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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Antoine Couëtte
Date: 2023 Oct 17, 06:10 -0700
Dale,
Circumference = Radius × 2 × π , or in other words : C₁ = 2 π R₁
If we contemplate a “new” radius R₂, such as R₂ = R₁ + Δ , then the “new” circumference C₂ is easily computed as : C₂ = 2 π R₂ = 2 π (R₁ + Δ).
The difference between C₂ and C₁ is simply : C₂ - C₁ = 2 π Δ. And, lo and behold, this difference in circumferences is no longer a function of R, but only a function of Δ.
Hence your example which deals with differences in circumferences related to differences in radius will yield the very same output irrespectively of the sphere considered. Hence:
(1) - If you add 3 feet to the cord, it will evenly “fly” just under a foot from the surface, whether it be on Earth, on the Moon, on the Jupiter, on the Sun or Betelgeuse (assume it plasma proof then).
(2) - If you tighten around the Earth a rope initially flying evenly at x feet over its surface, its apex will be at π × x feet for a total length equal to (Earth Circumference + π x feet) .
Much faster for mathematicians:
C = 2 π R , from which we immediately derive : dC = 2 π dR, a formula in which R has simply vanished.
Kermit
