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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’s how it currently works.
25 <p>Let’s suppose we start with square wire, with side $S$,
26 and we roll it to thickness $t$. Then we find that the
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 $D$ 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 $L$ and $l$ respectively, then
39 \[ L = \frac{l w t}{S^2} \]
40 Finally, determining the required initial stock length $L_0$ given
41 its side $S_0$ (for square stock) or diameter $D_0$ (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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