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