A rope surrounds the Earth adjusted to the equator. It then elongates exactly $2\pi$ meters and separates itself evenly from the ground around the entire planet. What is the new height of the rope above the surface? Does it depend on the size of the Earth?
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The rope around the Earth
Hints
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- If the radius of the Earth is R, the initial length of the string is: C = 2pi R.
- When lengthening it by 2pi meters: C' = 2pi R + 2pi = 2pi(R+1).
- The uniform spacing is 1 meter.
Solution
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Answer: The uniform spacing is 1 meter.
Explanation:
If the radius of the Earth is $R$, the initial length of the chord is:
$$C = 2\pi R.$$
By lengthening it by $2\pi$ meters:
$$C' = 2\pi R + 2\pi = 2\pi(R+1).$$
Therefore, the new radius is $R+1$, so the ground clearance is:
$$\Delta r = 1 \text{ metro}.$$
Conclusion: it does not depend on the size of the Earth; It depends only on the total increase in length.
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