1 <!DOCTYPE HTML PUBLIC
"-//W3C//DTD HTML 4.01//EN"
2 "http://www.w3c.org/TR/html4/strict.dtd">
5 <title>Rolling wire-strip calculator: equations
</title>
6 <meta name=viewport
content=
"width=device-width initial-scale=1.0">
7 <link rel=stylesheet
type=
"text/css" href=
"rolling.css">
8 <script type=
"text/x-mathjax-config">
10 tex2jax: { inlineMath: [['$', '$'], ['
\\(', '
\\)']] }
13 <script type=
"text/javascript"
14 src=
"https://www.distorted.org.uk/javascript/mathjax/MathJax.js?config=TeX-AMS_HTML">
19 <h1>Rolling wire-strip calculator: equations
</h1>
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.
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
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
$l$ respectively, then
38 \[ L S^
2 = l w t \,
\text{,} \]
40 \[ L =
\frac{l w t}{S^
2} \,
\text{.} \]
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 \[ L_0 =
\frac{S^
2 L}{S_0^
2} =
\frac{
4 S^
2 L}{\pi D_0^
2} \,
\text{.} \]
46 <p>[This page uses
<a href=
"https://www.mathjax.org/">MathJax
</a> for
47 rendering equations. It probably doesn
’t work if you don
’t