Angular Momentum Extended System

Consider an extended system of n particles. The total angular momentum of the system is defined as the vector sum of angular momenta of individual particles about the fixed reference point O. That is,

L = L₁ + L₂ + …. + L_n



where, m_i, r_i, v_i denote the mass, position and velocity of the ith particle about O. The co-ordinate system at O is an inertial system. At any instant, if the position vector of CM of the system is R_CM with respect to O, then we can write

R_i = R_CM + r_i’

and, v_i = V_CM + v_i’

where r_i’ and v_i’ are the position and velocity vectors of i^th particle with respect to CM. Hence, we have

L = Σ_i [m_i (R_CM + r_i’) × (V_CM + v_i’)]

= R_CM × M V_CM + R_CM × [Σ_i m_i v_i’] + [Σ_i m_i r_i’] × V_CM + Σ_i (r_i’ × m_i v_i’)

where M = Σ m_i is the total mass of the system.

Now, in CM frame, by definition,

Σ_i m_i r_i’ = 0 = Σ_i m_i v_i’

Hence, angular momentum of the system about an arbitrary but fixed, reference point of O is given as

L = L_CM + R_CM × M V_CM

where L_CM = Σ_i r_i’ × p_i’

is the ‘intrinsic’ angular momentum of all the particles (i.e. extended system) about its CM. the second term on right denotes the angular momentum associated with motion of CM about O (i.e. of a particle of mass M moving with CM).

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