Center of mass and center of gravity don´t match, why?

[Catire]

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.

 

[Orodruin]

There is no difference, the positions are equivalent. If you get different results it is a problem of computation.

 

[theodoros.mihos]

The conditions are: $\sum_i m_ir_i = 0$ and $\sum_i gm_ir_i = 0$.
Think about what can make this differnance on your calculations.

 

[Tom_K]

As other posters have said, the center of mass and the center of gravity are the same point.

However, I suspect that when you said the center of mass, you really meant to say the half-way mass point. That is the point where, if you cut the beam, both sections would weigh exactly the same. But that is not the point where the beam would balance on a fulcrum because the mass of the two sections in a tapered beam is not distributed the same way.