Question

the radius of a circle A is 1 by 3 of the radius of circle B. Circle A rolls around circle B one trip back to its starting point . How many times will circle A revolve in total​

Answer 1

The intuitive answer is 3, but the correct answer is 4. I understand the trick -- that the center of Circle A must travel a distance of 2(+)
2
π
(
r
B
+
r
A
)
, not 2
2
π
r
B
-- but I'm still confused on one item.

At the risk of sounding very un-mathematical, how do the (infinite set of) points on the circumference of each circle map to each other to accomplish this?

Consider Circle A rolling along a straight line the length of the circumference of Circle B. Then it will revolve 3 times. It's like the universe "knows" when to apply a different point mapping when you change the arrangement of matter.

3 times lol

Answer 2

Answer:

prefer to it child (The ratio is 3, but the relative turns is 4....)

Step-by-step explanation:

Large circle A has circumference 3 times that of small circle B. First, mark the circles where A and B touch. Then imagine that you drag B along A’s edge without changing the point of B that touches A. After sliding all the way around, B would have made one full rotation relative to itself.

Now roll B around A without any sliding. With A stationary, B must roll along A’s circumference 3 times to return to the initial position, for three full rotations. The total rotation of B is 1+3 = four full rotations. QED.

Another way to look at this question is to consider rotating B while pinning both A’s center and B’s center in place. Think of gears on fixed axes. To return to the initial position, B will rotate 3 times and A will rotate 1 time in the opposite direction. The ratio is 3, but the relative turns is 4....see I explained u in full details hope you understand...