Proof related to the center of mass

[patric44]

Homework Statement

proof that the distribution of mass around the center of mass is = 0.

Relevant Equations

∫Rdm = 0

hi guys
in the proof of the parallel axis theorem this equation is just put as it is as a definition of the center of mass :
$$\int[2(\vec{r_{o}}.\vec{r'})I-(\vec{r_{o}}⊗\vec{r'}+\vec{r'}⊗\vec{r_{0}})]dm = 0$$
is there is any proof for this definition ? and what is the approach for it

 

[hacivat]

If you calculate the position of center of mass with respect to a reference frame whose origin is fixed at the center of mass of the body then of course you get zero.

I cannot comment on your equation unless I know what are the nomenclature for the symbols but parallel axis theorem uses the setup I explained above.