Hello Swlabr

I'm not entirely sure what you're hoping to do here. I think you're talking about a solid formed by attaching a cone to the top of a cylinder (with the same radius and density as the cone), and you want the ratio of the heights of the cylinder and the cone, so that the resulting solid's centre of gravity lies in the plane where the cylinder and the cone meet.

If this is so, let the heights of the cylinder and the cone be $\displaystyle h_y$ and $\displaystyle h_o$ respectively. Then, with common radius $\displaystyle r$, their respective volumes are$\displaystyle \pi r^2h_y$ and $\displaystyle \tfrac13\pi r^2 h_o$

and the distances of their respective CG's from the plane interface are$\displaystyle \tfrac12h_y$ and $\displaystyle \tfrac14h_o$

If the solid's CG lies in this plane, the moments of the two separate parts about a line in this plane are equal. So:$\displaystyle \pi r^2h_y\cdot\tfrac12h_y = \tfrac13\pi r^2 h_o\cdot\tfrac14h_o$

$\displaystyle \Rightarrow \tfrac12h_y^2=\tfrac{1}{12}h_o^2$

$\displaystyle \Rightarrow h_y:h_o = 1:\sqrt 6$

Is that what you were looking for?

Grandad