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Answer :
If the distance between the Earth and the Moon were doubled, according to Kepler's Third Law of Planetary Motion, the lunar month would increase to about 77.2 days, as the relationship between orbital period and distance is T^2 proportional to R^3. Thus, the correct answer is (3) 77.2 days.
The student is asking about the effects of changing the distance between the Earth and the Moon on the lunar month's duration.
The lunar month is currently about 27.3 days.
According to Kepler's Third Law of Planetary Motion, the square of the orbital period of a planet (or moon) is directly proportional to the cube of the semi-major axis of its orbit.
Therefore, if the distance between the Earth and Moon were to be doubled, we would expect the lunar month to become significantly longer.
Let's call T the orbital period (lunar month) and R the average distance between Earth and Moon.
According to Kepler's third law, T2 is proportional to R3, or T2 / R3 = constant.
If the distance R is doubled, then T will increase by the cube root of 23, which is 2to the power of 3/2 or 2√2.
Thus, we have: New T = T * 2√2
Using the initial sidereal lunar month of 27.3 days and multiplying by 2√2, we find:
New T = 27.3 days * 2√2 ≈ 27.3 days * 2.828 ≈ 77.2 days
This implies that if the distance between the Earth and Moon is doubled, the number of days in a lunar month would increase to approximately 77.2 days.
Therefore, the correct answer from the provided options is (3) 77.2.
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