Very Wrong Math
6 comments
·January 10, 2025BalinKing
kurthr
The only issue I see with this is that as a classic physics trope, we've approximated the earth as a sphere.
If, instead we approximate it as a fractal... then the distance is infinite, or at least highly dependent on the thickness of the rope!
The error in the original is assuming that the radius is proportional to the height above the earth (Earthradius=0?).
Dylan16807
> infinite, or at least highly dependent on the thickness of the rope
The latter. But that's only if it's not somewhat taut. Some tension brings it closer to a circle and makes the actual thickness pretty unimportant.
But I like the idea overall. It means that lifting up the string makes it smoother and it actually needs less length. How's that for being unintuitive?
kurthr
Exactly, if you're only 1cm off the surface you follow every nook and cranny. If you're 10km off the surface only Everest is a blip.
shortrounddev2
(2pi * (n + 1)) - (2pi * n)
-> 2pi * (n + 1 - n)
-> 2pi * 1
-> 2pi
If I remember my algebra correctly. Someone else check my work I'm a dropout
freeopinion
For convenience, we set τ=2pi. :-)
x = τ(r+1) - τr = τ(r+1-r) = τ(1) = τ
Related Wikipedia article: https://en.wikipedia.org/wiki/String_girdling_Earth#Implicat....
The takeaway is that the extra length of the arc is likely much smaller than one would intuitively expect. The problem is usually framed like so: If you wrapped a rope around the earth, how much more rope would you need to add so that it would be 1 meter above the ground at all points? The answer is only 2π meters!