Angular Momentum Arbitrary Point

Let us consider a reference point P moving arbitrarily with respect to fixed origin O in space. If denotes the position of ith particle of the extended system with respect to point P, then the angular momentum of the system of P is given by,



We can split as,



where, is the position of CM of the system relative to P and r_i’ is the position of ith particle relative to CM. Also, we get



Now, following the steps exactly, we find the equation corresponding to,



Note that while and are the position and velocity of CM with respect to point P, R_CM and V_CM are that of CM with respect to 0.

We can take point P within the system itself. In case of a rigid body, P any be a point fixed within the body.

If P is taken as CM of the system, then = 0 and L_P = L_CM ; then reduces to,

Let us now write the torque equation about point P; the torque due to external forces about P are given by



where, and is torque about CM. Now, we get,



Note that,
    
If P is CM, then = 0,
    
If P is moving with constant velocity, RP = 0, and we simply get



Thus, in general, the relation d holds only if point P is not accelerating. However, if P is CM, then even if it is accelerating.

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