A rope surrounds the Earth adjusted to the equator. It is then lengthened exactly $2\pi$ meters and placed again forming a concentric circle, uniformly separated from the ground. How high does the rope rise above the surface?
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The rope alrededor of the Tierra
It seems that the size of the Earth should matter, but in this classic the surprise is that the radius disappears from the account.
Hints
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- You don't need to know the radius of the Earth.
- The longitud inicial can escribirse as $2\pi R$.
- The new length is $2\pi(R+h)$.
Solution
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Answer: it rises 1 meter. Let $R$ be the radius of the Earth. The initial chord measures: $$
2\pi R.
$$ By uniformly raising a height $h$, the new radius is: $$
R+h.
$$ The new length is: $$
2\pi(R+h).
$$ Since the rope has lengthened $2\pi$ meters: $$
2\pi(R+h)-2\pi R=2\pi.
$$ By simplifying: $$
2\pi h=2\pi.
$$ Therefore: $$
h=1.
$$ So the rope is 1 meter from the ground throughout the entire loop.
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