The moons gravitational influence on the earth causes the tides, which dissipate

Question

The moons gravitational influence on the earth causes the tides, which dissipate the kinetic energy of the earth-moon system. As a consequence the earth's rotation is slowing and the moon is moving away from the earth at a rate of 3.84 cm each year (same speed your fingernails grow at). At some point in the earths future it is possible that the moons orbit will correspond to the time it takes the earth to rotate - in other words a day will equal a lunar month - and the moon will be in geostationary orbit (the moon will only be visible from one side of the earth). By conserving angular momentum find how far into the future (in years) that this will occur (assuming the sun doesn't go supernovae before then). The moment of inertia of the earth is 8.008 x 1037 kg m2, the mass of the earth is 5.972 Times 10^24 kg, the mass of the moon is 7.347 Times 10^22 kg, and the angular momentum of the earth and moon combined is 3.468 Times 10^34 kg m^2 s^-1. The distance from the earth to the moon is currently 3.84 Times 10^8 m, although this will increase with time. Assume a circular orbit throughout.

Explanation / Answer

Applying conservation of momentum

= >        3.468 * 1034 = (8.008 * 1037 + 7.347 * 1022 * R2) *(2 * 3.14)/(29.53 * 24 * 60 * 60)

=>        R = 4.3667 * 108   m

=>     Increase in distance from Earth to Moon =   4.3667 * 108 - 3.84 * 108 m

                                                                           = 0.5267 * 108 m

=>    Time in future it will held =   (0.5267 * 108)/0.0384

                                               =   1.3716 * 109   years

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