# The center of mass->

1. Jan 23, 2004

### deda

This is attempt to address Arcon's page on
center of mass

He says: $$\vec{R}=\frac{\sum m_i\vec{r_i}}{M}$$
R is radius vector of the center;
ri are radius vectors of mi;
M is sum of all mi;

It can be generalized to vector mass too.

$$M\vec{R}=M\vec{e_M}\times\vec{R}\times\vec{e_M}=\vec{M}\times\vec{R}\times\vec{e_m}$$;

also just like
$$\vec{M}=\sum \vec{m_i}$$
under special conditions it can as well be
$$\vec{R}=\sum \vec{r_i}$$

=> $$(\sum \vec{r_i})\times(\sum \vec{m_i})=\sum (\vec{r_i}\times\vec{m_i})$$;

=> $$\sum_{i<>j} (\vec{r_i}\times\vec{m_j})=\vec{o}$$;

This way the position of the last mass depends on all the masses (including its own) and all the other positions;

I just can't say what are the terms for the last equation.
Can you?