Where Exactly Is the Weight of a Body Applied?

A common belief among undergraduates is that the gravitational force exerted by a homogeneous sphere of mass M on an extended body of mass m with arbitrary shape is always given by Newton’s law of gravity F = GMm/r_cc², where r_cc is the distance from the center of the sphere to the center of mass (c.m.) of the body. In this note, I introduce the simplest counterexample of a vertical dumbbell to show that, in general, this procedure does not return the correct gravitational force. I also show that not even the center of gravity (c.g.) of the body, determined according to the weighted-average formulae found in textbooks, leads to the correct force. Finally, I present an equation for the c.g. position whose solution always corresponds to the correct force.