Systems of Particles

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Physics Newtonian Mechanics Systems of Particles

Conservation of Momentum in a System of Particles:

When a group of particles interact then they exert force on each other and consequently, the
momentum. The exerted impulses are equal and opposite to each other. The momentum of the
individual particles might change. However, the momentum of the system remains constant. This
phenomenon of constancy of momentum describes the conservation of linear momentum. Thus,
the conservation of momentum for a system of particles states that the momentum of the system
before the interaction is same as that after interaction when there is no external force.

Consider a two particle system. The pair of forces exerted on each other are related
as: F₁₂ = -F₂₁. Thus, we can write:

F₁₂ + F₂₁ = 0

If we replace these forces in terms of rate of change of momentum,
we would get:

Thus, the time derivative of total momentum is zero. Hence, it can be concluded that total
momentum of the system must remain constant:

or we can write equivalently,

This equation implies that the momentum before the interaction of the two body system is equal to
the momentum of the system after interaction. Now, the momentum in all three perpendicular
directions of the rectangular coordinate system would also be conserved:

These considerations can also be extended to many particle system and the concepts very well
remain same. It should be noted that in all these considerations the mass of the particles remain
constant.