Question

Problem Four You should attempt this problem after Lecture 6 Cycle 3 The moons gravitational influence on the earth cause the tides, which dissipates the kinetic energy of the earth-moon system. As a consequence the earth's rotation is slowing and the moon moves 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 × 1037 kg m2 the mass of the earth is 5.972 x 102 kg, the mass of the moon is 7.347 x 102 kg, and the angular momentum of the earth and moon combined is 3.468 x 10w kg m2s-1. The distance from the earth to the moon is currently 3.84 x 10 m, although this will increase with time. Assume a circular orbit throughout QUESTION 1 At some point in the earths future it is possible that the moons orbit will correspond to the time it earth to rotate and the moon will be in geostationary orbit. By conserving angular momentum find how far into the future (n years) that this will ocur (assuming the sun doesn't go supernovae betore then). See pdt for more details

Answer 1

Use conservation of angular momentum

$L_i = (I_{earth}+ m_{moon}r^2)\omega$

here, final angular velocity is: $\omega = \frac{2\pi}{T}$

for which T = 1 month = 30 x 24 x 3600 seconds

so,

( ) 3.468 × 1034-(8.008 × 1037 + 7.347 × 1022 R2 30 × 24 × 3600

this gives, R = 4.40041 x 10⁸ m

this is the distance when a day on Earth is the same as a lunar month.

R_o = current distance between Earth and Moon = 3.84 x 10⁸ m

so, change in distance = L = R - R_o = 5.6041 x 10⁷ m

Moon is moving away from Earth with a speed of 0.0384 m per year

so, time it will take for Moon to cover a distance L will be: T = Distance / speed = (5.6041/0.0384) x 10⁷ = 1.459 x 10⁹ years

this is the time it will take for 1 day on Earth to become a lunar month.