$(1,1) \equiv 1 \hat{i} + 1 \hat{j}=\hat{i}+\hat{j}$

$\equiv 1 \hat{e}_x + 1 \hat{e}_y = \hat{e}_x + \hat{e}_y$

$\vec{R}_{cm}=(0,0)$

$\vec{R}_{cm}=\frac{2(1/2,1/2)+1(1/2,-1/2)}{6}$

$\frac{+2(-1/2,-1/2)+1(-1/2,1/2)}{6}$

$=(6,6)$

$m_1 + m_2 +...+m_n=M$

$\vec{R} \equiv \vec{R}_{cm}$

$= \frac{\sum_{i=1}^{n} m_i \vec r_{i}} {\sum_{i=1}^{n} m_i}$

$=\frac{\sum_{i=1}^{n} m_i \vec{r}_i}{M}$

$M \vec{R}= m_1 \vec{r}_1 + m_2 \vec{r}_2 + .. + m_n \vec{r}_n$

Differentiate w.r.t 't' on both sides

$M \frac{d \vec{R}}{dt}=m_1 \frac{d \vec{r}_1}{dt}+m_2 \frac{d \vec{r}_2}{dt}+..+ m_n \frac{d \vec{r}_n}{dt}$

$M \vec{v}= m_1 \vec{v}_1 + m_2 \vec{v}_2 + .. + m_n \vec{v}_n$

$\vec{V}=\vec{V}_{cm}=$ velocity of centre of mass

$M \vec{A}=m_1 \vec{a}_1 + m_2 \vec{a}_2 + .. + m_n \vec{a}_n$

$\vec{A}=\frac{d \vec{v}}{dt}; \vec{a}_i = \frac{d \vec{v}_i}{dt}$

$M \vec{A} = \vec{F}_1 + \vec{F}_2 + .. + \vec{F}_n$

$= \vec{F}_{ext}$

$M \vec{A}=m_1 \vec{a}_1 + m_2 \vec{a}_2 + .. + m_n \vec{a}_n$

$\vec{A}=\frac{d \vec{v}}{dt}; \vec{a}_i = \frac{d \vec{v}_i}{dt}$

$M \vec{A} = \vec{F}_1 + \vec{F}_2 + .. + \vec{F}_n$

$= \vec{F}_{ext}$

$M \vec{A}= \vec{F}_{ext}$ Covering equation

Forces

1) External

2) Internal

Linear momentum of a system of particles

$\vec{p}=m \vec{v}, \frac{d \vec{p}}{dt}=\vec{F}$

$\vec{P} = \vec{p}_1 + \vec{p}_2 + .. + \vec{p}_n$

$M \vec{V} = \frac{d \vec{P}}{dt}= \vec{F}_{ext}$

Suppose, $\vec{F}_{ext}=0$

$\frac{d \vec{P}}{dt} = 0 = \vec{P} = \vec{c}$, a constant vector

$\Rightarrow$ The linear momentum of the system of particle remains the same.

$\Rightarrow \vec{P}$ is a constant of motion.

$\vec{P}$ is constant.

i.e., the $\vec{v}_{CM}$ remains constant.

$\vec{F}_{ext}=0$

Motion of various parts of a system is separated (or spit) into:

(1) motion of CM

(2) motion about the CM

Question: What about the total KE of a two particle system relative to CM?

$v_{CM}=\frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 + m_2}$

Velocity of $m_1$ w r t CM, $\vec{v}_{1,CM}$

$=\vec v _ {1} - \vec v_{CM}$

$=\vec{v}_1 - (\frac{m_1 v_1 + m_2 v_2}{m_1 + m_2})$

$=\frac{m_1 \vec{v}_1 - m_1 \vec{v}_1 + m_2 \vec{v}_1 - m_2 \vec{v}_2}{m_1 + m+2}$

$\vec{v}_{1,CM}= \frac{m_2(\vec{v}_1 - \vec{v}_2)}{m_1 + m_2}$

$\vec{v} _ {2,CM} = \vec{v} _ 2 - \vec{v}_{CM}$

$=\vec{v}_2 -(\frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 +m_2})$

$\vec{v}_{2,CM} = \frac{m_1(\vec{v}_2 - \vec{v}_1)}{m_1 + m_2}$

$KE| _ {m _ 1,CM} = \frac{1}{2} m _ 1 (\frac{m _ 2 \vec{v} _ {1 _ 2}}{m _ 1 + m_2})^2$

$= \frac{1}{2} m_1 \frac{m_2^2 |v_{1_2}|^2}{(m_1 + m_2)^2}$

$KE| _ {m _ 2,CM} = \frac{1}{2} m _ 2 (\frac{m _ 1 \vec{v} _ {2 _ 1}}{m _ 1 + m_2})^2$

$= \frac{1}{2} m_2 \frac{m_1^2 |v_{1_2}|^2}{(m_1 + m_2)^2}$

$KE|_{system,CM}$

$= \frac{1}{2} \frac{m_1 m_2 |v_{1,_2}|^2(m_2 + m_1)}{(m_1 + m_2)^2}$

$KE |_{system,CM}$

$=\frac{1}{2} (\frac{m_1 m_2}{m_1 + m_2}) |v_{1,2}|^2$

Reduced mass

$\mu_0 = \frac{m_1 m_2}{m_1 + m_2}$

$KE| _ {Total} = \frac{1}{2} \mu|\vec{v} _ {rel}|^2 + \frac{M}{2} v_{CM}^2$

$KE|=\frac{1}{2} \mu |\vec{v}_{rel}|^2 + \frac{1}{2} (m_1 + m_2)(\frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 + m_2})^2$

RHS $=\frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (\vec{v}_1 - \vec{v}_2)^2$

$= \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (\vec{v}_1^2 + \vec{v}_2^2 - 2 \vec{v}_1 . \vec{v}_2)$

= $+ \frac{1}{2} \frac{(m_1 + m_2)}{(m_1 + m_2)^2} (m_1^2 v_1^2 + m_2^2 v_2^2 + 2m_1 m_2 \vec{v}_1 . \vec{v}_2) + \frac{1}{2} \frac{1}{(m_1 + m_2)} (m_1^2 v_1^2 + m_2^2 v_2^2 + 2 m_1 m_2 \vec{v}_1 . \vec{v}_2)$

$= \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2$

$KE|_{Total} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2$

$\mu = M = \frac{m_1 m_2}{m_1 + m_2}$

$\frac{1}{\mu}=\frac{1}{m_1} + \frac{1}{m_2}$

$\mu \leq m_1, \mu \leq m_2$

Reduced mass is always less or equal to mass of each body.