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> 0% selected the right answer on this SAT question: Circle A has 1/3 the radius of circle B, and circle A rolls one trip around circle B. How many times will c

> 0% selected the right answer on this SAT question: 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? That's fun. I of course immediately selected 3 which means I could have a bright career in test preparation ahead of me.

clon 5y ago
It was implied that the correct answer is 4.
- caf 5y ago
  Similarly, if you set out in a boat and circumnavigate the world in an easterly direction, ticking off a day on your calender every sunset, when you arrive back at your setting-off point your calendar will be one day ahead of everyone else.
  - bongoman37 5y ago
    Wasn't this a major plot point in Around the World in 80 Days?
kristjansson 5y ago
That one got me good, so my future at College Board is as bright as yours. However, I don't think the argument by demonstration in that video is particularly convincing. Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - it traces a circle whose radius is the sum of the two radii. So, if 3 * r_small = r_big, the center of the small circle has to travel 2 * pi * (r_s + r_b) = 4 * 2 * pi * r_s, then divide by 2 * pi * r_s per rotation to get 4 rotations.
- hinkley 5y ago
  Without the math: Radius is always proportional to circumference, so a circle twice the size is twice as big around. Take the case of two identical circles. To move a point on the first circle from 12 o'clock back to 12 o'clock, it only goes halfway around the other circle, which you can prove to yourself by imagining you've wrapped a string around the circle and marked it at 12 and 6 o'clock. If you unwrap half of the string and wrap it around the other circle, then the end of the rope is at 6 o'clock. To roll the string back up by moving the circle, the top of the circle will be pointing upward again when it reaches the bottom. 1 full revolution. Now wrap the other half of the string around the other side, 6 has to go back to the bottom again to roll the string back up. 2 revolutions.
  - kristjansson 5y ago
    It’s interesting to read others reasoning on this. I find it hard to follow and much harder to generalize without some notation (and a diagram which this comment box struggles to reproduce)
    - Linosaurus 5y ago
      I like to decompose it. Hold center of small circle still, rotate big circle once counter clock wise, small one rotates 3x clockwise. Glue them together, rotate big circle once clockwise, small circle also rotates once clockwise. Sum them together, 0 rotations for big circle, 4 for small. I'm not at all sure how to rigorously generalize.
- db_admin 5y ago
  You could also argue that it is a matter of perspective. From the perspective of either circle, A will only revolve 3 times. Only by introducing a larger frame of reference, a grid or in the video a table, you gain an outside perspective. From this outside perspective you redefine a revolution according to some new orientation and end up with n+1 revolutions. Or maybe the argument is backwards and I just try to justify answering 3.
- unholiness 5y ago
cousin_it 5y ago
I just thought of a simple argument: unroll the bigger circle into a line. Then as the smaller circle rolls from one end of the line to the other, it makes 3 revolutions. After that, roll up the line back into a circle (with the smaller circle still attached to the end). That adds one more revolution.
markc 5y ago
>A rolls one trip around circle B Without the diagram (which I didn't see until watching the video) this is ambiguous. In my visualization, the plane of the small circle (A) was perpendicular to the plane of the larger circle (B). (Think of circle B drawn on paper, while circle A is a coin on its edge) With that interpretation of "rolls one trip around", 3 is indeed is the correct answer.
dartharva 5y ago
I'm confused, I also immediately came to 3 when I read this question. Is that wrong? What's the correct answer?
- ryathal 5y ago
  the answer is 4. The reason it's 4 is because distance traveled is relative to the center of the circle. A circle will travel it's circumference in a rotation, but the distance traveled isn't actually the circumference of the inner circle, because that isn't where the center of the circle is. It actually travels the sum of the two circles radii.
raldi 5y ago
Imagine Circle B is reduced to infinitesimal size, like rolling a quarter around a needle. It still makes one full revolution, even though the ratio of the circumferences is effectively infinite.

clon 5y ago

It was implied that the correct answer is 4.

caf 5y ago
Similarly, if you set out in a boat and circumnavigate the world in an easterly direction, ticking off a day on your calender every sunset, when you arrive back at your setting-off point your calendar will be one day ahead of everyone else.
- bongoman37 5y ago
  Wasn't this a major plot point in Around the World in 80 Days?

kristjansson 5y ago

That one got me good, so my future at College Board is as bright as yours. However, I don't think the argument by demonstration in that video is particularly convincing. Instead, I think's its easier to note that that the _center_ of a circle of radius r travels 2 * pi * r distance over one rotation. In the problem, the center of the smaller circle has to travel further than the circumference of the bigger circle - it traces a circle whose radius is the sum of the two radii. So, if 3 * r_small = r_big, the center of the small circle has to travel 2 * pi * (r_s + r_b) = 4 * 2 * pi * r_s, then divide by 2 * pi * r_s per rotation to get 4 rotations.

hinkley 5y ago
Without the math: Radius is always proportional to circumference, so a circle twice the size is twice as big around. Take the case of two identical circles. To move a point on the first circle from 12 o'clock back to 12 o'clock, it only goes halfway around the other circle, which you can prove to yourself by imagining you've wrapped a string around the circle and marked it at 12 and 6 o'clock. If you unwrap half of the string and wrap it around the other circle, then the end of the rope is at 6 o'clock. To roll the string back up by moving the circle, the top of the circle will be pointing upward again when it reaches the bottom. 1 full revolution. Now wrap the other half of the string around the other side, 6 has to go back to the bottom again to roll the string back up. 2 revolutions.
- kristjansson 5y ago
  It’s interesting to read others reasoning on this. I find it hard to follow and much harder to generalize without some notation (and a diagram which this comment box struggles to reproduce)
  - Linosaurus 5y ago
    I like to decompose it. Hold center of small circle still, rotate big circle once counter clock wise, small one rotates 3x clockwise. Glue them together, rotate big circle once clockwise, small circle also rotates once clockwise. Sum them together, 0 rotations for big circle, 4 for small. I'm not at all sure how to rigorously generalize.
db_admin 5y ago
You could also argue that it is a matter of perspective. From the perspective of either circle, A will only revolve 3 times. Only by introducing a larger frame of reference, a grid or in the video a table, you gain an outside perspective. From this outside perspective you redefine a revolution according to some new orientation and end up with n+1 revolutions. Or maybe the argument is backwards and I just try to justify answering 3.
unholiness 5y ago

cousin_it 5y ago

I just thought of a simple argument: unroll the bigger circle into a line. Then as the smaller circle rolls from one end of the line to the other, it makes 3 revolutions. After that, roll up the line back into a circle (with the smaller circle still attached to the end). That adds one more revolution.

markc 5y ago

>A rolls one trip around circle B Without the diagram (which I didn't see until watching the video) this is ambiguous. In my visualization, the plane of the small circle (A) was perpendicular to the plane of the larger circle (B). (Think of circle B drawn on paper, while circle A is a coin on its edge) With that interpretation of "rolls one trip around", 3 is indeed is the correct answer.

dartharva 5y ago

I'm confused, I also immediately came to 3 when I read this question. Is that wrong? What's the correct answer?

ryathal 5y ago
the answer is 4. The reason it's 4 is because distance traveled is relative to the center of the circle. A circle will travel it's circumference in a rotation, but the distance traveled isn't actually the circumference of the inner circle, because that isn't where the center of the circle is. It actually travels the sum of the two circles radii.

raldi 5y ago

Imagine Circle B is reduced to infinitesimal size, like rolling a quarter around a needle. It still makes one full revolution, even though the ratio of the circumferences is effectively infinite.