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Q: Why does gravity pull things toward the center of mass? What’s so special about the center of mass?
Nothing and nowhere gets special treatment, and that includes the center of mass. The stuff at the center of the Earth does pull on you, but that’s nothing special; so does literally everything else. So, because the mass of the Earth is spherically symmetric, gravity around here points toward the center of mass. The center of mass of the Earth and Moon combined is a point just a little below the Earth’s surface. So the closer you are to the center, the less mass is below you, and the less gravity there is.
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