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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’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 |
| 29 | \[ w = \sqrt{\frac{S^3}{t}} \,\text{.} \] |
| 30 | Rearranging, we find that |
| 31 | \[ S = \sqrt[3]{w^2 t} \,\text{.} \] |
| 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., |
| 35 | \[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \,\text{.} \] |
| 36 | Volume is conserved, so if the original and final wire lengths |
| 37 | are $L$ and $\ell$ respectively, then |
| 38 | \[ L S^2 = \ell w t \,\text{,} \] |
| 39 | and hence |
| 40 | \[ L = \frac{\ell w t}{S^2} \,\text{.} \] |
| 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: |
| 44 | \[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \,\text{.} \] |
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