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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