Consider a system of N particles subject to a uniform gravitational field g acting verti- cally down. Starting with the work done by gravity on each mass, prove that the total gravitational potential energy of this system of N particles is the same as if all the mass were concentrated at the center of mass of the system; i.e., Ugrav = (Ugrav) a = MgY, (1) = where Mama is the total mass and R = (X, Y, Z) is the position of the center of mass, with the y-coordinate measured vertically up.