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5 | <title>Rolling wire-strip calculator: equations</title> | |
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18 | ||
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’s how it currently works. | |
25 | ||
26 | <p>Let’s suppose we start with square wire, with side $S$, | |
27 | and we roll it to thickness $t$. Then we find that the | |
28 | wire’s width is | |
6343cd5e | 29 | \[ w = \sqrt{\frac{S^3}{t}} \,\text{.} \] |
ef43c701 | 30 | Rearranging, we find that |
6343cd5e | 31 | \[ S = \sqrt[3]{w^2 t} \,\text{.} \] |
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32 | For round wire, we assume that the cross-section area is the important |
33 | bit, so a round wire with diameter $D$ ought to work as well as | |
34 | square wire with side $S$ if $S^2 = \pi D^2/4$, i.e., | |
6343cd5e | 35 | \[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \,\text{.} \] |
ef43c701 | 36 | Volume is conserved, so if the original and final wire lengths |
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37 | are $L$ and $\ell$ respectively, then |
38 | \[ L S^2 = \ell w t \,\text{,} \] | |
ef43c701 | 39 | and hence |
aca28a1d | 40 | \[ L = \frac{\ell w t}{S^2} \,\text{.} \] |
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41 | Finally, determining the required initial stock length $L_0$ given |
42 | its side $S_0$ (for square stock) or diameter $D_0$ (for | |
43 | round) again makes use of conservation of volume: | |
6343cd5e | 44 | \[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \,\text{.} \] |
ef43c701 | 45 | |
8a0fc4a2 | 46 | <p>[This page uses <a href="https://www.mathjax.org/">MathJax</a> for |
ea8c4d68 | 47 | rendering equations. It probably doesn’t work if you don’t |
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