Measure the length of a wrapping

Question

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!

Answer 1

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.

Answer 2

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.