Question

Ignoring external torques, the sum of these angular momenta must be constant. We know Earth is losing its spin angular momentum because of tidal friction. Show that the spin angular momentum lost by the Earth is gained by the moon as orbital angular momentum.

Answer 1

Consider the Earth-Moon system. Now the total angular momentum must be conserved. So the total angular momentum is the sum of the angular momentum of the earth and angular momentum of the moon. Thus,

before going into mathematics, lets first understand the phenomenon. Earth's large area is covered by water. These causes tidal forces i.e. when the earth rotates it drags the water along with it. Thus causing in bulging of Earth. it slows down the rotation of the earth. Slowing down the rotation of earth means decreasing angular momentum. Now, since the angular momentum is a conserved quantity, the amount lost by the earth is gained by the moon.

Now, this increases the velocity of the moon, which in turn increases the radius. Also, the angular momentum of the moon is due to two factors, L about the moon's axis and L about CoM. Let's see the math here,

since v²/r = a = F/m

therefore

where F = GmM/r²

thus

--------(1)

substituting the values, we get, L_M = 2.87 x 10³⁴ Kg m²/s

Angular momentum due to inertia is,

= 8.87 x   10³⁴ x 2.66 x 10^-6

= 2.35 x 10²⁸ Kg m²/s

Thus, this is the change in the angular momentum which is gained by the moon and lost by the earth.

Now if we differentiate the previous angular momentum equation (1), we get

substituting the values we get the rate of change of angular momentum with distance from earth to the moon.

Thus, the noon is receding away from earth by 4 cm/year