Imagine you are floating in space, far from any objects large enough to exert significant gravity on you. A car floats up to you, and you push it. The car will spin around its center of mass.
Now, you magnet some weights to the bottom of the car, changing the center of mass, and you push the car again: now the car spins around the new center of mass.
Changing the center of mass doesn’t rely on gravity to have an effect. The behavior of any force pushing the car will be dictated by the center of mass.
Center of gravity and center of mass are the same if you assume that gravity acts the same on all parts of the car. (This is approximately right, but with very large objects like planets, the center of mass and center of gravity are different because gravity will effect different parts of the planet differently depending on their distance from whatever is generating the gravity.)
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