You can’t fall through the Earth, but if you could use an X-Men power so that you’d fall through the ground, the center of gravity of the Earth is the point you’d fall to. It’s the “average” of all the gravitational force of an object.
When you do physics problems the center of gravity is useful because in most circumstances where you want to figure out gravitational force, you can ignore the object and just use the points that are the centers of gravity for each object. Instead of an irregular person shaped object to worry about, you focus on the single point that is their center of gravity.
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