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.
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.
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:
Remember C = πd, so the diameter of the earth is d = C/π: 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. 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:
Adding 1 yard slack will make that much difference on any planet. You could be on Jupiter, or the Sun, and this would work.