As per the question,
It is given that the mass of the 4 particles are m i = m 1 , m 2 , m 3 , m 4 m_i= m_1, m_2, m_3 , m_4 m i = m 1 , m 2 , m 3 , m 4
Position of the particles are r i = r 1 , r 2 , r 3 , r 4 r_i= r_1, r_2, r_3, r_4 r i = r 1 , r 2 , r 3 , r 4
Velocity of the particles are v i = v 1 , v 2 , v 3 , v 4 v_i=v_1, v_2, v_3, v_4 v i = v 1 , v 2 , v 3 , v 4
i)
ii) R c m = m 1 r 1 + m 2 r 2 + m 3 r 3 + m 4 r 4 m 1 + m 2 + m 3 + m 4 R_{cm}=\frac{m_1r_1+m_2r_2+m_3r_3+m_4r_4}{m_1+m_2+m_3+m_4} R c m = m 1 + m 2 + m 3 + m 4 m 1 r 1 + m 2 r 2 + m 3 r 3 + m 4 r 4
v c m = m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4 m 1 + m 2 + m 3 + m 4 v_{cm}=\frac{m_1v_1+m_2v_2+m_3v_3+m_4v_4}{m_1+m_2+m_3+m_4} v c m = m 1 + m 2 + m 3 + m 4 m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4
iii) Total momentum of the system P t o t = m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4 P_{tot}=m_1v_1+m_2v_2+m_3v_3+m_4v_4 P t o t = m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4
Total momentum about the center of mass be P c o m = ( m 1 + m 2 + m 3 + m 4 ) v c o m P_{com}=(m_1+m_2+m_3+m_4)v_{com} P co m = ( m 1 + m 2 + m 3 + m 4 ) v co m
= m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4 =m_1v_1+m_2v_2+m_3v_3+m_4v_4 = m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4
iv) Angular momentum L t o t = L 1 + L 2 + L 3 + L 4 L_{tot}=L_1+L_2+L_3+L_4 L t o t = L 1 + L 2 + L 3 + L 4
= m r 1 → × v 1 → + m r 2 → × v 2 → + m r 3 → × v 3 → + m r 4 → × v 4 → =m\overrightarrow{r_1}\times \overrightarrow{v_1}+m\overrightarrow{r_2}\times \overrightarrow{v_2}+m\overrightarrow{r_3}\times \overrightarrow{v_3}+m\overrightarrow{r_4}\times \overrightarrow{v_4} = m r 1 × v 1 + m r 2 × v 2 + m r 3 × v 3 + m r 4 × v 4
Angular momentum about the center of mass L c o m → = m t o t a l r c o m → × v c o m → \overrightarrow{L_{com}}=m_{total}\overrightarrow{r_{com}}\times \overrightarrow{v_{com}} L co m = m t o t a l r co m × v co m
= ( m 1 + m 2 + m 3 + m 4 ) ( m 1 r 1 + m 2 r 2 + m 3 r 3 + m 4 r 4 m 1 + m 2 + m 3 + m 4 ) × ( m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4 m 1 + m 2 + m 3 + m 4 ) =(m_1+m_2+m_3+m_4)(\frac{m_1r_1+m_2r_2+m_3r_3+m_4r_4}{m_1+m_2+m_3+m_4})\times(\frac{m_1v_1+m_2v_2+m_3v_3+m_4v_4}{m_1+m_2+m_3+m_4}) = ( m 1 + m 2 + m 3 + m 4 ) ( m 1 + m 2 + m 3 + m 4 m 1 r 1 + m 2 r 2 + m 3 r 3 + m 4 r 4 ) × ( m 1 + m 2 + m 3 + m 4 m 1 v 1 + m 2 v 2 + m 3 v 3 + m 4 v 4 )
v) Total momentum about the center of mass and the total momentum always be same but angular momentum about the center of mass and the angular momentum about the origin will not be same.
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