Expanding
Centroid, center of mass, and center of gravity are not exactly the same
they only match if
• the gravitational field is uniform
• the density of the object is uniform
for large objects in which the gravitational field varies with height, as is the case on Earth, the center of gravity will differ from the center of mass. in small objects of everyday life the difference is negligible,
Example: The center of mass of an object of negligible base S and height h and constant density would be calculated$$ \displaystyle \mathbf r _{\text {cm}} = \dfrac {\int \limits_V \rho \mathbf r dV} {\int \rho dV} = \dfrac {\int \limits_V \mathbf r dV} {V} = \dfrac {\int \limits_0 ^ H \mathbf hS dh} {SH} = \dfrac {\int \limits_0 ^ H \mathbf h dh} {H} = \dfrac H2 $$
and the center of gravity
$$\displaystyle \mathbf r _{\text {cg}} = \dfrac {\int \limits_V \rho \mathbf rg (r) dV} {\int_V \rho g (r) dV} = \dfrac { \int \limits_V \mathbf rg (r) dV} {\int \limits_V g (r) dV} =$$ $$ \dfrac {\int \limits_0 ^ H \mathbf hS g (h) dh} {S \int \limits_0 ^ H g (h) dh} = \dfrac {\int \limits_0 ^ H \mathbf hg (h) dh} {\int \limits_0 ^ H g (h) dh} $$as you can see if ## g (h) = constant ## both expressions matchbut generally ## g (h) = \dfrac {GM} {(R_{Earth} + h) ^ 2}## for the variation of gravity as a function of the mass and the terrestrial radius, it is not worth it consider the difference when ## H << R_{Earth}##
Nor is it the same, the centroid only takes into account the geometry of the object, if the object has a constant density then the centroid and the center of mass coincide, and if the gravity is also constant it also coincides with the center of gravity.
Following the example,
$$ \displaystyle \mathbf r _{\text {ce}} = \dfrac {\int \limits_V \mathbf r dV} {\int \limits_V dV} = \dfrac {\int \limits_V \mathbf r dV } {V} = \dfrac {\int \limits_0 ^ H \mathbf hS dh} {SH} = \dfrac {\int \limits_0 ^ H \mathbf h dh} {H} = \dfrac H2 $$
| Centroid | Center of mass | Center of gravity |
| ##\displaystyle\mathbf r_{\text{ce}} = \dfrac{ \int \limits_V \mathbf r dV}{ \int \limits_VdV}## | ##\displaystyle\mathbf r_{\text{cm}} = \dfrac{ \int \limits_V \mathbf r\rho(V) dV}{ \int \limits_V\rho(V)dV}## | ##\displaystyle\mathbf r_{\text{cg}} = \dfrac{ \int \limits_V \mathbf r\rho(V) g(V)dV}{ \int \limits_V\rho(V)g(V)dV}## |
