newtonian mechanics - Will centre of gravity coincide with centre of mass if density of object is non-uniform?

I read that for bodies of very large dimensions, but having non-uniform density, the centre of gravity does not coincide with centre of mass. I can understand that with large dimensions the strength of gravity can vary and hence affect the centre of gravity. But how does having a non- uniform density makes the centre of gravity not to coincide with the centre of mass?

Answer

The centre of gravity will always be the same as the centre of mass in a uniform gravitational field (constant in magnitude and direction). This applies for bodies with non-uniform density as well as those with uniform density. The Earth's gravitational field can be considered uniform if the dimensions of the object are much smaller than its distance from the centre of the Earth.

The reason for this is, when calculating CM constituent elements of mass are 'weighted' by their mass $m_i$ and vector distance $\vec{r_i}$ from an arbitrary fixed point :

$$\vec{r_{CM}} \times \sum m_i = \sum m_i \vec{r_i} \tag{1}$$

whereas when calculating CG the same elements are weighted by their gravitational weight $m_i g(\vec{r_i})$ which might vary with position $\vec{r_i}$ :

$$\vec{r_{CG}} \times \sum m_i g(\vec{r_i}) = \sum m_i g(\vec{r_i})\vec{r_i} \tag{2}$$

If $g(\vec{r_i})=g$ is the same for all elements (ie the gravitational field is uniform) then the calculation gives the same centre (CM=CG). If $g(\vec{r_i})$ is not constant (ie the gravitational field is not uniform) then the positions of CM and CG are likely to be different.

However, it is still possible for the CM and CG to coincide in special cases. For example, the effects of non-uniform field and non-uniform density could cancel out, making the CM and CG coincide.