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Let's consider that an object has a uniform mass

The center of surface is

$$\vec C_s=\frac{\iint_{\mathbb{S}} \vec rdS}{\iint_{\mathbb{S}} dS}$$

And the center of mass is

$$\vec C_v=\frac{\iiint_{\mathbb{V}} \vec rdV}{\iiint_{\mathbb{V}} dV}$$

For a sphere or a cube, both result the same point.

I am wondering,

1- For which objects are they necessarily the same?

2- For a CAD design which is described by surface triangles, calculating $\vec C_v$ is hard. Is $\vec C_s$ a good approximation?

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Center of mass and center of surface necessarly coincide for symmetric object like spheres, cubes, cylinder indeed in these cases the center of mass coincides with a center of symmetry. Otherwise it is not necessarly true and the approximation could not be so good.

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  • $\begingroup$ what do you mean by symmetry? is a human body symmetric as well? $\endgroup$ – ar2015 Feb 6 '18 at 6:51
  • $\begingroup$ the human body has (ideaaly) only a plane of symmetry, thus necessarly the two center S and V lie on the same plane but not necessarly coincides, whereas if you have multiple plane/axes of symmetri the center of mass must coincides with this center as for cubes, sphere and also a cylinder. $\endgroup$ – user Feb 6 '18 at 6:55
  • $\begingroup$ A cone also has multiple planes of symmetry, but its center of mass and center of surface do not coincide. A sufficient condition is central symmetry, which is satisfied by cubes, spheres, and cylinders, but not by cones. $\endgroup$ – user856 Feb 6 '18 at 7:14
  • $\begingroup$ @Rahul yes you are absolutely right, we need a center of symmetry, I've used your link in the answer, thanks $\endgroup$ – user Feb 6 '18 at 7:17

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