- #1
deda
- 185
- 0
This is attempt to address Arcon's page on
center of mass
He says: [tex]\vec{R}=\frac{\sum m_i\vec{r_i}}{M}[/tex]
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.
[tex]M\vec{R}=M\vec{e_M}\times\vec{R}\times\vec{e_M}=\vec{M}\times\vec{R}\times\vec{e_m}[/tex];
also just like
[tex]\vec{M}=\sum \vec{m_i}[/tex]
under special conditions it can as well be
[tex]\vec{R}=\sum \vec{r_i}[/tex]
=> [tex](\sum \vec{r_i})\times(\sum \vec{m_i})=\sum (\vec{r_i}\times\vec{m_i})[/tex];
=> [tex]\sum_{i<>j} (\vec{r_i}\times\vec{m_j})=\vec{o}[/tex];
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?
center of mass
He says: [tex]\vec{R}=\frac{\sum m_i\vec{r_i}}{M}[/tex]
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.
[tex]M\vec{R}=M\vec{e_M}\times\vec{R}\times\vec{e_M}=\vec{M}\times\vec{R}\times\vec{e_m}[/tex];
also just like
[tex]\vec{M}=\sum \vec{m_i}[/tex]
under special conditions it can as well be
[tex]\vec{R}=\sum \vec{r_i}[/tex]
=> [tex](\sum \vec{r_i})\times(\sum \vec{m_i})=\sum (\vec{r_i}\times\vec{m_i})[/tex];
=> [tex]\sum_{i<>j} (\vec{r_i}\times\vec{m_j})=\vec{o}[/tex];
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?