Puzzle #2

[Empirical]

This is another puzzle where the intuitive answer—a few millimetres, say—is wrong. We are confused by the vast size of the earth, and the small extension of the rope.

Let, C = circumference of earth (or any sphere), where R is the radius.

from Euclid, C = 2πR

add 2 metres (2m)

(C +2m) = 2π(R + x) where x is the distance between the sphere and the rope

substitute C = 2πR, and expand

2πR + 2m = 2πR + 2πx

so, 2m = 2πx

so, x = 2m/2π = m/π = m/3.14 (approx), = .318m , or 31.8 cms

Interestingly, the size of the Earth or sphere doesn't matter, a rope 2 metres longer than the circumference will always be 31.8 cms from the surface.