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5   <title>Rolling wire-strip calculator: equations</title>
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18 <h1>Rolling wire-strip calculator: equations</h1>
20 <p>The calculations performed by the <a href="rolling.html">rolling
21 wire-strip calculator</a> were derived by examining experimental data.
22 We might not have considered all of the necessary variables.  Anyway,
23 here&rsquo;s how it currently works.
25 <p>Let&rsquo;s suppose we start with square wire, with side&nbsp;$S$,
26 and we roll it to thickness&nbsp;$t$.  Then we find that the
27 wire&rsquo;s width is
28 $w = \sqrt{\frac{S^3}{t}}$
29 Rearranging, we find that
30 $S = \sqrt[3]{w^2 t}$
31 For round wire, we assume that the cross-section area is the important
32 bit, so a round wire with diameter&nbsp;$D$ ought to work as well as
33 square wire with side $S$ if $S^2 = \pi D^2/4$, i.e.,
34 $D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi}$
35 Volume is conserved, so if the original and final wire lengths
36 are&nbsp;$L$ and&nbsp;$l$ respectively, then
37 $L S^2 = l w t$
38 and hence
39 $L = \frac{l w t}{S^2}$
40 Finally, determining the required initial stock length&nbsp;$L_0$ given
41 its side&nbsp;$S_0$ (for square stock) or diameter&nbsp;$D_0$ (for
42 round) again makes use of conservation of volume:
43 $L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2}$
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