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How can I find the point of balance of an half ellipsoid with the equation

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1,\:x\ge 0$$ As point of balance I mean the point on the surface it stays when laid down on a flat surface. And I think this means that the surface normal of the surface at the point when the semi-ellipsoid is laid down must direct to the center of mass. The solution will probably be a set of points on the surface that form a circle.

After some integration I have found the center of mass of the ellipsoid as $\left(\frac{3}{8}a,0,0\right)$ .

Now I think I need to find the point on the surface of the semi-ellipsoid which will have a surface normal that points towards the exact point $\left(\frac{3}{8}a,0,0\right)$ .

Or maybe find the closest point on the surface to the point $\left(\frac{3}{8}a,0,0\right)$ because that will also be perpendicular to surface?

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  • $\begingroup$ You need to write each formula between dollar signs for them to render as math. Write \$formula\$ for inline mode, and \$\$formula\$\$ for display mode. $\endgroup$
    – Christian E. Ramirez
    Commented Aug 31, 2023 at 1:27
  • $\begingroup$ Your two options will lead — mostly — to the same points. But I don’t believe they are necessarily valid. Try a semicircle. I think just symmetry leads you to the answer, and the center of mass is irrelevant. $\endgroup$
    – Ted Shifrin
    Commented Aug 31, 2023 at 1:36
  • $\begingroup$ I think there might have been a confusion because my question wasn't clear enough. Sorry, my mistake. I edited it now its clearer what I need to do. $\endgroup$
    – Mehmet Umut Ata
    Commented Aug 31, 2023 at 2:02
  • $\begingroup$ There is still a stability question. What about the point $(a,0,0)$? $\endgroup$
    – Ted Shifrin
    Commented Aug 31, 2023 at 2:33

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