The center of mass is the point where the force of gravity would pull you if the force increased proportionally with distance. In reality the force decreases as

The center of mass is the point where the force of gravity would pull you if the force increased proportionally with distance. In reality the force decreases as 1/r^2, which changes things. Think about the more extreme case of a barbell -- a single object. If you're 2/3 of the way on the bar, you'll be attracted towards the closest endpoint, away from the center of mass (midpoint).