Rope around the Earth

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2022-09-23 00:00:31

In this puzzle, we treat the Earth as though it were a prefect sphere, even though it actually bulges toward the equator. Suppose a rope was tied taut around the Earth's equator. It would have the same circumference as the Earth (24,901.55 miles). The question is: by how much would the rope have to be lengthened so that, if made to hover, it would be one foot off the ground at all points around the Earth?

Despite the enormous size of the Earth, and the 1 foot gap around the entire circumference, the rope would have to be lengthened by a mere 2π feet, or roughly 6.28 feet.

Just as bizarre is what happens when one point on the lengthened rope is lifted up so that the rope is taut again, as in Image 1. The maximum clearance under the rope proves to be quite large. For the specific case of a rope looped around the Earth, a 2π foot extension would provide 614.771 feet of clearance if the rope were lifted. This is enough room to fit two Statues of Liberty under it, base and all. Unlike the previous question, however, this result is dependent on the size of the ball.

The puzzle states that we have to lengthen the rope and made the rope hover 1 foot of the surface of the earth. Lengthening the rope so that it is 1 foot off the ground at all points simply means changing the radius of the circle it forms from: Rrope 1= Rearth to Rrope 2= Rearth+1 ft.