a line of length 18 three times
Therefore the answer is 3.
+James Birkett The answer is 4 as illustrated in the video. Check out the sources and references for more information. (it's in the video description, I have copied them here)
Sources:
www.donaldsauter.com/rolling-circles.htm
www.nytimes.com/1982/05/25/us/error-found-in-sat-question.html
Also see the Coin Rotation Paradox
en.wikipedia.org/wiki/Coin_rotation_paradox
Update: some people have made really cool interactive worksheets at Geogebra. You should check these out!
Rolling Circles: tube.geogebra.org/m/787131
Rolling coin paradox: tube.geogebra.org/m/112822
A question about a rolling circle: tube.geogebra.org/m/107691
+MindYourDecisions OK i see how it works... Kind of... But is there a mathematical way of proving/showing this?
+James Birkett The fact of the matter is, that when circle A revolves, it doesn't just have to revolve around circle B, it has to revolve around it self too. The total distance; is the circumference of circle B + the circumference of circle A.
It's a mechanical problem, and it's just a given fact that the speed of revolutions is affected by the relation between the two circles. You use it in gears all the time.
+James Birkett Imagine it this way. As the circumference of the smaller circle is 1/3 of the big one, each 1/3 segment of the big one, the smaller one is spinning itself too, "travelling" 1/9 faster (1/3 of the 1/3). So, the third time it spins, the smaller circle has covered 3/3 of 1/3 of the bigger circle, so it has a full spin left.
Remember the smaller circle saves time reaching the same orientation by spinning itself.
+MindYourDecisions The answer is not 4... Here's my solution: Radius A = 1/3 Radius B, Circumference B =2*Radius B*Pi, Circumference A= 2*Radius A*Pi=2/3*Radius B* Pi, Circumference B/Circumference A=(2*radius B*Pi)/(2/3*radius B*Pi)=3
1 Revolution
3 Rolls
4 Rotations
Enough said.
The catch here is that there's one revolution relative to the bigger circle, and the question creators used revolve = roll. Anyway, you're absolutely most accurate.
Solar vs. Sidereal. ;-)
Men you are wrong .are you have takes correct frame of reference
Beautiful.
That's what I got. The question is ambiguous. Do they mean how many revs does the smaller circle make about its own axis? That would be 4, which wasn't one of the answer choices. It is def 3 rolls to roll around, which was one of the choices. In the SAT you're supposed to pick the best choice, so the answer you would pick is B, 3. But it's a bad question.
The fact that 4 is not an option removes any ambiguity from the question. It should be clear that the creators of the test expected 3 to be correct, and those who answered 3 got the mark. Anything beyond that is just interesting conversation.
They expected 3 to be correct, but they were wrong. The answer is clearly 4. Circle A revolves _around_ Circle B. Revolving _around_ Circle B gives Circle A an additional rotation. That cannot be ignored. Especially when the question specifically asks how many times will Circle A revolve _in total._
+Captain Quirk i never said that 3 is correct. I said that the creators wanted 3 to be correct, and the fact that 4 isn't an option should make it easy for anyone to infer that, despite being incorrect, 3 would get them the mark on this question.
@ Justin : Perhaps, but that involves some degree of mindreading capability. I would say that it's asking a lot to demand that a test-taker infer the "best" incorrect answer from a batch of incorrect answers.
@ downunder diva : If the writing starts out level, and ends up level, that is a complete rotation. As I said above, _revolving around Circle B_ gives Circle A an additional rotation. Circle B is not a straight flat line. It's a circle. So revolving around a circle gives Circle A an additional rotation. That's why the total number of rotations (which is what the question asks for) is 4 and not 3.
Captain Quirk I completely disagree. It doesn't take a mind-reader to reach the conclusion that they're comparing the circumference of each circle, and that the assumption is since one circle is 3 times the circumference, it would revolve 3 times. None of the other answers come even close to making any sense at all. It's certainly not as simple as selecting the best incorrect answer from a batch of incorrect answers.
The circumference that matters here is made by the path traveled by the center of coin A. So you just add the radii of both circles and solve for circumference of the new circle "C" and divide the circumference of C by A.
A = 1
B = 3
C = A + B
1*2*3.14 = 6.28
4*2*3.14 = 25.12
25.12 / 6.28 = 4
The above works for any fraction.
Any fraction with a numerator of 1 can be solved by simply adding the numerator and denominator together.
Example:
Instead of A being 1/3 of B, lets make A 1/17 of B.
A = 1
B = 17
C = 18
1*2*3.14 = 6.28
18*2*3.14 = 113.04
113.04 / 6.28 = 18
Yours is the most helpful way to understand the principle at work. Thanks.
@@rbeehner2 Thank you and I'm glad you found it to be a good explanation!
Aha! Now I get it! 🙂🙃🙂🙃
This is the real explanation
Real Answer:
'1'
Reason: Question says 'Revolve', not 'Rotate'!
Thanks.
You're right. The question asks revolution not rotation.
Exactly.
Definition of revolve
intransitive verb
3a : to move in a curved path round a center or axis
b : *to turn or roll round on an axis*
Applying rolls from statement 1 to circle A and revolve in statement 3 applying to C A they are the same as the touching points of CA and CB will separate and the point on CA will "rotate" for its rolling motion but the center point defining the location of CA will revolve around the center point of CB. This problem can be solved with both math and Geometry.
oof, mincecraft
The maths never lies. The answer is 3. It only LOOKS like it has completed 4 revolutions but in actuality, every time it APPEARS to complete one revolution (from our perspective), it has only rotated 270 degrees. Really, it rotates 360 degrees a third of the way around the circle, as expected.
And it is fairly well known that the images are not meant to be to scale
The math never lies...unless you use it wrong.
No, the answer is 4. You need to take into account the fact that the center of the circle itself is always rotating about the tangent point. Instead of having the small circle revolve around the outside of a circle, have the small circle revolve around the inside of a hoop that has exactly the same circumference as the big circle. You'll see that in that case, the small circle revolves only 2 times.
ft55555 yes I agree that the centre is also rotating, which is precisely why one revolution LOOKS like (but ISN'T) 1/4 of the way round the outside (and 1/2 of the way round the inside like you mentioned). I'm already allowing for the rotation of the centre of the circle by picking a reference point on Circle A and rotating it about Circle B. When I do this, one revolution occurs 1/3 of the way round.
Also, the maths never lies. One revolution is 360 degrees. Circle A rotates 360 degrees a third of the way round. Hence, the answer is still 3. QED
I don't think you understand what revolution means. "Revolution" is one object moving about another object (i.e. "orbit"). In both cases, whether the small circle revolves (or "orbits") around the outside or inside of the large circle, it completes one revolution. "Rotation", OTOH, is defined as "movement around an axis", and in this case, the axis is its own center. Depending on how the small circle revolves (i.e. outside or inside the large circle), it rotates about its own center 4 times or 2 times. It's not about whether or not the math lies. You're simply misunderstanding what's actually being measured. The only thing that is the same whether the small circle negotiates the larger circle on the outside or the inside is the distance that the TANGENT POINT moves--i.e. the tangent point moves a distance that is equivalent to the circumference of the larger circle in both cases. However, think about the distance that the center point of the smaller circle travels whether it's around the outside of the large circle or the inside of the large circle. In that case, there's an obvious difference. That distance dictates how many rotations the small circle actually completes. If you don't believe me, take an Eisenhower silver dollar and have it orbit a smaller coin like a dime. You'll see that even in that case with such a huge discrepancy in circumferences, the silver dollar completes at least one full rotation about its own center. In fact, any object that completes one revolution around any arbitrary fixed point will complete at minimum one rotation about its own center as well.
I interpreted the question as "how many times does the circumference of circle a fit into the circumference of circle b" which would be 3.
Circle a does rotate by 360° 4 times.
I'm not a native english speaker so I might misunderstand the word revolve but it feels as though my interpretation leading to 3 as an answer is more correct.
You can in fact prove it with a practical example. Instead of writing "circle a", draw an arrow on the smaller circle.
Initiate it with the small circle above the big one with the arrow pointing towards it. Then roll the circle around and count every time the arrow points towards the circle. It's 3 times.
What you counted is basically how many times the arrow would have pointed downwards isn't it?
I thought the same thing
Maybe I'm wrong, but I thought his mistake with the coins was that he counted each 180ﾟ turn as a rotation, and if you counted each 180ﾟ as a rotation it would be 6, not 4. Do you mind explaining a little more? I don't get where the idea of adding one came from.
@@LilyHLilyH no he was correct. But I don't know how to explain in words.
I believe this would be wrong. (Please watch 1:30 again to help you imagine it) Let's say that you draw an arrow pointing down and start at the top of big circle. When the small circle is at the right of the big circle, the arrow would point down again, right? The arrow had completed a 360 degrees turn and points down again. This is what we count as revolving. If we made the arrow point at the big circle before we count it as revolving, then it would be 480 degrees I believe, then the small circle would revolve (1 + 1/3) times.
@@LilyHLilyH think of the circle on the straight line example. From the beginning of that line, roll the circle along the line such that the circle makes 1 full revolution. Now, superglue the circle to the line. Then, try bending that line into the bigger circle shown in the initial question. You'll realize that the smaller circle that you rolled along the line isn't upright anymore. It will have made a little more than one full revolution, even though you clearly remembered rolling it exactly once. Because the straight line is now curved.
For your experiment with the coins at 2:42, the coin has, in actuality, only rolled half of its circumference by the time it reaches the bottom of the other coin.
Haha I hella missed that. Good eyes 👀
Actually half the circumference of each coin was contributed, thus one full circumference.
@@StatsJedi Sophistry. It doesn't work like that. That's like saying I walked half a mile and the ground underneath me walked another half, so I really walked a full mile.
I'm pretty sure the coin turned 360
@@dan1e1473 Yes, you can literally see the coin rotate 360° so it is upright again.
This question is wrong on even more levels.
From a physics standpoint, the answer is 1. The reason being is that the word "revolve" means to orbit something, and Circle A orbits around circle B once.
Except that the question doesn't ask how many times it will revolve around B. Instead it asks how many times will it revolve, so one has to assume that it means around its center. Remember, it's not a sphere, so revolution as you defined it would be difficult to attain.
+victor zeng But it orbits 0 times around the Earth so the answer is 0. Nowhere in the question do they ask how many times it orbits around something. Just how many times it revolves aka how many 360 turns it does.
+Matt Abrams
You only need 2 dimensions to define "going around something" (disjunction so it wouldn't be difficult to obtain.
But overall, that question is a horrible question. Because as you mentioned, they don't specify an object to revolve about, it is impossible to derive meaning from this question without making an assumption.
there is no reason to assume the physics or astronomical definition rules here - "revolve" has more usages - and the one the author of the question assumes - is the same one that most of the students who answered the question (for real - instead of just guessing) assumed - and myself - ie - the moving circle has made a full revolution/rotation/roll around the fixed circle - when the same point is touching that was touching when it started out - the answer is 3
the other interpretation - a revolution is when the rolling circle is upright again as it was at the start - (i believe this would generally be called a "rotation" instead) - of course - it hasn't played out the full circumference - but that is irrelevant in the new interpretation
If you draw a spot on Circle A at the point where Circle A touches Circle B at the start, the point will only make 3 full rotations with reference to the circle (in other words, touching the circumference of the circle).
However, it will turn a total of 1440 degrees (4 rotations) with respect to a non-moving horizontal line across the page).
The question asks how many times "around" Circle B so I'm inclined to say 3 times.
Andrew Wilson No, the question asks, "How many times will circle A revolve in total?" It's a stupid question to ask, because the word "revolve" is ambiguous: it can mean "orbit", or it can mean "rotate". If you interpret it to mean "orbit", then the answer is 1. If you interpret it to mean "rotate", then the answer is 4. There is no sensible interpretation that leads to the answer 3.
Some people are arguing that you can get the answer 3 if you look at circle A from a rotating frame of reference, but if you do that, you can get literally any answer you want, by just making your frame of reference rotate fast enough.
+omp199 I think (I'm not entirely sure though) that if the circumference of circle B was a straight line, as opposed to the outside of a circle, then circle A would rotate 3 times.
omp199 Basically, the Circle A's curcumference is 1/3 of Circle B.
omp199 "you can get literally any answer you want "
No, you can't. The rotating frame of reference, in this case, is entirely legitimate.
joe3000abc! Andrew Wilson If the question had asked about circle A rolling along a straight line of a length equal to the circumference of circle B, then yes, the answer would have been 3.
If the question had been different, then it stands to reason that the answer would have been different. No-one is disputing that. But we should be answering the question that was asked, not answering a different question from the one that was asked.
If you observe circle A from a frame of reference that does not rotate, then you get the answer 4.
If you observe circle A from a frame of reference that rotates once, clockwise, as circle A goes around circle B, then you get the answer 3.
If you observe circle A from a frame of reference that rotates twice, clockwise, as circle A goes around circle B, then you get the answer 2.
If you observe circle A from a frame of reference that rotates three times, clockwise, as circle A goes around circle B, then you get the answer 1.
If you observe circle A from a frame of reference that rotates four times, clockwise, as circle A goes around circle B, then you get the answer 0.
If you observe circle A from a frame of reference that rotates once, anticlockwise, as circle A goes around circle B, then you get the answer 5.
And so on.
These examples go some way towards illustrating that by changing the frame of reference, you can get any answer that you want.
3 is the correct answer. There is an error in the coin revolving experiment. One revolution of coin A is not completed yet when the label “coin A” is horizontal again. You need to keep track of the 6 o’clock point of coin A. Another way of thinking about this problem is that there are ropes rolled out on the edge of coin B. Each rope has the length 2 * Pi (radius of coin A is 1).
What I think you are missing here is that the origin of movement is actually displaced relative to the origine of disposition. In this case it is center vs end point of radius vector. Your logic would be correct if we think about a moving point. But the revolving motion is actually rotating, so the center of mass of the circle is going to slow down a bit. Just look at the problem directly and imagine the trajectory of motion of the center of mass of the small coin. It will draw another circle which will have radius equal to Rb+Ra. Once we have the correct path, then we can unravell it like a rope and we will get 2*PI*(Rb+Ra). Another way to think about that is like this: How much of a distance would a rotating circle with R=1 need, to move the center of the same circle 1 unit forward? and the answer is obviously not 1 unit, since the whole point of the circle is to increase the path, by decreasing needed force. That is how the shape works and that is the reason the wheel is used like everywhere in the modern world :D
I think it depends on viewpoint.
From the viewpoint of either circle the answer is 3. From our viewpoint, 4.
It does seem to be a frame of reference issue here
@@codblkops85adding 'outer perimeters' would've fixed it, but no.
@Dominic Amoe
This isn’t a “how many triangles” problem mate.
To be honest - I never understood what that additional revolution shall come from. I always thought it must be the same as if coin a would roll on a plain surface with the length of the circumference of coin B. Your explanation helped me a lot! I imagine moving coin A around B in a way its bottom point of its starting position constantly stays touching coin B - that would be equivalent to just SLIDING it over a plain surface instead of rolling. On a plain surface coin A would have 0 revolutions, but around a circle it already does complete one.
Thanks, great stuff!
Let the small circle have a radius r and the large circle radius of R. The center of the smaller circle travels a circular path with a radius of r + R and one revolution of the small circle about the large circle would give the length of that path as the circumference of that circular path. The circumference of that circle C = 2π(r + R). If we say the smaller circle is 1/n that of the larger circle we get the relation nr = R and substitute that into C = 2π(r + nr) or C = 2πr(n + 1). If the smaller circle does not slip, the distance the edge rolls (e) while the center of the small circle travels for C is the number of rotations (x) multiplied by the small circle circumference (A = 2πr) or e = x * 2πr. Since e and C are equal for one revolution of the small circle about the larger circle we get 2πrx = 2πr(n+1). Reduce to x = (n+1).
If both circles were wheels or gears, each on rotating axels, and both the same size, then each circle would rotate ONE time to return to their original orientation.
Because one circle in the stated scenario remains stationary, the circle rotating around the circumference rotates BOTH times before returning to its original position/orientation. That is where the "additional" revolution comes from.
This is the principle of a planetary gear configuration, the stationary circle is the "sun" gear and the rotating circle is the "planet" gear.
if both circles were the same size, would then the circle circumnavigating the other complete more than 360 degrees to complete the full circumference?
The easiest explanation I can think of is "its always PLUS ONE, because in addition to traversing the linear distance of the circumference of the other coin it is also making ONE rotation around the object. It is the other circle that is responsible for the +1, because it bends that linear line into a circle. The coins of the same size really do translate one to one with each other, its just orientation starts with the top of the coin opposite the edge touching the coin, and when it has gone one half a circuit, it now finally has the top in contact with the other coin, which had not been until HALF the circumference of both was translated through. Both orientations are the head is UP, even though only HALF the circumference has been rotated through. If you were translating it along a STRAIGHT LINE of the same length as the circumference it would rotate exactly ONCE, not twice. Orientation is doubled because the line it is translating along curves around the rotational dimension. The dimensional curve provides the other turn.
Oh yea that is a pretty easy explanation
Circle B is stationary and not turning along with circle A. If you turn both circles as though they are meshing gears on concentric shafts it will be n revolutions as expected. Fixing circle B in place forces circle A to make one additional rotation.
@@TheLightningStalker That helps as well!
That is only true if they are connected on the outside rim, as soon as they overlap or are not connected, it no longer applies.
@@alexvandenwollenberg1563 That's like saying _"multiplication only pans out if you're multiplying, as soon as you're subtracting multiplying no longer applies"_ Of course the solution to this problem wont' be the solution to all problems. Thanks for the laugh.
It depends from which system you watch. We see 4 revolutions in the but it made 3.If you draw a contact point it will only touches 3times meaning it effectively only makes 3 rotations. You would see 3 if you turn around B with A.
The problem is that he fails to understand that the term revolution confers the positions of the two circles relative to one another not to an external view point. The prime example of the term revolution being used correctly is planets revolving around stars, the term revolving confers relative positions of the planet to its star, and it is the same in this example which is why he is wrong.
Speaking of which, Earth rotates 366.25 times in a year, not 365.25.
@@marcusscience23 Compared to the stars, yes. Compared to the sun, no.
@@andrewhawkins6754 That's simply incorrect. Rotation is absolute, not relative, unlike translational movement.
The options were given keeping in mind the way vehicle gears work, i.e. the circles' centres are fixed.
However in the given problem, it is important to note that after each rotation while moving around the big circle, the point of contact on the smaller cirlce is not the same. Initially it was below. After one rotation it is on its left. Actually, the smaller circle does not cover the path equal to its circumference on the bigger circle in one rotation.
Man finally someone who cleared my doubt
His demonstration, the smaller circle only does 3/4 of a full revolution with respect to points of contact. 3/4 of his answer - 4 - is of course 3 which most people seem to be agreeing is actually the correct answer and was all along.
@@NeilMalthus you miss out on the last revolution, if it's 3/4
you can figure it out, simply like this: by finding the possible linear positions of circle A, circle A top (1) and circle A bottom (1) circle A (3), 3+1+1 = 5, now put circle A in a vertical, horizontal, and diagonal position, 1 vertical, 1 horizontal and 2 diagonal = 4
5*4 = 20, we get the total possible positions
20/5 = 4, we divide the total circular possible position by the possible linear positions
Precisely. A circular orbit is not the same as fixed center rotation.
The correct answer IS three because for me "revolve" means returns to the starting point it originally made contact with the larger coin in other words a full revolution.
The issue as I see it is in the paraphrasing. As it stands, here it is completely reasonable to assume that they mean circle A to revolve IN RELATION to circle B, the object it is said to roll around. But it is also reasonable to interpret circle A's revolution in relation to itself. These are then two completely different questions, but neither answers would be wrong, unless the question choses to specify otherwise.
...ok I'm going to go with 31 !
@@miles6671
That's very true. People have also made the point that the answer could also be one, since it orbits around once. That's more of a cheeky answer, but it's an answer none the less.
But math defines revolution, you don't' get to: completing 360 degrees of rotation about a point or axis IN ITS INITIAL FRAME OF REFERENCE. You refer it its starting point, not the other coin.
@@miles6671 - "tech-science" channel has a video called "Rotation Paradox" which animates the issue - and uses something called the "Willis Equation" - to explain the why both answers are correct - in it - he distinguishes between the POV of the fixed wheel - and an outside observer
That depends on which way you define the revolve. If you define one revolution as point P on circle A making contact with the ground, the answer is 3.
This is in no way a definition of "revolution" in math. There is no "ground" here, this is not a physics problem.
The question could have been worded better though, replacing "revolve" by "revolve around its center" would have been enough to remove any doubt.
@@alestane2- there is a ground - the edge of the fixed circle is the ground - - for me - the description of the problem would not have caused any confusion - i understood them to mean 1 full "roll" of the moving circle - the orientation of the lettering (or image on a coin) would not have affected my calculation
You’re counting it before it completes a revolution though… when you were counting you were referring to the number of times a becomes upright relative to you, which isn’t correct.
That's the rub with relative motion. You get different answers depending on where you're watching from.
You are the only one who noticed this. Great job!
Seems like you discovered the theori of relativity... but for geometry instead of physics! Well done!
@@Terza15 Thanks
That's what I saw, too. If you put a spot on Circle A where it initially touches Circle B then every time that spot comes around to touch Circle B is one revolution.
Therefore, the way the question is worded, the correct answer is "3"
You just need to figure out what the testing authority wishes to hear. That is the most important skill for life that the schools teach the kids these days.
Very sad but true ! This is why I disagree with the method of learning at school : it is designed to make people obey to authority instead of developing critical thinking.
I never question my teacher, although I think they would accept.
@@capitaopacoca8454 I understand why you do this but I think it is a part of the problem of the education system.
In my opinion everybody would benefit if the system was requiring students to question what the teachers and books say.
At my university we had a terrible teacher who ridiculed a student in front of the whole class because he dared to ask a completely relevant and legitimate question. Really bad experience and I was just witnessing it.
These days the circle would have to identify as only having turned around three times.
@@adamae.7246 You had lots of opportunities to think critically. You might not have taken advantage of those opportunities but you did have them
From an engineering prospective, imagining the circles as gears, the correct answer is 3, as if the big circle is driven, the little circle rotates 3 times.
yeah that's what I thought too, but that's wrong, the circle actually travels around the circle B
you're only rotating the circle until the word "circle A" is upright, not when the point of contact is the same. So, you're not actually fully rotating the smaller circle until you count it as 1.
Make the original "intersecting" point on CA come around to touch CB again. What do you get????
I have read some of the answers and yours makes complete sense
Bro that still counts as 1 rotation
Anirudh SilverKing 1 revolution would be equal to the point where the starting point of circle a (in this case, the bottom of the circle) comes in contact with circle b again.
"you're only rotating the circle until the word "circle A" is upright, not when the point of contact is the same"
??? how do you figure that these aren't one and the same??
Wohoo! I got 4 right away! Then... I remembered I've watched this a few months ago. :(
LOL
@justsomeguy Did you actually watch the damn video?
@justsomeguy The fact that circle B is rotating *around* an object (as opposed to just in a straight line) adds 1 total rotation. One rotation by your math is a rotation with respect to Circle A's surface, but Circle A's surface curves as well. Did you actually watch the video, or do you just disagree? If so, why?
haha
You are wrong answer is 3
We used to have to design a new "test" to locate our best field technician each year. Writing totally unambiguous questions is pretty hard to do. Sometimes, we'd even have to give up when writing a question meant to address a real issue from the field. The result was that many of our questions ended up being pretty simple.
One of the issues with test writing is, in my opinion, a lack of enough proofing of the test.
I had one field tech who wasn't all that bright, but he was a hard worker.
EVERYTHING that went to the field was tested first by him. If he "got it" we were good to go.
Having spent over two decades in graphics, including typesetting, we knew "never proof your own work," so it may not be lack of proofing, it may be the wrong person performing it! As a teacher now, I hate recycling others' questions, always preferring to come up with my own. And of course this is fraught - often better to emulate than innovate, or risk memeworthy answers like "Here is is!" for "find this angle..."
Is anyone here after Veritasium's video?
"Only using paper and pencil"
Me: *starts ripping paper into circles*
Think outside the box
@EG to think out of it
@@dannygee_6051 Kono Dio Da!
(Sorry for being late)
I just feel bad for those SAT students, logically speaking, at least one of them got the correct answer (4) on paper, but then couldn't find it on the sheet so they just guessed.
Even then, the "best' answer to select from is still 3, and that's what the SAT asks you to do.
@@spikeconley 9/2=4.5 is closer to the correct answer though!
Yea i also choose 9/2 cause it was closest to the 4 that i initially got
4 is not the right answer though. Presh has mistakenly claimed that the label 'circle A' being upright again is one full revolution of circle A, but it's not. If you put a dot on circle A where it touches circle B and revolve it until the dot touches again, it's 3 revolutions.
Just imagine rolling it on a straight line. You get three revolutions. Then take the end of that line with the circle at it and curve it around into a circle. It rotates an extra time as you do this.
Or imagine the smaller circle sliding around, without rolling, just keeping the same point (bottom of the text) touching the larger circle as it slides. It will rotate once despite not having rolled over any of its circumference.
That's a very intuitive explanation!
Yeah! I thought this way.
I counted 3. Draw a line where the two circles meet. Roll the smaller coin around the larger until it's back at its original position. The line on the smaller coin will have touched the larger coin 3 times.
I saw it roll around three times.
I understand what you're saying, but you're using the wrong frame of reference. The coin is rolling around a curved surface, but you're using an invisible flat plane for reference. If you watch the text on the circle relative to the surface of the circle it is rolling around you will see it rolls around three times.
I understand the paradox, but it's not a great mystery. You're just looking at it wrong. The answer is, of course, 3.
The question does not specify the frame of reference, so the correct frame of reference is not the circle itself, but the environment that it's in. The question asks how many rotations does Circle A make in total. Not in reference to Circle B, but in reference to space. Clearly, if the circle is rotating _around_ Circle B, it's making an additional rotation.
This is a high school geometry question, and by default is using euclidean geometry. The background piece of paper is the reference point. Watch the video, it rotates 4 times in reference to the piece of paper.
When did you ever change your coordinates mid problem in a high school class?
This isn't rocket science, and your eyes aren't fooling you.
Tim Hodson -
'This isn't rocket science', you say?
So you've got no excuse when you get the answer wrong.
The test writers admitted as such.
www.nytimes.com/1982/05/25/us/error-found-in-sat-question.html
Get your head out of the sand.
@ Mark, who said to me: "I think you should just shoot yourself."
I consider that to be a total and unmitigated concession of defeat. I accept your concession.
The ambiguity in this problem parallels the definition of a day in astronomy. One way gives you the solar day, which is 24 hours. Counting it the other way gives you the siderial day, which is about 23 hours and 56 minutes.
Yes, it's the same. If you see the day from the outside of the solar system and calculate based on some place on the equator, then you would get an extra day per year, which translates in a 4 minutes difference each day.
But if you take the definition of a "revolution" as something that moves around it's own axis, then the real answer is the solar day, because if you were to take out the Sun, the Earth would do just that and the sideral day would be the same, 24 hours. So I believe 3 is the right answer, 4 is decent in context as seen by an external observer. If it's seen from either circles of planets, the revolution is 3 or 24 hours for the planets.
This video is in error, you aren't accounting for the motion of the circle. It will complete one revolution only when the text on the smaller circle is parallel to the tangent line. Otherwise, you are seeing a visual illusion where it looks like the circle is completing a revolution sooner than it really is, because you are judging one rotation based on the text's original orientation.
Put a red dot on it and calculate the number of times it touches the surface. This is the standard of how any revolution is measured when moving along a curve. Imagine that the little ball is a soccer ball on a planet, and you're standing on the planet's surface watching it roll. You will observe one revolution of the ball when the point that originally touched the surface touches it again.
+thatoneguy1349 I just did the "experiment" with one quarter and I got one revolution, as expected, as the red dot only touched once. I think you're mistaken.
+James Pedid oh fuck.. you're right. I'm gonna delete my comment now.
+thatoneguy1349 don't dismiss your comment right away! by the time I'm typing this you already deleted it, but your experiment still applies to the correct answer, which is 4. The question on the SAT is "how many times will circle A revolve in total?" It doesn't ask "how many times will circle a completes a revolution around circle B." What I mean is that if you were to mark your quarters (and start with quarter A above quarter B), and ROTATE circle A about circle B, you will find that the mark on the quarter has returned to its initial state when its below quarter B (ie: the mark is on the bottom). that is one revolution. When you continue to ROTATE Quarter A back to its initial starting position, it will have made 2 revolutions. Reference points do not matter. Lastly, there are 2 misconceptions that many people have this problem:1.When some people are trying to solve this they immediately "unravel" or "flatten out" circle B in their head, which is not the correct way to engage this problem, because that skew the revolutions made by circle A. 2. the definition of a revolution. In this case, since we are saying circle A rolls around circle B, 1 revolution is the movement of circle a in a circular course around its own center, or when a point on circle A revolves around its center and reaches returns to its original position. So, if I mark circle A on the bottom, and revolve it around its center, it will complete 1 revolution when the mark returns to the bottom.
+Brandon Kekahuna is correct. The small circle revolves once when the text is parallel to the big circle's text again.
It made 3 rotations. If you are counting from where it is contacting the other circle the original point of contact only touched 3 times. It is an illusion that it rotated 4 times due to the fact we are not standing in the middle of the large circle. If we were we would only see it revolve 3 times.
Yet another example where thinking about limiting cases is helpful. Consider the limit as the radius of circle B goes to zero. We can picture this as circle B being reduced to the width of a pin. Circle A still has to perform a full rotation to go around the pin, despite the circumference of the pin being approximately zero.
Or, as another way to simplify the reasoning: Consider taking circle B and squishing it into an increasingly elliptical shape, while keeping the circumference fixed. Eventually, it becomes basically a thin toothpick with length equal to half the circumference (because half the circumference is on its top side and the other half on its bottom side). To go around this shape, circle A has to roll forward half the circumference, do a 180 degree turn to round the corner, then roll forward the other half the circumference and do another 180 degree turn. So in total it has performed one extra 360 degree rotation over what was needed to cover the distance.
Maybe it's just me, but I find that to be simpler because it separates out the rotation needed to complete the revolution from the rotation needed to cover the distance, so they happen at different times.
Thanks, that helps make it seem more intuitive. I was completely mind blown as to why having thrice the circumference does not equal three revolutions of the smaller circle.
This is very helpful. Another part of your illustration could show that if you took a shoelace with a length the same as the circumference of the larger circle, you could roll the smaller circle from 1 end to the other in THREE rotations, but if you folded the shoelace in half so that the smaller circle had to get to one end, roll over and come back the other way, it would need FOUR. thanks for helping me understand.
The question is not well posed, it has some ambiguity and we could argue that 1, 3 or 4 could all be correct answers. This would be a problem if you just had to guess the asnwer. But the given solutions a,b,c,d,e are, in my opinion, an integrating part of the answer and the only correct answer there is 3. If you understood the problem there was no way to get the wrong answer.
I've looked at a few of these. MindYourDecisions relies upon intentional ambiguity.
Answer is 4, through calculations
My 1st thought was a ring and pinion gearset at 3 to 1 the small gear is the pinion and the large gear the ring. Pinion gears are stationary at the end of a drivesaft. So its,the ring gear that turns. Took a test for miltary entrace once. My shop class was the only ones to get it right.
N a page full of gears and gear #1 is rotating clockwise. What direction is the last gear turning.
The title of the video is then misleading, since some (maybe all?) must have selected 3 as the answer. If the question had not been multiple choice, all three possibilities would have been used as answers and the test creators would have realized the question was ambiguous.
@@roog49 Yes, their questions are obviously intentionally ambiguous.
Revolve is Revolution, in order to complete one Revolution, all 360 degrees of A has to touch. Therefore the answer is 3. You can't stop short because you are trying to use the background as a reference point. That was where you failed. All 360 degrees of A has to touch B before it even makes one revolution. Or in other words revolve once. As in every single degree on A has to have touch B before it even revolves once, to make a complete rotation, revolution. That is what you are not seeing or thinking about. You are just going off of if the text is upright, of if the coin is upright.
Edited, The word you are really looking for is Trips. Circle A makes 4 trips, yet completes 3 full rotations / full revolutions of 360 degree turns as in surface to surface contact.
Revolve means one object moving about another. The small circle completes one revolution. Rotation is the movement about an axis, and in this case we're interested in movement about its own center. When the small circle completes one revolution around the outside of the large circle, it completes 4 rotations about its own center. Still not convinced? Flip the problem around and have the large circle revolve around the small circle. How many rotations does it complete? Does it complete only 1/3 of a rotation? Better yet, have it revolve around a point. Does it complete no rotations? According to your reasoning it should rotate zero times since a point has a circumference of zero. However, the actual answer is that it completes one full rotation about its own center in the case of a point and 1.33 rotations in the case of a circle that's 1/3 the diameter of the circle doing the revolving.
What part of all sides of a circle must of had touch before a revolution counts. It is on a curved path. The question never stated B is completely Stationary as in never moves to the observer. Therefore if they both were on an physical Axis, A was made to be on top and B on bottom. A would rotate 'revolve' 3 times. Again the question in the video never said B had to be stationary to the observer.
exactly
JackFreedomcis It's pretty obvious by "revolve", they actually meant "rotation" (i.e. the number of times the words "Circle A" return to the original position from the viewer's perspective). As is often the case, scientific terminology is bastardized by colloquialism. But let's assume that they meant your definition of "revolve" and change the problem around a bit. Instead having the small circle roll around the big circle, flip it around. Make the big circle roll around the small circle (i.e. "Circle A" is now the large circle and "Circle B" is the small circle). AND let's shrink Circle B down to .0001r where r=radius of Circle A. How many times will the label "Circle A" return to the original position before YOUR definition of "revolve" is met.
Nightwriter: you mean area not circumference
So the answer is 3+1. The roll distance travelled is 3 times the circumference of the smaller circle PLUS one rotation to remain in contact with the edge of the larger circle.
Image not drawn to scale?????? 😂😂
Prometheus area does not increase proportionally to the radius, since it depends on the square of the radius rather than the radius itself. If you place the small circle inside the large one and line it up with the diameter, you would see that the diameter of A is roughly the size of 1/3 the diameter of B, even though the area of A is 1/9 of B.
Mate its circumference you absolute scrub
Why on Earth would you even need the areas of the circles?
You don't. That's exactly what I was explaining to Prometheus; The image is drawn nearly to scale.
close enough
Prometheus 😂
it'd be hilarious if one of the guys just cut out the circles during the test and started counting
The diagrams in the exam are not necc to scale.
@@davidho2977
I encountered a very similar problem in high school, where the radius' of the two circles were equal. My classmates made the same mistake as the SAT test-makers. I literally figured it out and proved it to my classmates, by demonstrating it with two oranges.
We got in some very heated math debates in my senior high school years (grade 11 and 12), which was pretty fun.
We actually do it sometimes. I remember doing it in the National Math Olimpics finals (team game). There was a 3D geometry problem which was just impossible to take on. 10 minutes before the end I saw that no one got it right, so I took it up myself to go for the big bounty. I took a snack, all the pens we had and a lot of imagination and BOOM. We got first place with that trick, it was unbelievable.
Rotation:
The spinning motion of an object on it's axis
Revolution:
The orbital motion of an object around another object
So, Circle A will revolve 1 time, rotate 4 times and roll 3 times
Revolution around means what you said but the word around isn't in the question. In that case revolution and rotation are synonyms.
You didn't define roll, brother
there was no reason to assume astronomical definitions were assumed - so using colloquial ones would suffice - which is what the vast majority of students who answered the question correctly did (not including those that guessed the correct answer) - in that case - the playing out of the entire circumference of Circle A would indicate a revolution/rotation/roll - and the answer would be "3" - - altho the alternate interpretation was also valid - and the answer "4" was also correct
The question is poorly written, it doesn't say a complete revolution or 360 degrees anywhere. Therefore, the answer could be 1 or even infinity.
Read both paragraphs of the question next time.
@@Hhhh22222-w I did, my point still stands
It would be interesting to see the EXACT text of the original question. I'm wondering if the scenario they had in mind was that circle B is rolling along a surface while circle A remains on top of it until circle B has completed a full revolution. That changes things!
''In the figure above, the radius of circle A is one-third the radius of circle B. Starting from position shown in figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A first reach its starting point?''
The question makes a clear distinction between a "roll" and a "revolution". It revolves 4 times having rolled 3 times over the circumference of circle B.
@CipiRipi00 Yes, the original question uses the words "revolutions of circle" with respect to the "centre of circle A" returning to its original position, which I would probably have described as rotations if I'd written the question.
Nice try, but you're picking an arbitrary frame of reference.
CapAnson12345 What other frame of reference makes sense? The people arguing it is actually 3 are taking a very odd frame of reference, which considers the edge of the larger circle to be flat. Such a frame of reference makes no sense to take, unless you are microscopic organism living on the edge of the larger circle, and the horizon is so far that you can't tell that you are on a circle.
Not really.. the video example shows circle B remaining in place while A rotates around it. But suppose A and B are flat on a surface and you stand at the center. You turn as Circle A rotates... and it rotates 3 times. You can see a little more what the difference is if you keep A stationary and rotate B three times around it.
"You turn as Circle A rotates" -- Why would you do this? It's a very strange thing to do. And it's even more strange to then subtract your own rotation from the rotation of Circle A, and pretend you didn't rotate at all.
Would you say that 0 rotations is an acceptable answer? Because as easily as you could rotate once, you could rotate 4 times. Then, from your perspective, A didn't rotate at all. Or, if you rotate 96 times in the opposite direction as A, you could say that A rotated 100 times.
Rotating yourself once, 4 times, or 96 times are all arbitrary frames of reference. Rotating yourself 0 times, because you're sitting at a desk, taking a test (rather than riding a merry-go-round), is not an arbitrary frame of reference. It is the only frame of reference which is sensible to assume.
Why? The problem itself says the frame of reference is B, as in A rolls AROUND B - ergo the point of view is inwards towards B. I know what you're saying.. and I get the point of this whole video... I just think it's a more a clever taking advantage of the wording rather than the problem being strictly wrong.. at least how it was intended.
Well, since B is stationary relative to the paper, the answer is still 4, even with B as the frame of reference.
Guys, look at 1:03 when he is explaining what the test makers thought. The thing is that the circle IS rolling around 3 times the length of it's circumference... When it is going straight. When it is in a circle, you also have to account for the fact that the center of the circle is traveling around a circle with three times a radius ( the big circle b) PLUS the radius of circle a. If circle a radius is r, and circle b radius is 3r, the circle a is going around a 3r+1r circle, or 4r, hence the answer is 4
Well explained... Thanks
The correct answer IS in fact 3, sometimes. The difference is the point of observation. You see 4 revolutions in the demo because its perspective is perpendicular to the plane of the circles. But the true answer is 3 when viewed from the center of either circle.
You are fooling yourself thinking it's gone a full revolution when the label is horizontal again, but that's not a full revolution relative to the circles because circumference A has not completely rotated in contact with circumference B.
So if you are creating a visual effect to satisfy human perspectives for reading the label you can say the label rotates 4 times. But when practically applied, as in planetary gears, it's 3 revolutions of A to revolution of B.
It's in line with the wheel paradox, which is not really anything mysterious.
It's not unusual for theoretical types to smarmily claim they've outsmarted everyone else, but you're only seeing it from a limited visual perspective (much like an optical illusion) and you're not properly defining what a revolution really is.
Well I got three like the test preparers intended, so I count it as a win.
lol me 2
Bear in mind that test preparers quite often get things wrong.
Same
The circle has not rotated fully when the words are horizontal. This is because it is spinning and moving in a wider circle. It has rotated fully (360 degrees) only when the bottom of the letters is facing the midpoint of the big circle. It did this 3 times. Look at the demonstration, when he stops the circle to show it rotated what seems to be 360 degrees, he only rotated it 270 degrees.
A way to prove this is by putting a dot on the small circle just next to where it touches the big circle in stage one. If this dot is touching the big circle again, it has rotated 360 degrees. This would happen three times. Just imagine the dot, then you see it. The fact that in a rotating circle that DOES NOT MOVE it has rotated 360 degrees when the text is horizontal again, throws you off.
It all comes down to peoples different definitions of "a full rotation". My definition would be as follows:
If we were to draw a fixed x, y coordinate plane on the paper (we could say that the origin is the center of circle B). Then we draw another x and y axis on circle A and let it stay fixed to circle A (ie it will rotate with it). Now, at the start, the two y axes and the two x axes are respectively parallel. My defenitions of a full rotation is when the x axis of circle A has gone through a full 360 rotation with respect to the x axis fixed to the page and circle B.
Using this definition, circle A would rotate 4 times by the time it returned to its original position. Can anyone tell me why this definition would be invalid?
For the answer to be 3 rotations, circle B (and its axes) would have to rotate along with circle A as if they were gears. For this to happen we would have to assume both circles were fixed at their centers and simply rotated around them like axles. Given the problem description, I would say that this scenario is far fetched to assume. Instead, when it says that that circle A revolves around circle B, it seems to imply circle B is stationary and circle A is doing the revolving. In which case my definition would hold true.
I can tell you why this definition is invalid. This would only work if the entire circle (or the center) does not move, like circle B. However, circle A is rotating and revolving around circle B at the same time (just like the earth around the sun).
You could draw another circle if you were to trace the centre of circle A. This circle is the way circle A makes when revolving around circle B. Lets call this circle D with centre P.
Then, if you were to draw a x,y coordinate plane on circle A like you suggested, centre P would be on the y axis, if you were to extend the y axis.
Start rotating and revolving circle A like shown in the video, and you would see that circle A has only rotated 360 degrees when centre P is on the y axis again if you were to extend the y axis. This occurs a total of three times, at 4 o'clock and at 6 o'clock from centre P. This clearly shows that circle A rotated 3 times while revolving around circle B and centre P.
Achillez
I dont think you proved it invalid, you just provided your own definition which is equally valid within itself, however my argument was that the question seemed to fit with my definition better.
Why do you say mine would only work if the entire circle does not move? Are you talking about circle A?
It sounds like in both of our definitions circle B is stationary and does not rotate. It seems reasonable to set up a coordinate plane fixed to circle B. It also seems reasonable to call this our reference since circle B is not changing at all. If we also set up a coordinate plane on circle A that is fixed in relation to circle A, then we can measure things between these two coordinate planes. Does all of this seem reasonable so far?
With this set up, if circle A was placed on top of circle B so their centers were at the same point, then both coordinate axes would be perfectly overlapping. If we were to keep circle B fixed and rotate circle A, we would see the axes rotating relative to each other. This angle of rotation could just be measured between the two positive x axes. Call this case #1, would you agree that, in this case, a full rotation would be when the angle between the x axes rotates to 360°?
Now for case #2, say you moved circle A off to the side of circle B. If you again keep B fixed and rotate A about its own center (spinning it in place), I would still define a full rotation of A by the angle between both x axes. Of course when this angle gets to 360° it has completed a full rotation. Would you agree?
Finally, in the case of the problem, instead of spinning in place, circle A is moved around circle B while it rotates so that they are in contact at all times. At the end of one full revolution of A around B, A has conpleted four full revolutions about its own center and relative to the fixed axes of circle B. Obviously this case you dont agree with.
For your definition to work, you would have to rotate circle B(and its axes) to follow circle A as it revolves around. That way the angle between the positive x axes would only reach 360° when circle A is one third of the way around circle B. OR if you were only considering the surfaces of the two circles and considered a full rotation when a point on circle A touches the surface of circle B twice. But this only works if A was rolling on a flat surface.
Another example is if we said that circle A always stays upright as it revolves around B (ie. The "A" keeps its orientation). In this case you would say it has rotated once and I would say that it didnt rotate at all since the x axes were constantly parallel. Similarly, if circle A didnt roll around circle B but simply "slid" around circle B (The bottom of the "A" would always point towards the center of B). Then I would say it has completed one full rotation after completing its revolution around B, and you would say it didnt rotate at all as it revolved around B.
Please do not act as if his definition is valid. It is not. Ben Watson youre definition is correct. Mathematics does not actually make the same distinction between rotation and revolution since "rotation" is just a special form of revolution (revolving about its centroid). In fact a rev and a rot are measures of angles: www.mrbigler.com/moodle/pluginfile.php/10239/mod_resource/content/1/125_page_Notes-AP-Physics-1-2016-17.pdf
You didn't show the calculations for why it is 4 though. That demonstration didn't convince me at all.
The calculations are easier than people realize. They make a mistake in assuming the PATH the smaller circle takes is equal to the circumference of the larger circle, when in fact if you drew the path the small circle takes from the CENTER of that circle, you would see it actually travels a distance farther than the larger circle, which is why it takes an extra rotation.
If you want to see the math, then I give that in my latest comment.
+Travis Tidwell yes!
The smaller circle's center moves through a path that has a diameter of 4d (where d is the diameter of the smaller circle). This means that a point of the circumference of the smaller circle moves 4 times the distance of its center as it rotates. CirA = PI * d, and Cir Path = 4 * PI *d. Dividing you get 4.
+gary1168 The ratio is 1 to 3, not 1 to 4. It's in the first line of the problem.
+breakingglass27 the radius of the path taken by the center of cir A is the sum of the 2 radii 3r + r.
No, it hasn't!
You're counting a roll wrongly. If you simply mark a dot (.) on the starting touch point on the rolling circle, 1 roll will be completed when the same point touches the boundary again. And that will only happen thrice in a full round.
It is like a train crossing a platform.
If the length of both is the same, still, the train will have to travel twice its length, to completely clear the platform.
Basically, it will have to clear the platform, and then cover its own length.
No, it's not like it at all
Awesome, this just clear it for me
This is so easy. How could no none get it?
[3 minutes later] Wait, what!? How?!
The point is reference frame.
Label the point at which the circles initially intersect as point P on the small circle.
A revolution of the small circle is defined NOT as when point P touches the large circle again, but when point P is pointed directly downward.
Tricky!
When he explain in wrong way to get some views clearly the two similar coin revolved only one time...
@@TheLearningFrontier A "revolution" is defined as when the revolving coin is turned right-side up (because it starts right-side up and ends right-side up, meaning it rotated completely). You can see with the similar coins that it clearly turns right-side up when it's at the bottom, and then it's right-side up again when it's at the top, so it's 2 revolutions. He was correct.
We are same bro😁
I think that the question stating "revolve" here means to "rotate" and thus we are allowed to consider the rotation of circle A.
I think that you're changing the question.
It only "revolves" once. The questions asks about revolutions but provides answers for rotations. It's just a terrible question.
colloquially - circle can revolve around the center - imagine taking slices out of a cylinder which has an axis - it can be said to "revolve around its axis" or "rotate around its axis" - this is why dictionaries have numbered definitions - words can have more than one meaning - and that's why this question caused problems for a few students - they had another interpretation of "how many times will Circle A revolve in total?" - it was a valid interpretation - but a better one - the fact that a large number of other students had no problem and answered the question "3" - indicates to me that the author's intended definition was just fine
The edge travels a circle with a radius of 3, the center travels a circle with a radius of 4. But the question didn't ask about a 3rd circle with a radius of 4, it asked about Circle B with a radius of 3. Circle A rotates 3 times to travel around Circle B, but the guy in the video is looking at a different circle with a radius of 4 which intersects the center of Circle A.
I made a video with the camera fixed to the perspective of the rotating circle, and the illusion of the 4th rotation disappears. ruclips.net/video/06O1_xOS3Bc/видео.html
Your visual demo was explained incorrectly. When you do a “full rotation,” you have to move the entire circumference of the circle around the bigger one. Therefore, the only time the circle should ever be upright from a full rotation is at the very top as the words should always be expressed towards the center of the bigger circle (when you claimed you did “one full loop,” notice how 1/4 of the circumference of the smaller circle still hadn’t even touched the bigger one yet). Think about it mathematically: the distance the circle travels in one full rotation is d(pi). In order to fully go around the big circle, it has to go around the circumference of the bigger circle, which is 3d(pi). 3d(pi)/d(pi) is 3.
this
You forget that the circle is also rotating around itself once as well. I was very skeptical bc of the math but I tried it with two quarters and it actually works.
The mathematics does actually work. Let r be radius of B, and so (1/3)r is the radius of A. Consider the centre of A, which is at a distance of (4/3)r from centre of B. The centre of A moves in a circle and the circumference is (8/3)rpi. The circumference of A is (2/3)rpi. Then by division we can simply write 8/2 = 4 and this will work in the ''general 1/n case''.
If I have made any mistakes such as false assumptions etc. please let me know.
Sarah Greenwood I'm afraid you made a slight mistake. To put it simply, you can solve this question either mathematically or by demonstration. Mathematically, you can take the ratio of the circumference of the two circles and that proves to be three. By demonstration, you can refer to the part in the video where he used the cardboard circles. Before he starts moving Circle A, I want you to imagine a red colored dot on Circle A where the two circles are touching, at 1:34 . Then, you just simply count the number of times that red dot touches Circle B again. This, by definition of rotation/revolution, shows that the answer is in-fact 3. It's a 2-D diagram so revolution and rotation have the same meaning in this question. Y'all over-complicating it by starting to differentiate between the two terms.
P.S. if you read the 5th line of the question, you can see that it is referring to 'Circle A' specifically and not relative to B. If it were relative to B, then the "astronomy guy's" answer of 1 would make sense. Aaaand I just realized I made this too long lol.
Watch Walter lewin problem 13, it's mystery of pure roll, answer is indeed 4. Requires some physics (pure rolling).
If you put a mark at the bottom of circle A, you can then sketch the position of this mark (and circle A) each time the mark touches the circumference of circle B. You can see the second point of contact will subtend an angle of 120 degrees at the centre from the starting position, meaning the mark will have rotated 120 degrees beyond the bottom of circle A. Thus circle A must have rotated 480 degrees. It does this 3 times, so the total rotation must be 3x480=1440 degrees. i.e. 4 full rotations.
Now place yourself in the center of circle B and count how many times the mark on circle A touches circle B... three times. However, you yourself must rotate once to witness the experiment!
@@wesleyc.4937 Exactly, this is less a math test than a test of perspective.
120 x 3 = 360, not 480 as you mentioned. So it rolls 3 times not 4 times.
@@stocks9662 No. Circle A will have rotated a full 360 degrees - the mark which was at its bottom will be at its bottom again, but this time not in contact with circle B. It will rotate a further 120 degrees before it next touches circle B. 360+120 = 480.
@@stocks9662 Circle A rolls 3 times across the circumference of circle B but that's not the question.
''In the figure above, the radius of circle A is one-third the radius of circle B. Starting from position shown in figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A make to reach its starting point?''
The question makes a clear distinction between a "roll" and a "revolution". It revolves 4 times.
The number of revolution vary on from where you are observing (reference frame). If you are at the centre of circle B then the revolution is 3 only and if you are away from the circle then it is n + 1. ie. 4.
I refuse to believe that no one selected the right answer. There must have been thousands of people who determined it was too difficult and just guessed.
The center of the revolving corkle travels in a circle whos diameter is 3+1, as in the radius of the stationary sircle plus its own radius. So the revolving sircle travels 3 + 1 = 4 units.
Wow, a simple problem like this is actually more complicated than we think. I love it.
The correct answer is 3. The correct way to solve this problem is algebraically by comparing the linear (rolling) distance around circle A to the distance around circle B. This linear distance is of course the circumference. By using X and 3X for the respective radii .. circumference of A is 2(PI)X and B is 6(PI)X. Diving B by A gives the correct answer of 3.
@@elmoremundell9450 yes, we also know that, but have you watched the video to see what he has to say as well?
@@kaan8964 just becouse he has a reasoning, doesnt mean he is correct.
One resolution isnt when the word "circle A" is readable.
Next time make a mark on circle A and do it again. You'll quickly see that A revolves (not rotates) 3 times.
I used algebra to show the maths behind the answer, it's hard to follow in the RUclips format but bear with me. If you write it out yourself it should become clear. I'm afraid I also don't have the symbol for Pi on my IPad thus why I use 'Pi'.
rA means radius of A, rB, radius of B. DA diameter of A, DB, diameter of B.
In order to go from rA to DA one must do the following:
2*rA*Pi=DA
rA*3*2*Pi=rA*6*Pi=DB
The Pi's can be cancelled to give
2*rA=DA
6*rA=DB
DA*3=DB
Thus the answer is 3 times.
The numbers don't lie.
Fuck me, it's 1 am. When I say "diameter" I mean circumference.
I might be retarded.
+조진혁
It revolves 3 times and rotates 4 times. These words don't mean the same thing.
Revolve:
"move in a circle on a central axis."
Rotate:
"move or cause to move in a circle around an axis or center."
???
+Tae Kyu Kim
Revolve means to move about an axis so that a point on the circle faces the centre of the thing it orbits around. It is really only used correctly in Astronomy. Rotate means to circumnavigate the thing you orbit.
My algebra shows this mathematically.
That's what I suggested! Wow, but anyway. You don't even need algebra, if you have the actual circles, you have to make sure that the same place on A is touching B at the end of the rotation.
A revolves 3 times around it's own axis.
It is also 3 revolutions around the A axis from B's point of view because B also rotates once relative to A (or A rotates once around B). If you were an observer on the surface of B and followed A as it rolls, you would do one revolution and would see A revolve 3 times. In the same time, someone stationary on A would see one revolution around B, and three around itself, it would stop seeing B three times and would eventually see the same starting point once. So it's like 3 days and a year on Earth, if a year was only 3 days.
It's 4 from an outside perspective C where you add the 3 rotations of A around it's own axis, plus if you count the rotation around B as 1 (does not matter if it is from A or B's perspective). But 4 cannot work, because you are basically using two different axis and adding them up, which simply cannot be a "revolution", which is always based on one axis, not many. So the answer is either 1 or 3. It cannot be 4 strictly speaking since it requires two different axis.
The actual answer is 1 because if you ever studied astronomy you know there’s a difference between rotation and revolution. If it asked for the number of rotations the answer would be n+1. But it asked for the number of revolutions which means how many times it went around the bigger circle so the answer is simply 1.
This is not astronomy. Your argument is invalid.
U r right bro
"The word flip is synonymous with an airborne somersault in a number of countries. In contrast, in Britain and some other countries, a flip must rely on the arms to induce body revolution."
A body revolution... Around the Earth I guess. Or Moon?
If you spin both circles without changing their position, the answer is 3.
But that's not what the question is asking.
+ft55555 True. Just an observation.
True it is 3 times.
HEY ft55555
The question never stated that circle B is stationary to the observer as well. Therefore if both were stationary and spun on an axis, A still rolls on B.
For A to do a full rotation, revolution, all 360 degrees of it side needs to have touch B first. Now if A was separated from B and spun on its own. You would be using reference points in space to determine if it did a 360 degree spin or another form of reference point.
A is touching B, For A to successfully revolve once, all sides must of have touched B since A is on a curved path along B surface. Its called Curvature Math, and the Answer is 3. That is where everyone is failing at who answer with 4.
JackFreedomcis You're making up your own definition of what rotation is. How many times a given point on the small circle touches the large circle is actually irrelevant to the definition of rotation. Rotation is an object's movement about its own center. This isn't a math problem or "curved math" problem. This is a physics problem. That is where YOU'RE failing. Make the "circle" that the small circle roll around a point rather than a large circle. How much of the circumference on the circle ends up touching the point as it revolves around the point? Think real hard. But is the circle actually "rolling"? You bet your ass it is. If you stuck an axle in the center of the circle and applied a force to the axle to move the circle around the point, the circle will definitely move relative to the axle.
Revolutions and Rotations are the same thing. Think about your Vehicles Dash, you see the RPM "Revolutions Per Minute" A revolution is a full rotation of the crankshaft.
Imagine you were a tiny person on the Circle A. You were standing near the edge looking onto B. Now count how many times you see the edge of A touch B again.
It is 3.
2-Base, 8-base, 10-base, 16-base, math.
11111111,377,255,FF
How many times does the smaller circle go around the bigger: 1
How many times does the smaller circle rotate (round it's own center): 3
How many times does the smaller circle achieve its initial orientation: 4
It seems to me that rotating around it's own center and turning to achieve it's initial orientation are two different things. If we observe the scene where one coin turns around another, the turning can be viewed as establishing a 1-1 mapping between the points of the coins that touch each other. When the coin has moved from the top to the bottom, only half circumferences of the coins have touched with each other pointwise. How come the moving coin has done a full rotation around its own center?
This is the best explanation.
Great and logical explanation.
I was so confused.... Now I got the point.
You really rocked me.
the smaller circle rotates around it's own center 4 times. That's the whole point the video is making
Max Max,
Your second and third statements say exactly the same thing.
The answer is 4, as Martin Lebeau said, above.
Well. I am actually confused about my interpretation above. LOL. On the other hand, I now do think it's 4 full rotations around its own center.
Got to love when a demonstrator is making a common mistake and has no clue that he is wrong. Hey look the label is parallel with the bottom of the page so that is a complete revolution; despite it only travelling 3/4 a revolution in relationship to the other circle.
Gotta love when a math expert knows the mathematics definiton of revolution: Completing 360 degrees about a point or axis *IN ITS INITIAL FRAME OF REFERENCE* You refer it to ITSELF, not the other circle. Pat pat pat.
If you observe the whole thing from the center of Circle B while turning to constantly face Circle A, then for you as the viewer Circle A only revolves 3 times 🙂
Yeapp
That's some relativity sheeit right there.
That Time also you will see 4 times. Because gravity is always upward. Even when you are in the circle. It's same as seeing it from outside the circle.
The way you see it doesn't matter. The fact is that the circle made a 360 degree rotation 4 times.
With same logic, you can say, if you stand on CircleA, it revolves 0 times or of you stand on a fast rotating platform, ir revolves 10000000000.... times. There is no reason for such an assumption.
1:41 circle didn't rotated completely as 360 degrees . . better mark a point on the circle for making perfect rotation of it to get 3 times (absolute true)
agree, the answer should be 3. the coin didn't rotate a 360 degree, it only did for 270 instead. so its 270x4 which equals to 1080, 3 complete circles
Well I guess he fits in perfectly with the crowd who got this wrong.
It’s absolutely 4 rotations, as indicated in the video and described throughout these comments...
@justsomeguy Your math is perfect. However, you failed to understand what he meant. The test makers made that question with your solution in mind, yes, but the problem is in the way they phrased the question "How many times will circle A revolve in total?" because they didn't say to what reference and that made the question ambiguous. One possible definition of "revolve" is to rotate 360° around its own central axis. And with that logic the answer is 4. Here's something to think about: How many times does the moon revolves around Earth? If you consider only your interpretation of the problem the answer will be zero! Because we always see the same side of the moon.
@justsomeguy No one is lying. He said nobody picked the correct answer simply because he thinks 4 is the only possible answer, but it wasn't among the options. I think both 3 and 4 could be correct since it has two possible interpretations. However, since only 3 is among the options it is clear how they wanted it to be interpeted. Anyway, if you "know" you're right then good for vou.
In this example, to demonstrate properly,put a small perpendicular line where the 2 circles meet at the starting point. Then roll circle A around circle B ,starting point 1:33-count how many times that line touches circle B. (1) at 1:45- (2) at 1:57-(3) at 2:08- we are back to the starting point- Ok 3 times!!!
And why isn't it correct to observe the circumference of B is 3 times the length of the circumference of A so if you laid two strings of each length out, Length A X 3 = Length B? Oh, I get it! I did not understand the question: revolutions of A without respect to B.
The line touched 3 times, yes, but the circle revolved 4 times. 4 would be the correct answer.
If you want to be even More technical the coin only Circled One time!!!
I thought about it this way: If circle B was slowly turning so that A would stay on top, it would be 3. But now B would've "wound up" 1 revolution, and when it unwinds (with a steady Circle A "glued" onto it), it'll revolve A for another turn.
Or thought differently: A is turning 3x for the distance, 1x for going around once.
If A and B were the same size, the answer would be 2. You can really easily visualize this in your mind if you imagine both circles have their radius drawn onto them to the right. This helped me further confirm my suspicion.
Click bait. Technically, people choosing 3 didn't get this SAT question wrong, because their SAT mark would have been correct. The SAT examiners got this question wrong.
I thought 3 as well. But as soon as I saw your experiment, I remembered trying to understand the difference between a sidereal and a synodic day. The same problem arises, a rotating Earth travelling around its orbit of the sun has a day relative to the sun (synodic) of 24 hours but relative to the changing position of the stars (sidereal) it is 23 hours 56 min (approx). This problem is asking for the 'sidereal' answer, which is not in the list of answers.
The question did not specify with the 'sidereal' rotation, therefore the correct answer was listed, it's 3.
Presh's circles are not to correct scale. The correct answer is 3. The correct way to solve this problem is algebraically by comparing the linear (rolling) distance around circle A to the distance around circle B. This linear distance is of course the circumference. By using X and 3X for the respective radii .. circumference of A is 2(PI)X and B is 6(PI)X. Diving B by A gives the correct answer of 3.
@@elmoremundell9450 no that's not the issue. the issue is because the question assumes 1 full revolution is a full revolution in reference to the point of contact (i.e. the larger circle), whereas in Presh's physical example(s), counting 1 revolution in reference to the observer, it completes an extra rotation.
With the second example, where each circle has the same radius, this is more obvious. As the revolving circle makes it halfway, and you look at it from the observer's perspective, then it makes 1 revolution half way through the overall rotation.
However, if you look at it from the perspective of the center circle, it will only look like one revolution. This is what you are effectively doing when you compare the circumferences algebraically, you are effectively considering the central circumference as a straight line.
It is not a scale issue, it is an issue in how you define a "full rotation," which, when given a diagram like the one in the test, implies that Presh's interpretation is actually correct.
How did you determine that the problem intended to ask for the sidereal rotations?
Frame of reference and the definition of revolve seem important. If I draw 2 lines on 2 quarters that are aligned and rotate one quarter around the other, the line on the moving quarter will re-contact the stationary quarter just once.
You are right... He was counting 1 rotation when coin is upright which is only halfway...
Mentally, I was thinking about circumference and that’s 3 circumference matching rolling segments, but after the 1st segment where Circle A is in the bottom-right, it’s not back to upright- what was initially down to Circle A is now up-left (120° CW), so its circumference has been rolled over once, but it’s revolved 1 1/3 times. Multiply by 3 and bam, 4.
I think the answer should be 3 because when the smaller circle has gone 1/3 the way around, the point touching the bigger circle is the same point that was touching it at the start
Correct, the edge of Circle A that was touching it when it starts has to stop on that exact edge / spot. It can't stop short like it did in the video. I understand their error because they were using the background as a reference point. Since B was at A 6 o'clock position, when that position touches B again, it would then completed a revolution.
Everyone is making the wrong assumption here. The amount of rotations the small circle does is determined by "length of path of small circle" / "circumference of small circle".
Let me explain by asking a simple question... How far does the small circle travel?
People are making the wrong assumption that it only travels the circumference of the larger circle, when that is not right (unless if the small circle overlapped the big circle). If there were a hole in the middle of the small circle and you put a pencil through that hole, then went around the big circle, the circle that you would draw would be larger because you have to also account for the radius of the small circle. Therefore the path is longer than people think it is.... So rotations = (Radius A + Radius B) / (Radius A). In this specific case, that is 4.
Negative on the 4 Travis.
For a full rotation / revolution of circle A, It can only count if the point it of A touching B, touches B again. Just imagine if both of them where stationary, and they only spun on an axis. You would see the real truth It is 3.
That would only be correct if they were both stationary and both were spinning (big circle clockwise, little circle counter-clockwise, etc). That is not what they are showing here. What they show and explicitly ask is how many times does Circle A spin when traveling around the perimeter of Circle B while Circle B is stationary. In that case, Circle A does spin 4 TIMES. And the reason it does is because Circle A is traveling a distance of 2*pi*(Ra + Rb), it requires Circle A to spin 4 times to travel that distance without slipping.
To think of this another way... If you are counting how many times a point on the small circle touches the large circle, the answer to that is 3. However, if you look at the orientation of the face of the small circle when it makes first contact again, the face is not pointing up... It is actually pointing at 120 degrees (vs 0 degrees) having already spun once completely around already. The reason for this is because that is the angle on Circle B it completes one revolution. It does this 3 times, so by the 3rd time, you get an extra rotation out of the face of Circle A.
So this is really a point of reference problem. If you were standing on Circle B and looking at Circle A, Circle A would spin 3 times. However, that is not what they are showing and illustrating in the video. The point of reference is stationary, so therefore it spins 4 times.
Based on the question and mathematical simple calculation the correct answer is that circle A revolves 3 times relative to the center of circle B
Yep. It’s just an illusion that it rotates 4 times.
@Eran Ohayon
"The small circle have to keep his face in the same direction in order to call it one revolve. each turn you said you make is actually 270 degrees.. so the only problem is with your understanding..."
Of course! At first I was really confused. I saw everyone talking about frames of reference and I just couldn't see this till this comment! Thanks!