Q7.33     Separation of Motion of a system of particles into motion of the centre of mass and motion about the
               centre of mass :

        (c) Show L=L^{'}+R\times MV
            where L=\sum r_{i}^{'}\times P_{i}^{'}
is the angular momentum of the system about the centre of mass with velocities taken relative to the centre of mass. Remember r_{i}^{'}=r_{i}-R ; rest of the notation is the standard notation used in the chapter. NoteL^{'}  and MR\times V can be said to be angular momenta, respectively, about and of the centre of mass of the system of particles.

Answers (1)

The position vector of the ith particle (with respect to the center of mass) is given by :

                                                                    r_i'\ =\ r_i\ -\ R

or                                                                  r_i\ =\ r_i'\ +\ R

From the first case we can write :

                                                                   p_i\ =\ p_i'\ +\ p

Taking cross product with position vector we get  ;  

                                                          \sum r_i\times p_i\ =\ \sum r_i\times p_i'\ +\ \sum r_i\times p

or                                                                L\ =\ L'\ +\ \sum R\times p_i'\ +\ \sum r_i\times m_iv\ +\ \sum R\times m_iv

or                                                                  L=L^{'}+R\times MV