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[dep-ui] / rolling-eqn.html
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1<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"
2 "http://www.w3c.org/TR/html4/strict.dtd">
3<html>
4<head>
5 <title>Rolling wire-strip calculator: equations</title>
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6 <meta name=viewport content="width=device-width initial-scale=1.0">
7 <link rel=stylesheet type="text/css" href="rolling.css">
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8 <script type="text/x-mathjax-config">
9 MathJax.Hub.Config({
10 tex2jax: { inlineMath: [['$', '$'], ['\\(', '\\)']] }
11 });
12 </script>
13 <script type="text/javascript"
f65560a6 14 src="https://www.distorted.org.uk/javascript/mathjax/MathJax.js?config=TeX-AMS_HTML">
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16<head>
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18
19<h1>Rolling wire-strip calculator: equations</h1>
20
21<p>The calculations performed by the <a href="rolling.html">rolling
22wire-strip calculator</a> were derived by examining experimental data.
23We might not have considered all of the necessary variables. Anyway,
24here&rsquo;s how it currently works.
25
26<p>Let&rsquo;s suppose we start with square wire, with side&nbsp;$S$,
27and we roll it to thickness&nbsp;$t$. Then we find that the
28wire&rsquo;s width is
29\[ w = \sqrt{\frac{S^3}{t}} \]
30Rearranging, we find that
31\[ S = \sqrt[3]{w^2 t} \]
32For round wire, we assume that the cross-section area is the important
33bit, so a round wire with diameter&nbsp;$D$ ought to work as well as
34square wire with side $S$ if $S^2 = \pi D^2/4$, i.e.,
35\[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \]
36Volume is conserved, so if the original and final wire lengths
37are&nbsp;$L$ and&nbsp;$l$ respectively, then
38\[ L S^2 = l w t \]
39and hence
40\[ L = \frac{l w t}{S^2} \]
41Finally, determining the required initial stock length&nbsp;$L_0$ given
42its side&nbsp;$S_0$ (for square stock) or diameter&nbsp;$D_0$ (for
43round) again makes use of conservation of volume:
44\[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \]
45
8a0fc4a2 46<p>[This page uses <a href="https://www.mathjax.org/">MathJax</a> for
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47rendering equations. It probably doesn't work if you don't enable
48JavaScript.]
49
50</body>
51</html>