When people talk about the moon orbiting the Earth, that’s not a precise statement. They actually both orbit a common centre of mass. Because the Earth is much more massive than the moon, that point is within the body of the planet but not at its centre. The fact that Earth orbits the moon the same way that the moon orbits the Earth, just less dramatically, is important here.
If you study a little bit about the math behind centripetal acceleration, you’ll find that, if two objects are going around in a circle at exactly the same rate, the one that’s farther from the centre gets a greater outward force than the one closer in. What does this have to do with what I’m talking about? Well, Earth’s orbit technically only describes the path the centre of the planet takes. The part of the Earth that’s farthest from the moon is therefore farther from the common centre of mass than the middle of the Earth. It receives a slight force away from the centre, causing an imbalance that compels fluids to gather there until that imbalance is corrected. The most convenient fluid in this case is water which then appears to us to rise and fall, depending on where the moon is in the sky.
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