Rolling paper cup

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Given a conic geometry, like that of a paper cup. What is the radius of a circle made by the rolling made by the cup. It actually makes two circles so either radius as long as it is specified will do.

Given:

The cup's large radius = R
The cup's small radius = r
The length of the cup or height = h


So far I know that the circumference of the circle made when the cup rolls is inversely proportional to the slope of the cup:
Circumference ~ 1/slope or h/(R-r)
 

HallsofIvy

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When you are "given" the dimensions of the cup, you are given the radius of the top and bottom of the cup (or can calculate them from the information you have). From that you can calculate the circumference of the top and bottom. Since the cup rolls the same degrees on both of those the circles that the two parts must be will have circumferences the same multiple if the radius you are looking for.
 
I guess I just don't know how many turns the cup will make in one complete circle.
 

mathman

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What you have to get is the height of the cone by extending the small end to a point. Let a=the addition to the height. then a/r=(a+h)/R. Solve for a, then the radii you want are a and a+h.
 
I get

a= (rh/R)/(1-r/R)

That seems right. As the height of the cup gets larger so does the turn ,which makes sense since the turn should be inversely proportional to the slope (R-r)/h


Thanks.
 

mathman

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I made a mistake in my previous note. After you get a as described, the radii are given by
r1=sqrt(a2+r2)
r2=(a+h)r1/a
 
have you tried rolling a paper cup to confirm the theoritical values?
 

mathman

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When I was in junior high, the math teacher introduced us to pi by having us cut out circles from cardboard and measure the diameters and the circumferences. We quickly learned that the ratio was approximately independent of the size of the circle (one smart aleck drew them on paper and knowing about pi, calculated the circumferences - the teacher was annoyed). In any case we quickly got the idea. I put your question about experimenting with paper cups in the same category. It might be a good exercise for a high student geometry student, but I don't think it is necessary here.
 

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