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I'm interested in learning how to find the length of a wrapping.

Let's say I'm going to be wrapping some flat fabric webbing around a pole. I'd like to find the amount (length of fabric) i'll need to buy to wrap a portion of the pole.

  • Desired length of wrap: 30 in.
  • Outer diameter of pole: 2 in.
  • Width of webbing: 0.5 in.
  • Angle of spiral: 15° (arbitrary value I came up with)

Thanks!

dhulihan
  • 173

2 Answers2

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Basically, you want to cover a cylinder with $h = 30\text{ in}$ and $\Phi = 2R = 2\text{ in}$. The area of a cylinder without the top and bottom is $A = 2R\pi h = \Phi \pi h$.

Imagine that you cut this "open" cylinder after you have wrapped it: you obtain a rectangle with sides $h$ and $2R\pi$ and this rectangle is covered by slightly tilted "$2D$ strips" of fabric.

The width of each strip is $w = 0.5\text{ in}$ and the area covered by a fabirc of length $\ell$ equals $w\ell$ (which must equal $A$). Note that the angle doesn't matter. So you need $$\ell = \frac{\Phi \pi h}{ w} \doteq \frac{2\cdot3.14 \cdot 30}{0.5}\text{ in}\doteq 377\text{ in}$$ of fabric.

Antoine
  • 3,439
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Consider unwrapping the wrapped strip, keeping the angle at $15°$. The final result is a right triangle with acute angle of $15°$ where the edge of the strip is the hypotenuse, and the side opposite the angle is the $30$-inch segment of the pole. Thus the length of the cloth strip is the length of the hypotenuse of that triangle: $30\,/\left(\sin 15°\right)$ inches, somewhat close to $115.9$ inches. As long as the $"\!15°\!"$ is fixed, the width of the strip and the diameter of the pole determine the number of times you loop around the pole but do not affect the length in the question.

Vlad
  • 6,710
Pete
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  • From what you are saying it follows that $116$ inches suffice to wrap a pole with $\Phi = 1 \text{ mile}$, am I right? – Antoine Aug 16 '15 at 00:56