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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
  22 wire-strip calculator</a> were derived by examining experimental data.
  23 We might not have considered all of the necessary variables.  Anyway,
  24 here&rsquo;s how it currently works.
  25
  26 <p>Let&rsquo;s suppose we start with square wire, with side&nbsp;$S$,
  27 and we roll it to thickness&nbsp;$t$.  Then we find that the
  28 wire&rsquo;s width is
  29 \[ w = \sqrt{\frac{S^3}{t}} \]
  30 Rearranging, we find that
  31 \[ S = \sqrt[3]{w^2 t} \]
  32 For round wire, we assume that the cross-section area is the important
  33 bit, so a round wire with diameter&nbsp;$D$ ought to work as well as
  34 square wire with side $S$ if $S^2 = \pi D^2/4$, i.e.,
  35 \[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \]
  36 Volume is conserved, so if the original and final wire lengths
  37 are&nbsp;$L$ and&nbsp;$l$ respectively, then
  38 \[ L S^2 = l w t \]
  39 and hence
  40 \[ L = \frac{l w t}{S^2} \]
  41 Finally, determining the required initial stock length&nbsp;$L_0$ given
  42 its side&nbsp;$S_0$ (for square stock) or diameter&nbsp;$D_0$ (for
  43 round) 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
  46 <p>[This page uses <a href="http://www.mathjax.org/">MathJax</a> for
  47 rendering equations.  It probably doesn't work if you don't enable
  48 JavaScript.]
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