Distance round a circle
Discussion
Ok usually I'm quite good at things like this and I'm the one who ends up explaining things to other people when they are convinced its black magic but this one has me a bit stumped and its probably primary school engineering really.
The beginning bit of this video where he is rolling the wheel along a linear surface and then around a wheel.
I get that the centre of the wheel is travelling the same distance along its length when its a flat surface but it has to travel a lot farther when its going round in a circle so the wheel has to rotate twice to cover that greater distance but.... how does the tyre know its going round in a circle and not along a flat track, it has traction on the surface and the outer edge is traveling along the exact same length but it isn't skidding along to do the extra rotation.
The only thing I can think of is because the surface is now curved then there is less contact area so it has to do more revolutions but that could be well wide of the mark for all I know. I just can't get my head round this for now.
The beginning bit of this video where he is rolling the wheel along a linear surface and then around a wheel.
I get that the centre of the wheel is travelling the same distance along its length when its a flat surface but it has to travel a lot farther when its going round in a circle so the wheel has to rotate twice to cover that greater distance but.... how does the tyre know its going round in a circle and not along a flat track, it has traction on the surface and the outer edge is traveling along the exact same length but it isn't skidding along to do the extra rotation.
The only thing I can think of is because the surface is now curved then there is less contact area so it has to do more revolutions but that could be well wide of the mark for all I know. I just can't get my head round this for now.
It's an interesting one, did my head in at first just like yourself. The explanation for the centre of the wheel seems quite straightforward, until you think about it like you say!
His explanation of the teeth and gears helps I think. Imagine one tooth on the round gear running along the straight gear, the round gear has to turn a certain amount for the given tooth to reach the next tooth on the straight gear. Take the same tooth on the round gear, but now moving around the big circular gear. The tooth doesn't just have to travel 'right a bit' like it did for the straight gear, it also has to travel 'down a bit'. That 'across a bit and down a bit' is further to travel, which I think accounts for the difference. At least that's how I can get it to make sense in my head.
His explanation of the teeth and gears helps I think. Imagine one tooth on the round gear running along the straight gear, the round gear has to turn a certain amount for the given tooth to reach the next tooth on the straight gear. Take the same tooth on the round gear, but now moving around the big circular gear. The tooth doesn't just have to travel 'right a bit' like it did for the straight gear, it also has to travel 'down a bit'. That 'across a bit and down a bit' is further to travel, which I think accounts for the difference. At least that's how I can get it to make sense in my head.
Replace 'wheels' with 'circles on paper'.
Draw 2 circles touching at the perimeter. Stick the point of a compass through the centre (axis) of the fixed circle, and the pencil through the centre (axis) of the moving one. Draw that circle, and while you're there draw a straight line between the two axes. You'll notice that straight line is just the radius of the first circle, and the radius of the second circle.
Double the radius, you get double the circumference. From the first example, 1 rotation = 1 circumference of distance traveled. So if the circumference is doubled, double the rotations are needed.
Draw 2 circles touching at the perimeter. Stick the point of a compass through the centre (axis) of the fixed circle, and the pencil through the centre (axis) of the moving one. Draw that circle, and while you're there draw a straight line between the two axes. You'll notice that straight line is just the radius of the first circle, and the radius of the second circle.
Double the radius, you get double the circumference. From the first example, 1 rotation = 1 circumference of distance traveled. So if the circumference is doubled, double the rotations are needed.
Mr Pointy said:
Still confused, you can see it with the triangle though how it needs extra rotations and length to go round the corners, as opposed to a straight line.It is because the moving circle is rotating around the fixed circle as well as rotating about it's centre. If both circles stayed in place and just rotated against each other like gears, they would each do one rotation.
If you draw a line between the circles, and rotate the moving circle around the fixed circle without rolling it, IE with the line on the moving circle remaining pointed at the centre of the fixed circle(the circles sliding on each other), the moving circle rotates once. If you do the same, but also rotate the moving circle so the line drawn on it rotates away from the centre of the fixed circle, and rotates as the moving circle goes round the fixed circle,(as if they were gears) then the moving circle rotates both around the fixed circle, and around its own centre making two revolutions .
https://youtu.be/yyFLpN_E4y8?si=v2msw5jXVPp5aDQ7
If you draw a line between the circles, and rotate the moving circle around the fixed circle without rolling it, IE with the line on the moving circle remaining pointed at the centre of the fixed circle(the circles sliding on each other), the moving circle rotates once. If you do the same, but also rotate the moving circle so the line drawn on it rotates away from the centre of the fixed circle, and rotates as the moving circle goes round the fixed circle,(as if they were gears) then the moving circle rotates both around the fixed circle, and around its own centre making two revolutions .
https://youtu.be/yyFLpN_E4y8?si=v2msw5jXVPp5aDQ7
Edited by Super Sonic on Monday 2nd February 18:49
Was thinking about this last night.
I resolved it in a way that makes my tiny brain happy, I think.
Imagine the cog on the linear toothed rail, if you roll it along and do one revolution it stops at the end of the rail. Now weld that in place.
That rail instead of being straight was actually on the surface of a wheel, so starting where the cog started its travel, wrap that rail (with the welded cog at the other end) around the wheel, without doing anything to the cog. As you wrap the rail around the wheel the cog will rotate once.
So, one rotation when it travelled in a linear fashion plus the one rotation when making the rail conform to the surface.
My ability to explain this may have different mileage...
I resolved it in a way that makes my tiny brain happy, I think.
Imagine the cog on the linear toothed rail, if you roll it along and do one revolution it stops at the end of the rail. Now weld that in place.
That rail instead of being straight was actually on the surface of a wheel, so starting where the cog started its travel, wrap that rail (with the welded cog at the other end) around the wheel, without doing anything to the cog. As you wrap the rail around the wheel the cog will rotate once.
So, one rotation when it travelled in a linear fashion plus the one rotation when making the rail conform to the surface.
My ability to explain this may have different mileage...
Super Sonic said:
It is because the moving circle is rotating around the fixed circle as well as rotating about it's centre. If both circles stayed in place and just rotated against each other like gears, they would each do one rotation.
If you draw a line between the circles, and rotate the moving circle around the fixed circle without rolling it, IE with the line on the moving circle remaining pointed at the centre of the fixed circle(the circles sliding on each other), the moving circle rotates once. If you do the same, but also rotate the moving circle so the line drawn on it rotates away from the centre of the fixed circle, and rotates as the moving circle goes round the fixed circle,(as if they were gears) then the moving circle rotates both around the fixed circle, and around its own centre making two revolutions .
https://youtu.be/yyFLpN_E4y8?si=v2msw5jXVPp5aDQ7
That video actually makes a bit more sense. It did explain this in the Veritasium video as well. If you watch the line at the outer edge of the wheel the contact point only makes one revolution in relation to the other wheel so the distance travelled is just the same as if it were flat, the extra revolution comes from the wheel itself going round in a circle.If you draw a line between the circles, and rotate the moving circle around the fixed circle without rolling it, IE with the line on the moving circle remaining pointed at the centre of the fixed circle(the circles sliding on each other), the moving circle rotates once. If you do the same, but also rotate the moving circle so the line drawn on it rotates away from the centre of the fixed circle, and rotates as the moving circle goes round the fixed circle,(as if they were gears) then the moving circle rotates both around the fixed circle, and around its own centre making two revolutions .
https://youtu.be/yyFLpN_E4y8?si=v2msw5jXVPp5aDQ7
Edited by Super Sonic on Monday 2nd February 18:49
Is that right?
Frane Selak said:
That video actually makes a bit more sense. It did explain this in the Veritasium video as well. If you watch the line at the outer edge of the wheel the contact point only makes one revolution in relation to the other wheel so the distance travelled is just the same as if it were flat, the extra revolution comes from the wheel itself going round in a circle.
Is that right?
That's absolutely correct. It didn't show it but if you can imagine the line on the moving wheel staying pointed at the centre of the fixed wheel, the moving will would only rotate once.Is that right?
Edited by Super Sonic on Monday 2nd February 19:09
Imagine you have one massive wheel rolling around the outside of a tiny, tiny, tiny wheel. In fact imagine making that tiny wheel arbitrarily tiny. In the limit it just becomes an infinitely small pin, pinning one point on the side of the big wheel in place. Now the big wheel just spins around that point on its edge where it is pinned and it is intuitively obvious that spinning it around that point once will turn the wheel through 360. That's the "extra" rotation you get when you roll one wheel around the outside of another wheel.
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