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### Calculating the Length of String on a Reel

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Date: 12/07/2001 at 10:25:54
From: Larry Schworer
Subject: Calculate string on a reel

What is the formula for calculating the length of wire that can go on
a reel (spindel) where the reels may have different diameters, hub
diameters, and widths, and the wires may have different diameters?

Once a reel and wire combination are selected they do not change for
that reel combination.

I wish to write a program to make this calculation.

Thanks,
La
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Date: 12/07/2001 at 12:06:15
From: Doctor Peterson
Subject: Re: Calculate string on a reel

Hi, Larry.

The answer will be similar to that for a similar question we have
answered many times, about the amount of material (such as carpet) on
a roll:

How Much Carpet is Left?
http://mathforum.org/dr.math/problems/gillette.03.27.99.html

We find the length by equating the volume of material on the roll to
its volume laid out flat.

There is always a caveat in this calculation, however: we have to
assume that the material retains the same thickness as it bends around
the roll, and is not either compressed or wound loosely. The
calculation is very sensitive to the actual thickness of the material
on the roll.

The same is even more true in your case. The wire may be wound in
different ways that will incorporate different amounts of air space
between turns, so it is hard to be sure how much of the volume on the
reel consists of wire. The best thing to do would be to measure an
actual reel of known length in order to determine this ratio and
adjust the formula, rather than blindly trust theory.

Having said that, let's look at the problem. For simplicity, I'm going
to pretend that the wire has a square cross section, since then there
would be no need for space between turns. Your measurements will
determine how closely this corresponds to reality.

Suppose the reel has hub diameter Di, outside diameter Do, and width
W. Then the volume of wire on it is

V = pi(Do^2 - Di^2)W/4

Suppose you have a wire with length L and square cross-section with
side Dw (the diameter of the wire). Its volume is

V = LDw^2

Setting these equal, we get

LDw^2 = pi(Do^2 - Di^2)W/4

and we can solve to find

L = pi(Do^2 - Di^2)W/(4Dw^2)

Now suppose you measure an actual reel, and the actual length is X
times this calculated length (let's hope not too far from 1). Then you
can take X as the typical correction factor, and multiply any
calculated length by X to estimate the actual length when you don't
know it ahead of time. This correction factor may well be more or less
constant for all reels wound by the same method.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Practical Geometry

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