From 6343cd5e2317ed6c774e2d6704a6ceb09d7dd6aa Mon Sep 17 00:00:00 2001 From: Mark Wooding Date: Sat, 9 Jan 2021 02:16:08 +0000 Subject: [PATCH] rolling-eqn.html: Add trailing punctuation to equations. As noted elsewhere, I've decided that Ian Stewart was wrong about this. --- rolling-eqn.html | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/rolling-eqn.html b/rolling-eqn.html index 9588d54..4b5a973 100644 --- a/rolling-eqn.html +++ b/rolling-eqn.html @@ -26,22 +26,22 @@ here’s how it currently works.

Let’s suppose we start with square wire, with side $S$, and we roll it to thickness $t$. Then we find that the wire’s width is -\[ w = \sqrt{\frac{S^3}{t}} \] +\[ w = \sqrt{\frac{S^3}{t}} \,\text{.} \] Rearranging, we find that -\[ S = \sqrt[3]{w^2 t} \] +\[ S = \sqrt[3]{w^2 t} \,\text{.} \] For round wire, we assume that the cross-section area is the important bit, so a round wire with diameter $D$ ought to work as well as square wire with side $S$ if $S^2 = \pi D^2/4$, i.e., -\[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \] +\[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \,\text{.} \] Volume is conserved, so if the original and final wire lengths are $L$ and $l$ respectively, then -\[ L S^2 = l w t \] +\[ L S^2 = l w t \,\text{,} \] and hence -\[ L = \frac{l w t}{S^2} \] +\[ L = \frac{l w t}{S^2} \,\text{.} \] Finally, determining the required initial stock length $L_0$ given its side $S_0$ (for square stock) or diameter $D_0$ (for round) again makes use of conservation of volume: -\[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \] +\[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \,\text{.} \]

[This page uses MathJax for rendering equations. It probably doesn’t work if you don’t -- 2.11.0