One extra meter of rope around the Earth

Question

Imagine a rope tied snugly around the Earth's equator, touching the ground the whole way around (pretend the Earth is a perfect smooth sphere). Now you splice in just one extra meter of rope and arrange the slightly longer loop so it hovers the same height above the ground all the way around.

How high off the ground does the rope now sit? Guess before you calculate.

Accepted Answer

The rope rises about 16 centimeters off the ground, enough to slide your hand under, easily. And astonishingly, the answer doesn't depend on the Earth's size at all. The one-line calculation. A circle of radius r has circumference 2\pi r. The original rope has length 2\pi R where R is the Earth's radius. After adding 1 meter, the new circumference is 2\pi R + 1, so its radius R' satisfies: 2\pi R' = 2\pi R + 1 \quad\Rightarrow\quad R' = R + \frac{1}{2\pi}. The gap is the increase in radius: R' - R = \frac{1}{2\pi} \approx 0.159 	ext{ m} \approx 16 	ext{ cm}. Why the planet's size is irrelev…

One extra meter of rope around the Earth

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