Please show all steps. thank you 5, Ryden and Peterson # 4.8 (2 points) 4.8. The

Question

Please show all steps. thank you

5, Ryden and Peterson # 4.8 (2 points) 4.8. The earth will be in synchronous rotation with the Moon once its rotation period has increased to 47 days. (a) How far away will the Moon be from the Earth when this happens? (b) How long wil it be until the Earth attains this synchronous ro- tation, assuming that dProt/dt is approximately constant? This problem deserves a little explanation. Eventually, the tidal forces from the Moon on the Earth will lock the Earth in synchronous rotation with the moon. When this happens, the periods for 1) Moon going around the Earth, 2) Moon rotating on its axis once, and 3) Earth rotating on its axis once will all be the same (47 days). In current terminology: an Earth "day" will become the same as a "month" and both will last 47 days. This also means that the Moon will only be observed from one hemisphere on Earth since Earth will rotate at the same rate that the Moon will orbit. The other side of the Earth will NEVER see the moon!

Explanation / Answer

time period , T = 47 days

= 47 * 24 * 3600 s

mass of earth , Me = 5.98 * 10^24 kg

let the distance from earth be R

time period ,

T = sqrt( (R^3 * 4 * pi^2) / G * Me)

47*24*3600 = sqrt((R^3 * 4 * pi^2) / (6.673 * 10^-11 * 5.98 * 10^24))

R = 5.49 * 10^8 m

distance from earth is 5.49 * 10^8 m

b)

the initial period of rotation , Ti = 27.32 days

so, the change in rotation period ,

T = 47 - 27.32 = 19.68 days

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