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Linear Momentum of a System of Particles

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Linear momentum of a system of particles:

Let us recall that the linear momentum of a particle is defined as

p = mv

Let us also recall that Newton’s second law written in symbolic form for a single particle is

`"F" = ("dp")/"dt"`

Where F is the force on the particle. Let us consider a system of n particles with masses m1, m2,.... mn respectively and velocities V1, V2,.... Vn respectively. The particles may be interacting and have external forces acting on them. The linear momentum of the first particle is m1 v1, of the second particle is m2 v2 and so on. For the system of n particles, the linear momentum of the system is defined to be the vector sum of all individual particles of the system,

`"P" = "p"_1 +"p"_2 +... +"p"_"n"`

`"m"_1"v"_1 + "m"_2"v"_2 + ... + "m"_"n" "v"_"n"`

comparing this wiht eq `"MV" = "m"_1"v"_1 + "m"_2"v"_2 + ... +"m"_"n" "v"_"n"`

P = MV

Thus the total momentum of a system of particles is equal to the product of the total mass of the system and the velocity of its centre of mass. Differentiating Eq. P =MV with respect to time,

`("dP")/"dt" = "M"("dV")/"dt" = "MA"`

`("dP")/("dt") = "F"_"ext"`

This is the statement of newtons second law of motion extended to a system of particles.

suppose now, that the sum of external forces acting om a system of particles is zero. Then from eq. `("dP")/("dt") = "F"_"ext"`

`("dP")/("dt") = 0`  or P = Constant           

Thus, when the total external force acting on a system of particles is zero, the total linear momentum of the system is constant. This is the law of conservation of the total linear momentum of a system of particles.

This also means that when the total external force on the system is zero the velocity of the centre of mass remains constant. (We assume throughout the discussion on systems of particles in this chapter that the total mass of the system remains constant.) Note that on account of the internal forces, i.e. the forces exerted by the particles on one another, the individual particles may have complicated trajectories. Yet, if the total external force acting on the system is zero, the centre of mass moves with a constant velocity, i.e., moves uniformly in a straight line like a free particle.                

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