Clackson scroll formula

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The Clackson scroll formula [1] [2],

l = \pi \cdot s \cdot n^2

is used in blacksmithing for estimating the length l of stock required to produce a scroll of n turns with a spacing s between the turns.

[edit] Derivation

The equation of an archimedean spiral is r(φ) = a + bφ. The spacing is s = r(φ + 2π) − r(φ) = 2πb.

In our case we want r(0) = 0. Therefore a = 0 and the equation becomes r(\phi) = \frac{s}{2\pi}\phi.

The arc length of the spiral from angles 0 to θ is given approximately by l(\theta) \approx \int_{0}^{\theta}r(\phi)\, d\phi = \int_{0}^{\theta}\frac{s}{2\pi}\phi\, d\phi = \frac{s}{4\pi}\theta^2.

For n turns, θ = 2πn and l = l(2\pi{}n) \approx \frac{s4\pi^2n^2}{4\pi} = \pi{}s{}n^2.

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[edit] References

  1. ^ S.G. Clackson, "The Trinity Church Screen", SCAT Report 1981
  2. ^ "MSC Craft Based Training - Forging and Hand Skills
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