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It is often easier to look at these kinds of problems using conservation laws. The Moon raises tides (water and rock) on the Earth. If the Earth were rotating at the same rate as the Earth-Moon orbital rate, the tidal bulge would line up with the Earth-Moon line. However, the Earth's rotation advances the tidal bulge so the bulge on the Moon side of the Earth leads the Moon. This leading bulge exerts a slightly greater gravitational force on the Moon than does the trailing bulge on the opposite of the Earth. The Moon receives a small prograde boost, increasing its angular momentum. Conservation of momentum dictates that the Earth lose angular momentum.