PX155 - B1 - systems of particles I

consider an object/body as $N$ particles at positions ${\vec{r}}_{i}$ relative to some origin $O$
force on particle $j$ due to particle $i$ is ${\vec{F}}_{i j}$
external force on particle $i$ is ${\vec{F}}_{i}$
total force on one of the particles $i :$

{\vec{F}}_{i} + \sum_{j \neq i}^{N} {\vec{F}}_{j i} = m_{i} {\vec{a}}_{i}

sum over all particles:$$\sum_i^N{\vec F_i}+\sum_i^N{\sum\limits_{j\neq i}^{N}{\vec F_{ji}}}=\sum\limits_i^N{m_i \vec a_i}$$
2nd term = 0 : for every ${\vec{F}}_{j i}$ there is an equal and opposite ${\vec{F}}_{i j}$
${\vec{F}}_{j i} = - {\vec{F}}_{i j}$ by PX155 - A3 - newton's third law
$\vec{F} = \sum_{i}^{N} m_{i} {\vec{a}}_{i}$
$\vec{F} = (\sum_{i}^{N} m_{i}) \frac{\sum_{i}^{N} m_{i} {\vec{a}}_{i}}{\sum_{i}^{N} m_{i}} = (M) \frac{\sum_{i}^{N} m_{i} {\vec{a}}_{i}}{M}$
$\vec{F} = M (\frac{\sum_{i}^{N} m_{i} \frac{d^{2} {\vec{r}}_{i}}{d t^{2}}}{M})$
$\vec{F} = M (\frac{\frac{d^{2}}{d t^{2}} \sum_{i}^{N} m_{i} {\vec{r}}_{i}}{M})$
$\vec{F} = M \frac{d^{2}}{d t^{2}} {\vec{r}}_{c m}$
where, ${\vec{r}}_{c m} =$ centre of mass $\vec{F} = M {\vec{a}}_{c m}$
PX155 - A2 - newton's second law holds provided all points on the body accelerate at ${\vec{a}}_{c m}$ or if we only care about ${\vec{a}}_{c m}$