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I came to a question some years ago, and have made computer-calculation to solve it, not knowing, is it's answer correct... I formulated this: A tape (or belt) is wired up (on a cylinder) over itself. We know the diameter of the cylinder and the thickness of the belt, and how long it is.

With Excel I made program that gave;

(tape length: 1m, tape thickness 0.1mm, cylinder diam. 20mm): 13 whole laps. (tape length: 10m, tape thickness 0.1mm, cylinder diam. 20mm): 103 whole laps. (tape length: 11m, tape thickness 1mm, cylinder diam. 15mm): 51 whole laps.

I'm know very little about math, came here from stackoverflow (programming). Screen shot, Excel

Valter Ekholm
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1 Answers1

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Let $\ell$ be the length of material, $r$ the radius of the cylinder, $w$ the width of the material. Then, if the number of laps is exact, and ignoring the flexing that happens on the overlap in each lap, \begin{align} \ell&=2\pi r+2\pi(r+w)+2\pi(r+2w)+\cdots+2\pi(r+Lw)\\ \ \\ &=2\pi r(L+1)+2\pi w(1+2+\cdots+L)\\ \ \\ &=2\pi r(L+1)+2\pi w\frac{L(L+1)}2\\ \ \\ &=2\pi r(L+1)+\pi w L(L+1)\\ \ \\ &=\pi w L^2+(2\pi r+\pi w)L+2\pi r. \end{align} Then \begin{align} L&=\frac{-2\pi r-\pi w+\sqrt{\left(2\pi r+\pi w\right)^2-4\pi w(2\pi r-\ell)}}{2\pi w}\\ \ \\ &=\frac{-2 r- w+\sqrt{\left(2 r+ w\right)^2-4 w(2 r-\ell/\pi)}}{2 w}. \end{align} To allow for the number of laps to not be an integer, one would then have to take the closest smaller integer, i.e. $\lfloor L\rfloor$.

Martin Argerami
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  • OP: This ignores (as I suspect most answers would have to ignore) the flexure in the material when it overlaps the starting point. Still deserves a +1. – Brian Tung Oct 19 '16 at 23:19
  • Good point. I wouldn't exactly know how to deal with it; it certainly depends on properties of the material. – Martin Argerami Oct 19 '16 at 23:27
  • One possible approach is to assume that the bottom surface of the first revolution stays along the cylinder most of the way, and then extends along a tangent to the top surface of the start; further revolutions then sit directly upon this more or less smooth surface. Whether one address this or not, incidentally, the top surface is longer than the bottom surface, and one has an issue of how to address that. I think your approach is fine as it is! :-) – Brian Tung Oct 19 '16 at 23:34
  • I could use the formula, after all, it gave same results as my program (at least two first examples). Thanks – Valter Ekholm Dec 09 '16 at 23:51
  • Also same as the last example – Valter Ekholm Dec 10 '16 at 15:13