Question

A many-particle system

A system of four(4) free particles each has a mass m_i and initial position r_i and initial velocity v_i .

a) Describe your own four-particle system in detail.

b) Determine the position R_cm and velocity V_cm of the centre of mass of the system.

c) Determine the total linear momentum P_tot and the linear momentum P_cm of the centre of mass of the system.

d)Determine the total angular momentum L_tot and the linear momentum P_cm of the centre of mass of the system.

e) Discuss briefly about your answers in part b), c) and d).

Answer 1

As per the question,

It is given that the mass of the 4 particles are $m_i= m_1, m_2, m_3 , m_4$

Position of the particles are $r_i= r_1, r_2, r_3, r_4$

Velocity of the particles are $v_i=v_1, v_2, v_3, v_4$

i)

ii) $R_{cm}=\frac{m_1r_1+m_2r_2+m_3r_3+m_4r_4}{m_1+m_2+m_3+m_4}$

$v_{cm}=\frac{m_1v_1+m_2v_2+m_3v_3+m_4v_4}{m_1+m_2+m_3+m_4}$

iii) Total momentum of the system $P_{tot}=m_1v_1+m_2v_2+m_3v_3+m_4v_4$

Total momentum about the center of mass be $P_{com}=(m_1+m_2+m_3+m_4)v_{com}$

$=m_1v_1+m_2v_2+m_3v_3+m_4v_4$

iv) Angular momentum $L_{tot}=L_1+L_2+L_3+L_4$

$=m\overrightarrow{r_1}\times \overrightarrow{v_1}+m\overrightarrow{r_2}\times \overrightarrow{v_2}+m\overrightarrow{r_3}\times \overrightarrow{v_3}+m\overrightarrow{r_4}\times \overrightarrow{v_4}$

Angular momentum about the center of mass $\overrightarrow{L_{com}}=m_{total}\overrightarrow{r_{com}}\times \overrightarrow{v_{com}}$

$=(m_1+m_2+m_3+m_4)(\frac{m_1r_1+m_2r_2+m_3r_3+m_4r_4}{m_1+m_2+m_3+m_4})\times(\frac{m_1v_1+m_2v_2+m_3v_3+m_4v_4}{m_1+m_2+m_3+m_4})$

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.