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hw5ans - Phys 341 Homework#5 Solutions 1(a If the Earth and...

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Phys 341: Homework #5 Solutions 1. (a) If the Earth and Moon are tidally locked and the Earth’s rotation period is 47 days, the the Moon’s orbital period is also P = 47 days = 4 . 1 × 10 6 s. Then from Kepler’s Third Law, the semimajor axis of the Moon’s orbit is a = GM P 2 4 π 2 1 / 3 = (6 . 67 × 10 - 8 g - 1 cm 3 s - 2 ) × (5 . 974 × 10 27 g) × (4 . 1 × 10 6 s) 2 4 π 2 1 / 3 = 5 . 5 × 10 10 cm = 550 , 000 km For comparison, the Moon’s current semimajor axis is about 384,000 km. (b) The Earth’s rotation period is increasing by about dP/dt = 0 . 0016 seconds per century. If the rate of change remains constant, the amount of time it will take for the period to increase from the current value of P 1 = 1 day to the final value of P 2 = 47 days is T P 2 - P 1 dP/dt (47 - 1) × 86 , 400 s 0 . 0016 s/century 2 . 5 × 10 9 centuries 2 . 5 × 10 11 yr A different approach is to say that the Moon’s semimajor axis is increasing at a rate of da/dt 3 cm/yr. If that rate of change remains constant, then the amount of time it will take for the semimajor axis to increase from the current value of a
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