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1. Tidal forces exerted by the Moon on the Earth are causing the Earth’s

rotation rate to gradually decrease. As a result, the length of the day is increasing by about 2.3 milliseconds per century. Thus, Earth is loosing rotational energy and also rotational angular momentum. Much of the rotational energy lost by the Earth is converted to heat, but some of the energy, and all of the angular momentum, is transferred to the Moon by the tidal forces. As a result, the Moon gains both orbital energy and orbital angular momentum, causing it to spiral outward in its orbit.
(a)  Estimate by how many seconds a clock synchronized to the Earth’s rotation would fall behind an atomic clock (measuring true time) in one century.
(b)  Using the law of conservation of angular momentum, calculate the rate at which the distance to the Moon is increasing, in cm per year. You may assume that the Moon’s orbit is circular.
(c)  Suppose that the rate of increase of the Moon’s distance has been constant since the Solar System formed, about 4.7 billion years ago. How much closer to the Earth would the moon have been at that time? Do you think that the assumption of a constant rate is a good one? Why?


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