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5 | <title>Rolling wire-strip calculator: equations</title> | |

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18 | <h1>Rolling wire-strip calculator: equations</h1> | |

19 | ||

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. | |

24 | ||

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 | |

27 | wire’s width is | |

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 | |

37 | \[ L S^2 = l w t \] | |

38 | and hence | |

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 | \[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \] | |

44 | ||

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