
What wrapping rope around the Earth reveals about the limits of human intuition - lladnar
https://aeon.co/videos/what-wrapping-a-rope-around-the-earth-reveals-about-the-limits-of-human-intuition
======
deathanatos
So, my intuition was off, like the article describes. The article doesn't
mention it, but the rope would be nearly 6 inches off the ground. After I did
the math, this makes trivial sense, and I feel rather moronic for both knowing
the equations and not intuiting it. π is the ratio between circumference and
diameter. The planet is a red herring.

If you're still not seeing it, I'll try to explain:

Let's say the circumference of the earth in yards, c, is 43,825,760:

    
    
      >>> c = 43_825_760
    

Remember C = πd, so the diameter of the earth is d = C/π:

    
    
      >>> import math
      >>> d = c / math.pi
    

Add 1 to c (our 1 yard); this is the circumference of the rope plus slack,
then divide by π to get the diameter of the rope. Subtract earth's diameter
above.

    
    
      >>> ((c+1) / math.pi) - d
      0.3183098863810301  # yards. ~11.4 inches
    

Remember that this is the change in _diameter_. From the perspective of a
person looking at this rope, you'll only see half of that (the other half is
on the other side of the earth), so it's really only ~5.7 inches (The "nearly
6 inches" above).

Here's the thing that I missed thinking this through. That 0.318 number? If we
take the "1 yard" we added in slack (circumference) and we make a ratio
between it and the change in diameter (0.318) — the ratio between
circumference and diameter is called π. The math:

    
    
      >>> 1/0.3183098863810301
      3.1415926516431183
    

Adding 1 yard slack will make that much difference on _any_ planet. You could
be on Jupiter, or the Sun, and this would work.

------
chansiky
A different way to think about this is if you pulled an inelastic chain tight
to the earth, and closed the loop you would have the starting condition. Then,
standing where you are you cut the chain, grabbed an extra yard of the chain
and spliced it to the touching ends of the starting chain lying on the ground,
laying that extra yard folded perpendicular to the starting chain wrapped
around the earth. That extra chain added is now sticking out half a yard from
the starting chain. When you think about it that way, and I ask you “would
that extra yard make a difference?”, where does your intuition lead you? With
that visual, I would say, that is a ton of slack, I could loop it around my
body. I think this is really more about how you frame the question than some
flaw in intuition, although I’m not trying to say humans have flaws with our
intuition.

------
esotericn
Pretty clever.

My instinctual reaction was to say, well, no, it's trivial, of course not,
it's a tiny length compared to the Earth's size.

But mathematically, well, c = 2πr, right? It's another 15cm or so on the
radius, if it were perfectly rigid.

In practice though it'd flop down (perpendicular to the plane of the circle)
and if that were fairly well distributed I'd assume the 15cm wouldn't be
noticeable, and friction+gravity would make it difficult to pull up?

------
craftinator
My brain went the other direction, but my intuition still went to a correct
answer. A rope wrapped around the Equator, with the exact length required to
be around it, would rest on the ground and be tight. Adding an extra yard
would make it loose. Assuming it was frictionless and completely inelastic,
anyone could pull it and amass a yard length of length of the rope at their
location, and that yard could be distributed to any point along the Equator,
which would be noticeable.

~~~
ramblerman
No your intuition is still wrong. It would be almost 6 inches off the floor
_all_ the way round the earth.

~~~
craftinator
Why would it not lay on the ground with some slack? There is not enough
structure in a normal rope to hold one end of it horizontally, 6 inches off
the ground, and have the rest extend out horizontally, 6 inches off the ground
as well. I understand that gravity would be pulling equally on all portions of
it, but would not some variation in the material or surrounding air cause it
to crumple at one point (or shear, given the weight), or just be compressed
until it was laying on the ground?

What is it about being a loop that protects it from gravitational induced
motion?

