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dep-ui.js: Add some debugging machinery.
[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>
6 <script type="text/x-mathjax-config">
7 MathJax.Hub.Config({
8 tex2jax: { inlineMath: [['$', '$'], ['\\(', '\\)']] }
9 });
10 </script>
11 <script type="text/javascript"
12 src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML">
13 </script>
14 <link rel=stylesheet type="text/css" href="rolling.css">
15<head>
16<body>
17
18<h1>Rolling wire-strip calculator: equations</h1>
19
20<p>The calculations performed by the <a href="rolling.html">rolling
21wire-strip calculator</a> were derived by examining experimental data.
22We might not have considered all of the necessary variables. Anyway,
23here&rsquo;s how it currently works.
24
25<p>Let&rsquo;s suppose we start with square wire, with side&nbsp;$S$,
26and we roll it to thickness&nbsp;$t$. Then we find that the
27wire&rsquo;s width is
28\[ w = \sqrt{\frac{S^3}{t}} \]
29Rearranging, we find that
30\[ S = \sqrt[3]{w^2 t} \]
31For round wire, we assume that the cross-section area is the important
32bit, so a round wire with diameter&nbsp;$D$ ought to work as well as
33square wire with side $S$ if $S^2 = \pi D^2/4$, i.e.,
34\[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \]
35Volume is conserved, so if the original and final wire lengths
36are&nbsp;$L$ and&nbsp;$l$ respectively, then
37\[ L S^2 = l w t \]
38and hence
39\[ L = \frac{l w t}{S^2} \]
40Finally, determining the required initial stock length&nbsp;$L_0$ given
41its side&nbsp;$S_0$ (for square stock) or diameter&nbsp;$D_0$ (for
42round) again makes use of conservation of volume:
43\[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \]
44
45<p>[This page uses <a href="http://www.mathjax.org/">MathJax</a> for
46rendering equations. It probably doesn't work if you don't enable
47JavaScript.]
48
49</body>
50</html>