Question

The tidal forces between the Earth and the Moon slowed down the Moon's rotation about its own axis until the rotation period became equal to the Moon's orbital period around the Earth as we observe today. The same effect is also slowing down the Earth's rotation about its own axis and increasing the separation \(D\) between the Moon and the Earth at a rate of \(\Delta D / \Delta t=3.8 \mathrm{~cm}\) per year. In this problem, you can ignore the effects of the Sun and treat the Earth-Moon system as a closed system so that the total angular momentum is conserved. However, the total mechanical energy is not conserved because there are energy dissipations inside the two objects. For simplicity, you can assume that the rotation axes of the Earth and the Moon are perpendicular to the Moon's orbital plane. You can also assume that the center of mass of the system is at the center of the Earth. Let \(M, I_{e}\), and \(\Omega\) be the Earth's mass, moment of inertia, and rotational angular velocity, respectively. The corresponding quantities for the Moon are \(m, I_{m}\), and \(\omega\). You can model the Earth and the Moon as homogeneous spheres. [Note: You need to look up some astronomical data yourself such as the orbital period of the Moon etc.]

(a) Assume that the orbit of the Moon is circular. Write down the total angular momentum of the Earth-Moon system in terms of the given variables.

(b) Estimate the change of the rotation period of the Earth per year.

(c) After a sufficiently long time, assuming that the Earth-moon system is not disturbed, the rotation of the Earth would be synchronized with the orbital motion of the Moon (i.e., the rotational and orbital periods of the two objects become the same) so that the Earth would always show the same face to the Moon. Estimate the rotation period of the Earth when this synchronization occurs.