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  • Separation of Motion of a system of particles into motion of the centre of mass and motion about the centre of mass(c).

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.

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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

Posted by

Devendra Khairwa

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