All circles are proprtionate to each other, by having a fixed centerpoint for both you will have a fixed length for a single rotation because the length between the centerpoint and the surface being rolled upon is the same, youre essentially fixing a varible wich changes the path of movement, try taking to circles, one large one small that can fit in it, roll them both one rotation seperately with thier edges both touching the surface, you will absolutely have 2 seperate lengths
I think the distance is not the same. Rather the displacement is the same. So the distance is different for same time period, hence different angular velocity.
Before watching the video, I don't know how to prove it mathematically but I think it's because the small circle is within the larger circle and you're talking the dot through a shorter path since it's not rolling flat. It's going to be slight higher when the the larger circle is on the table and when the dot on the larger circle reaches the peak, the smaller circle dot is lower. Therefore slightly shorter path
Yes, the cycloids. But, you forgot to see that by a rolling of 360° the smaller perimeter is covering the same linear distance as the bigger one. It means, both perimeters have the same length as the straight distance they cover with a 360° rolling. This is a big problem.
I've never understood the cycloid explanation, although I've heard it goes back centuries. Just to have a concrete set-up, let's divide the wheels into 360 sectors of 1 degree each. Let's further suppose that the larger wheel has a circumference of 360 cm and the smaller a circumference of 180 cm. So each 1-degree sector corresponds to 1 cm of circumference on the outer wheel and 0.5 cm of circumference on the inner wheel. The outer wheel rolls along a 360 cm long track and, at the same time, the inner wheel rolls along its own (real or imaginary) 360 cm long track. Let's mark off 1cm intervals along both tracks. We might even number the intervals from 1 to 360 (from left to right, with the wheel starting from the left end), and also number the sectors of the circumference (in the counterclockwise direction starting from the initial contact point) from 1 to 360. Here's how I see the paradox (and I think this is essentially how it is stated in the original Mechanica text): as the outer wheel rolls, first the 1 cm long sector #1 of the outer circumference contacts the 1 cm long interval #1 of the track, then the 1 cm long sector #2 of the outer circumference contacts the 1 cm long interval #2 of the track, then sector #3 contacts interval #3, and so on for all 360 sectors. At the same time this is happening, first the 0.5 cm long sector #1 of the inner circumference contacts (the whole of) the 1 cm long interval of its track, then the 0.5 cm long sector #2 of the inner circumference contacts (the whole of) the 1 cm long interval #2 of its track, then sector #3 contacts (the whole of) interval #3, and so on for all 360 sectors. I think a paradox arose in people's minds because they could perfectly well visualize the matching up of 1 cm outer circumference sectors with 1 cm track intervals as the wheel rolls, but they had a hard time understanding how the 0.5 cm inner circumference sectors could match up with the whole of the 1 cm track intervals in a smooth and continuous way. After having thought through the problem in detail I personally don't see any difficulty, but I think I do understand people's initial discomfort. Now if you agree with my framing of the paradox -- and I do believe it's true to the original -- how does drawing cycloids and curtate cycloids help resolve it? The paradox is concerned with what's going on in at the points of contact of the two wheels with their surfaces. If we put red dots in sector #1 on both circles, the cycloid and curtate cycloid show the movement of the red dots, but most of that movement takes place far from the contact point. (The red dots come back to the surface only once per revolution.) To understand the paradox don't you have to examine carefully how things are moving at the point of contact? This involves a succession of different sectors, not one single sector. Also I don't understand how you can go from "the red dot on the inner circle travels a more direct path through space than does the red dot on the outer circle" to "the inner circle travels a more direct path than the outer circle". I don't see the path of the circle and the path of a point on the circumference as the same thing. But the more important point is that the paradox is concerned with matching, not paths. It's concerned with the matching between circumference intervals and track intervals.
stopped at 1:05 Because it doesn't travel the same distance on the same plane. The inner ring never touches the same surface as the outer ring. It is traveling along an imaginary plane above the table. Oh yeah, and the inner circle is traveling faster along the imaginary plane than the outer circle is along the table top. Edit: I see what you;re saying but I think my explanation is better lol
It is true that the inner ring never touches the table. The inner ring and outer ring are attached; therefore, they both have to travel the same distance. I don't think the inner circle is traveling faster, because both circles take the same amount of time to travel the same distance. I think I understand your imaginary plane explanation. I just disagree. This is the nature of the problem. It's a paradox, after all.
@@marcuskoseck98 But I don't believe it to be a paradox, just an unintuitive problem as you pointed out. Do the experiment differently and you will see what I mean. Get two rings equal to the inner and outer rings. Then move them across a plane. You could put two planes at the relative heights or just use your imagination that the same plane is at two different heights. In order for the smaller ring to move the same distance in the same amount of time, it must move faster across the plane. Now, if both planes were being used, and each had a mark, those marks would still be aligned as both rings travel. Basically, it is how gearing works. (Edit: faster as in rotation)
Yes. You explained it much better this time and everything you said is true. (The gear explanation really helped). Perhaps I didn't frame the question properly. The question assumes that the two rings are glued together the entire time and form a rigid body. So, the situation that you describe for the smaller circle to detach cannot happen in this problem. The nature of the problem is that the smaller circle rotates at the same rate as the larger circle and travels the same distance. It's not supposed to make sense, hence the paradox.
@@marcuskoseck98 Yes, exactly. It's like the rim of a tire. It may be a 15" rim, but if it's attached to a 17" tire, the car will move slower than if it were attached to a 19" tire although the rotation speed of the rim remains the same. Since as you increase the surface area of the tire, that same rotation is causing more surface area to move faster in order to maintain the same rotation speed.
The inner circle douse not travel the same distance. Note the inner circles distance of travel is it’s own distance Plus both the outside distance both Sides to the out side circle.distance. Measure both the inside circle and out side circle you will See that I am right Simple Yes!
The bigger circle is taking the snaller one for a ride. Its being carried forward at a faster rate than it would if it was driving the rotation.
Codependent motion. Since the larger circle encompasses the smaller, the distances become equal.
All circles are proprtionate to each other, by having a fixed centerpoint for both you will have a fixed length for a single rotation because the length between the centerpoint and the surface being rolled upon is the same, youre essentially fixing a varible wich changes the path of movement, try taking to circles, one large one small that can fit in it, roll them both one rotation seperately with thier edges both touching the surface, you will absolutely have 2 seperate lengths
I think the distance is not the same. Rather the displacement is the same. So the distance is different for same time period, hence different angular velocity.
Before watching the video, I don't know how to prove it mathematically but I think it's because the small circle is within the larger circle and you're talking the dot through a shorter path since it's not rolling flat. It's going to be slight higher when the the larger circle is on the table and when the dot on the larger circle reaches the peak, the smaller circle dot is lower. Therefore slightly shorter path
Yes, the cycloids. But, you forgot to see that by a rolling of 360° the smaller perimeter is covering the same linear distance as the bigger one. It means, both perimeters have the same length as the straight distance they cover with a 360° rolling. This is a big problem.
I've never understood the cycloid explanation, although I've heard it goes back centuries.
Just to have a concrete set-up, let's divide the wheels into 360 sectors of 1 degree each. Let's further suppose that the larger wheel has a circumference of 360 cm and the smaller a circumference of 180 cm. So each 1-degree sector corresponds to 1 cm of circumference on the outer wheel and 0.5 cm of circumference on the inner wheel. The outer wheel rolls along a 360 cm long track and, at the same time, the inner wheel rolls along its own (real or imaginary) 360 cm long track. Let's mark off 1cm intervals along both tracks. We might even number the intervals from 1 to 360 (from left to right, with the wheel starting from the left end), and also number the sectors of the circumference (in the counterclockwise direction starting from the initial contact point) from 1 to 360.
Here's how I see the paradox (and I think this is essentially how it is stated in the original Mechanica text): as the outer wheel rolls, first the 1 cm long sector #1 of the outer circumference contacts the 1 cm long interval #1 of the track, then the 1 cm long sector #2 of the outer circumference contacts the 1 cm long interval #2 of the track, then sector #3 contacts interval #3, and so on for all 360 sectors. At the same time this is happening, first the 0.5 cm long sector #1 of the inner circumference contacts (the whole of) the 1 cm long interval of its track, then the 0.5 cm long sector #2 of the inner circumference contacts (the whole of) the 1 cm long interval #2 of its track, then sector #3 contacts (the whole of) interval #3, and so on for all 360 sectors. I think a paradox arose in people's minds because they could perfectly well visualize the matching up of 1 cm outer circumference sectors with 1 cm track intervals as the wheel rolls, but they had a hard time understanding how the 0.5 cm inner circumference sectors could match up with the whole of the 1 cm track intervals in a smooth and continuous way. After having thought through the problem in detail I personally don't see any difficulty, but I think I do understand people's initial discomfort.
Now if you agree with my framing of the paradox -- and I do believe it's true to the original -- how does drawing cycloids and curtate cycloids help resolve it? The paradox is concerned with what's going on in at the points of contact of the two wheels with their surfaces. If we put red dots in sector #1 on both circles, the cycloid and curtate cycloid show the movement of the red dots, but most of that movement takes place far from the contact point. (The red dots come back to the surface only once per revolution.) To understand the paradox don't you have to examine carefully how things are moving at the point of contact? This involves a succession of different sectors, not one single sector. Also I don't understand how you can go from "the red dot on the inner circle travels a more direct path through space than does the red dot on the outer circle" to "the inner circle travels a more direct path than the outer circle". I don't see the path of the circle and the path of a point on the circumference as the same thing. But the more important point is that the paradox is concerned with matching, not paths. It's concerned with the matching between circumference intervals and track intervals.
Why is he dividing by 2pi. It's just divide by 2 to get the radius, why are you involving pi
Circumference=2πr
stopped at 1:05 Because it doesn't travel the same distance on the same plane. The inner ring never touches the same surface as the outer ring. It is traveling along an imaginary plane above the table. Oh yeah, and the inner circle is traveling faster along the imaginary plane than the outer circle is along the table top.
Edit: I see what you;re saying but I think my explanation is better lol
It is true that the inner ring never touches the table. The inner ring and outer ring are attached; therefore, they both have to travel the same distance. I don't think the inner circle is traveling faster, because both circles take the same amount of time to travel the same distance. I think I understand your imaginary plane explanation. I just disagree. This is the nature of the problem. It's a paradox, after all.
@@marcuskoseck98 But I don't believe it to be a paradox, just an unintuitive problem as you pointed out.
Do the experiment differently and you will see what I mean. Get two rings equal to the inner and outer rings. Then move them across a plane. You could put two planes at the relative heights or just use your imagination that the same plane is at two different heights.
In order for the smaller ring to move the same distance in the same amount of time, it must move faster across the plane. Now, if both planes were being used, and each had a mark, those marks would still be aligned as both rings travel.
Basically, it is how gearing works.
(Edit: faster as in rotation)
@@marcuskoseck98 Not sure if I explained it that well.
Yes. You explained it much better this time and everything you said is true. (The gear explanation really helped). Perhaps I didn't frame the question properly. The question assumes that the two rings are glued together the entire time and form a rigid body. So, the situation that you describe for the smaller circle to detach cannot happen in this problem. The nature of the problem is that the smaller circle rotates at the same rate as the larger circle and travels the same distance. It's not supposed to make sense, hence the paradox.
@@marcuskoseck98 Yes, exactly. It's like the rim of a tire. It may be a 15" rim, but if it's attached to a 17" tire, the car will move slower than if it were attached to a 19" tire although the rotation speed of the rim remains the same. Since as you increase the surface area of the tire, that same rotation is causing more surface area to move faster in order to maintain the same rotation speed.
Nice video! Thanks! : )
The simplest explanation!
Aristotle musta been drinking the day he got stumped on this. The inner circle is attached to the outer circle that's why.
Brilliant!
The inner circle douse not travel the same distance.
Note the inner circles distance of travel is it’s own distance
Plus both the outside distance both Sides to the out side
circle.distance.
Measure both the inside circle and out side circle you will
See that I am right Simple Yes!