3 ms·

Yea, I think the difference between “how many rotations” and “how far is the center of the small disk traveling” is pretty big. I was completely lost as to how

by omeze 3y ago

Yea, I think the difference between “how many rotations” and “how far is the center of the small disk traveling” is pretty big. I was completely lost as to how you “add one” until I watched what the video called a “rotation”. It would have never occurred to me to call the coin being upright a rotation… the Art of Problem solving question in another comment thread was more clearly worded for that scenario

HarHarVeryFunny 3y ago
The question wording was about "revolutions" of the small disk. The distance traveled by the small disk center (= 4x it's own diameter) was only used as a way to calculate the number of revolutions, but I'm still not getting the intuition for how each diameter of distance traveled by the disk center equals one rotation regardless of the shape of path followed (it's obvious for a flat line of course).
- jncfhnb 3y ago
  If a coin rolls such that its complete perimeter touches the other surface exactly once it will move the distance of its perimeter. In this case the total distance is 3x the perimeter so the edge answer is 3. Shape doesn’t matter. If you’re wondering how many times the traveling coin 360s you have to account for the fact that it is rotating both because it’s rolling and because it’s going around the the other thing. You can simplify this by just imagining it’s the edge again, but the edge is going along the radius of the inner and traveling coin (3+1). Dumb it down. How far does a point travel? X units. X units / coin perimeter = coin rotations required to travel X. Ergo the only question is whether the distance traveled is along the perimeter of the 3r coin or if it’s based on the center of the traveling coin (3+1).
- jodrellblank 3y ago
  (as I commented elsewhere) I found it useful to picture a unicycle; a straight line track means the wheel rolls forwards the same distance as the length of the track. Curve the track around to the right into a circle and when the unicycle rolls forward in a straight line it's getting away from the track (at a tangent) - it would have to slide sideways to get to where the track is. That is some extra movement for the curved track compared to the straight track which has to be accounted for somewhere, somehow, and of course the unicycle turns right while rolling forwards to follow the curve in the track. When it gets back to the start, it must have done 1 complete turn to the right as well as all the rolling forwards. As long as the track is simple (no loops, bridges, crossings), e.g. a square with rounded corners, going around and back to the start needs the left and right turns to balance out to get to 1 complete turn - too much turning right spirals in, too much turning left spirals out, only a balanced amount of left and right turning that ends up as 1 complete turn can meet up back at the beginning. Seeing the turns in two axes made it clear to me that the forwards rolling is always the length of the track because changing the shape doesn't change the length, and the sideways turn is always one for any simple loop or it wouldn't be a circuit. That fixes your question "how each diameter of distance traveled by the disk center equals one rotation regardless of the shape of path followed" - you say it's obvious for a straight line, and curving the line can't change the length of the line, the extra rotation to go round a circuit must be one, so there's nothing which can vary. (If you lie the unicycle down on its side, the 1 complete turn to the right needed to follow the curve around to the start doesn't go away, it has to happen, and now it has to happen in the same axis as the rolling forwards happens so it's much less clear (to me). If the only thing the wheel can do is roll forwards, this extra movement from earlier to avoid getting away from the track can only be extra rolling - the extra one turn for the loop, spread over the N turns for the distance).

by omeze 3y ago

HarHarVeryFunny 3y ago

The question wording was about "revolutions" of the small disk. The distance traveled by the small disk center (= 4x it's own diameter) was only used as a way to calculate the number of revolutions, but I'm still not getting the intuition for how each diameter of distance traveled by the disk center equals one rotation regardless of the shape of path followed (it's obvious for a flat line of course).

jncfhnb 3y ago
If a coin rolls such that its complete perimeter touches the other surface exactly once it will move the distance of its perimeter. In this case the total distance is 3x the perimeter so the edge answer is 3. Shape doesn’t matter. If you’re wondering how many times the traveling coin 360s you have to account for the fact that it is rotating both because it’s rolling and because it’s going around the the other thing. You can simplify this by just imagining it’s the edge again, but the edge is going along the radius of the inner and traveling coin (3+1). Dumb it down. How far does a point travel? X units. X units / coin perimeter = coin rotations required to travel X. Ergo the only question is whether the distance traveled is along the perimeter of the 3r coin or if it’s based on the center of the traveling coin (3+1).
jodrellblank 3y ago
(as I commented elsewhere) I found it useful to picture a unicycle; a straight line track means the wheel rolls forwards the same distance as the length of the track. Curve the track around to the right into a circle and when the unicycle rolls forward in a straight line it's getting away from the track (at a tangent) - it would have to slide sideways to get to where the track is. That is some extra movement for the curved track compared to the straight track which has to be accounted for somewhere, somehow, and of course the unicycle turns right while rolling forwards to follow the curve in the track. When it gets back to the start, it must have done 1 complete turn to the right as well as all the rolling forwards. As long as the track is simple (no loops, bridges, crossings), e.g. a square with rounded corners, going around and back to the start needs the left and right turns to balance out to get to 1 complete turn - too much turning right spirals in, too much turning left spirals out, only a balanced amount of left and right turning that ends up as 1 complete turn can meet up back at the beginning. Seeing the turns in two axes made it clear to me that the forwards rolling is always the length of the track because changing the shape doesn't change the length, and the sideways turn is always one for any simple loop or it wouldn't be a circuit. That fixes your question "how each diameter of distance traveled by the disk center equals one rotation regardless of the shape of path followed" - you say it's obvious for a straight line, and curving the line can't change the length of the line, the extra rotation to go round a circuit must be one, so there's nothing which can vary. (If you lie the unicycle down on its side, the 1 complete turn to the right needed to follow the curve around to the start doesn't go away, it has to happen, and now it has to happen in the same axis as the rolling forwards happens so it's much less clear (to me). If the only thing the wheel can do is roll forwards, this extra movement from earlier to avoid getting away from the track can only be extra rolling - the extra one turn for the loop, spread over the N turns for the distance).