Wierzycaz

2020-10-31

Imagine a rope tied around the Earth at the equator. Show that you need to add only feet of length to the rope in order to lift it one foot above the ground around the entire equator. (You do NOT need to know the radius of the Earth to show this.)

jlo2niT

Expert

Because the Earth is (roughly) spherical, if you tied a rope around the circumference of the Earth, it would be 2πr2πr feet long where rr is the radius of the Earth. Now imagine that you lift the rope up everywhere by 1 ft. Then it's still going around a circle, but the circle has a radius 1 foot longer. I.e. the length of the rope would need to be 2π(r+1) to make it all the way around.
So the amount of extra rope you'd need is just
$2\pi \left(r+1\right)-2\pi r=2\pi$
That is, about 6.28 feet of extra rope is all you'd need.

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