For a tapered beam of constant density the center of mass is calculated as the mean of the mass distribution along the central axis, the center of gravity is calculated by getting the equilibrium of the angular moments to the left and right of a fulcrum moving along the horizontal beam. The resulting positions of those points don´t match. The difference is about 5 percent, hence not trivial. Its not a problem of the computation, several different approaches give the same result (Integrals, sums, one or three spatial dimensions, high working precision). I had assumed those points to match and in the literature I found no obvious reason why they should not. Why this difference? A Mathematica program of a sample calculation can be given.