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# wheels moving and stuff watch

1. Find the ratio, over one revolution, of the distance moved by a wheel rolling on a flat surface to the distance traced out by a point on its circumferance.

So I had a think about it, and I guessed the wheel is going to trace the following (diagram attached).

Now I guessed the easiet thing to do would be to do it in parametric equations in theta, where theta is the angle it's moved through.

The x distance moved by the point on the circumferance would be x = cos theta + k.theta, the k.theta term I guessed from the wheel translating a certain distance at a fixed speed whilst it revolves.

The y distance would be y = sin theta

So you could find the length of this arc using S = integral of [(dy/dt)^2 + (dx/dt)^2]^0.5, but then I get something I couldn't integrate, and judging by the output of integrals.com nobody could easily.

Where have I gone wrong? Cheers. lex
Attached Images

2. hmm.

Over one whole revolution, the distance moved by the wheel will be equal to it's circumference... so why can't the ratio just be 1? Or I'm misinterpreting the question...
3. (Original post by mik1a)
hmm.

Over one whole revolution, the distance moved by the wheel will be equal to it's circumference... so why can't the ratio just be 1? Or I'm misinterpreting the question...
well think about it, if you role a bottle along the floor, it doesn't just revolve around its centre of mass, it actually translates along the floor.
4. Ah, I was thinking displacement. So you mean like travelling around a circle (say radius r) AND travelling the distance?

Wouldn't it just be 2 then, as the distance travelled is the sum of the distance travelled by the bottle and the circumference of the bottle (as in one revolution the bottle moves its circumference along)?

Ehh, probably not if you need sin and stuff for it.
5. I could have gone mad, but isn't the point tracing out a spiral?
6. (Original post by mik1a)
Ah, I was thinking displacement. So you mean like travelling around a circle (say radius r) AND travelling the distance?

Wouldn't it just be 2 then, as the distance travelled is the sum of the distance travelled by the bottle and the circumference of the bottle (as in one revolution the bottle moves its circumference along)?

Ehh, probably not if you need sin and stuff for it.
Yes. I think you can see it as like a rotating disc, on top of a lorry. But I don't think you can just add the circumferance and the distance travelled together.
7. (Original post by Mysticmin)
I could have gone mad, but isn't the point tracing out a spiral?
Don't think so?
8. (Original post by fishpaste)
Don't think so?
but it's like an electron moving in a circle and a force pushing it forwards in a horizonal direction as well...
9. It would look like this I think. How you can represent the length of this, I don't know,
Attached Images

10. (Original post by mik1a)
It would look like this I think. How you can represent the length of this, I don't know,
Yup, that's what I mean by a spiral
11. (Original post by mik1a)
It would look like this I think. How you can represent the length of this, I don't know,
Yes, I think it can be done using parametric equations. Where x=cos theta + k.theta, and y = sin theta. If you've not done P5 yet, when you do you'll learn an expression that the length of an arc S is
let t = theta
S = integral of [(dy/dt)^2 + (dx/dt)^2]^0.5 dt between the two t values, so in this case it would be the integral of

dy/dt = rcos t
dx/dt = -rsint + k

subbing that into S, mosst cancels, and I'm just left with integral of (k^2 -2krsint)^0.5 dt between 2pi and 0, whcih I can't evaluate.
12. Well the good news is that I plotted that parametric graph and it is like identical to the trace you guys sketched.
13. (Original post by fishpaste)
Well the good news is that I plotted that parametric graph and it is like identical to the trace you guys sketched.
I should recognise those equations...good thing it didn't come up in the P5 exam. btw, how did you find it fishpaste?
14. (Original post by Mysticmin)
I should recognise those equations...good thing it didn't come up in the P5 exam. btw, how did you find it fishpaste?
Did you do it on OCR? I found myself floundering for no good reason. Frustrating when you know you should be doing it with your eyes closed and you struggle. you?
15. (Original post by fishpaste)
Did you do it on OCR? I found myself floundering for no good reason. Frustrating when you know you should be doing it with your eyes closed and you struggle. you?
Edexcel, better than expected, the questions were easier than most of the practise ones we did. However I left out the 4 mark question at the end because I got the wrong answer for the 5 mark one before, possibly because I factorised my algebraic quadratic wrong
16. (Original post by Mysticmin)
Edexcel, better than expected, the questions were easier than most of the practise ones we did. However I left out the 4 mark question at the end because I got the wrong answer for the 5 mark one before, possibly because I factorised my algebraic quadratic wrong
Glad it went well overall.
17. (Original post by fishpaste)
Glad it went well overall.
I'm sure you did much better than you thought!
18. (Original post by Mysticmin)
I'm sure you did much better than you thought!
Well it all comes down to "Lex doesn't do substandard grades," so regardless of reality I'm just going to hope for the best
19. (Original post by fishpaste)
Well it all comes down to "Lex doesn't do poor grades," so regardless of reality I'm just going to hope for the best.
Likewise here. Hey at least you didn't mess up up P2
20. If the point is on the rim of the wheel, then the locus of its travel will be a cycloid - looks a bit like the arches of a bridge. See here
You can get the parametric eqns for cycloidal travel here

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