chiark / gitweb /
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4<head>
5 <title>Rolling wire-strip calculator: equations</title>
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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$S = \sqrt{w^2 t}$
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
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