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Newton Extended Object

Consider an extended system of n-particles. The total force on any individual particle, say the ith particle, consists of two parts (i) an external force F_i arising from some source external to the system, and (ii) the sum of all the internal forces arising due to mutual interactions between the ith particle and the remaining (n – 1) particles. Hence, according to Newton’s law of motion as applied to the ith particle, we find

where f_ij is the (internal) force on the i^th particle due to j^th particle; in the summation, we also get a term for j = i, viz. f_iiwhich is identically zero.

We get equations similar for all other particles, i.e. for all i = 1, 2, …, n. Adding all these equations, we find

The second term on the right hand side contain pairs like (f₁₂ + f₂₁), (f₁₃ + f₃₁), etc; however, according the Newton’s III law, f_ij = –f_ji. Hence, the entire double summation vanishes:

The remaining eq. reads as,

The above is Newton’s II law reformulated for an extended system. It says that the net, total ‘external’ force F_ext acting on a system is equal to rate of change of momentum of center-of-mass of the system. In terms of acceleration, we have

The entire extended system behaves as if it is a point mass M (centered at center-of-mass) whose acceleration a_CM can be determined from above relation if F_ext is known.

The principle of conservation of momentum for an extended system therefore implies that in absence of external force, the total momentum of system, or the momentum of center-of-mass of the system, remains conserved. That is, if F_ext = 0, then

P_CM = M V_CM = constant

The momenta of individual particles may still change because of the internal forces acting on respective particles.