I understand why the side closest to the moon would have a tidal “bulge,” but why on the opposite side of the planet as well?
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The moon is not orbiting the earth. Both Earth and the moon is orbiting the barycenter. That is the common center of mass for both earth and the sun it is 75% of earth’s radius from the center.
In most situations, it is enough to consider the smaller object is just orbiting the larger one that does not move. But that is strictly never true, it is just negligible when the smaller object is a lot smaller than the larger one like for satellites we launch. The mass of the moon is 1.2% and the result is that is it not negligible but very important in some situations like for tides.
It is just like a hammer thrower that balances out the hammer with its own body and rotates around the common center of mass. Hold something heavy in your hand and spin around and you notice to need to lean back. You and the object rotate around your common center of mass.
The rotation causes a centrifugal force away from the moon on all of the earth, it will be the same size and direction everywhere on earth
The gravity force between the earth and the moon is what keeps both together. The force depends on the distance and decreases the farther away you get. It will at the earth’s center of mass be identical to the centrifugal force.
The force of gravity from the moon will be stronger than the centrifugal force on the closest point on earth and lower than the centrifugal force on the point farthest away.
The result is a diagram like https://rwu.pressbooks.pub/app/uploads/sites/7/2019/05/figure11.1.4.jpg where Fg is the gravitational force and Fr is the centrifugal force. The moment of water is a result of the net force, the sum of them, that is the red arrows on earth and the will point away from the center of the earth for both the closed and farthest points. They do point towards the center of the earth on the side
So the key to the understanding is that you can just look at the gravity you need the orbit to. The obit part needs to ensure that both earth and the moon orbit barycenter
https://rwu.pressbooks.pub/webboceanography/chapter/11-1-tidal-forces/
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