Consider a closed system of particles. Since, by virtue of spatial homogeneity, all configurations occupied by it as a whole in space ar equivalent, we may assert that its properties will not change if it is moved parallel to itself to any distance.
A consequence of this circumstance is the conservation of a certain vector quantity characterizing the system. Namely, there exists a vector characterizing every material point, such that the sum of all the vectors over all the particles of a closed system is independent of time. It is called the vector of linear momentum.Momentum is related to velocity by a direct proportionality. The coefficient of proportionality, which is different for different material particles, is called mass. The law of conservation of the momentum of a system is thus expressed by the formula
m1v1 + m2v2 + ............. = `Sigma` m`alpha` v`alpha` = P = constant, ------(1) where `alpha` is the number of particles.
Mass of a particle : The law of conservation of momentum suggests a mode for correlating the masses of material points. Thus, for two colliding particles we can rewrite (1) as
m1 `Delta` v1 = -m2 `Delta` v2 , where `Delta` v1 and `Delta`v2 represent the change in the velocities of the respective particles.
Hence, `Delta` v1 = - (m1 / m2)`Delta` v2.
Knowing `Delta` v1 and `Delta` v2 , it is possible to correlate the masses of both particles. Obviously the basic mass cannot be measured and it must be taken for unity. It can be chosen arbitrarily : choice carries no deep physical meaning.
The law of conservation of momentum in a closed system can be regarded as a generalization of the inertia law (Newton's first law). In fact, for a free moving particle the momentum `rho` = mv remains constant. In the case of a system of material points interacting in any way the momentum of each particle is not constant, but the sum of the momenta of all the particles is conserved , and P = `Sigma` `rho` `alpha` .
Evidently, the quantity d`rho` `alpha` / dt = F `alpha` .---(2) , which expresses the change in momentum of a particle in unit itme, is a measure of the external action on that particle. This quantity is the force which the particles of a system exert on the particle `alpha` .
Some properties of mass : The concept of momentum can be used to formulate the concepts of rest and velocity of a system as a whole. A system of material points is at rest in that frame of reference in which its momentum is zero. The velocity of a system of material points relative to some reference frame O is defined as the velocity of a frame of reference O0 in which the system is at rest.
Denoted by v0 and v0`alpha` the velocities of a material point `alpha` relative to reference systems O and O0 , and by v the velocity of system O0 relative to system O, the Galilean transformations yield v`alpha` = v0`alpha` + V. Multiplying through by the mass of the corresponding material points and summing over all the points, we obtain
P = P0 + v `Sigma` m`alpha` . The system of material points is at rest relative to the reference frame O0., hence P0 = 0 and
P = v `Sigma` m`alpha` . As P is the momentum and v the velocity of the system as a whole, the latter relationship expresses the additivity of mass : the mass of a composite body is equal to the sum of the masses of its component parts. In other words, we have the law of conservation of mass.
We have seen that the law of conservation of linear momentum follows from the homogeneity of space with respect to a closed system of particles. Space is also isotropic; and any direction is as good as the other. Hence the properties of a closed system should not change if the whole system is turned through an arbitrary angle about an arbitrary axis. This condition implies the conservation of some vector quantity characterizing the system. Namely, every material point of a system is characterized by a vector, such that the sum of all the vectors over all the particles of a given closed system is independent of time. This is the vector of moment of momentum, or angular momentum . It is related in a specific way with the linear momentum vector: the angular momentum of a particle is equal to the product of the radius vector and the linear momentum of the particle
Thus the law of conservation of angular momentum takes the form
`Sigma` r`alpha` x p`alpha` = M = constant.
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