Conservation of linear momentum

The idea that in the absence of external forces the momentum remains constant is called the principle of the "conservation of linear momentum". This principle, like the conservation of energy or work-energy theorem, seems to be near the heart of the laws of nature. It applies in all kinds of situations, even those outside the realm of classical dynamics. In order for us to apply the conservation of linear momentum, there must be no net external force acting on the system.

Suppose that external forces F _1e and F _2e also act on m ₁ and m ₂, respectively. From Eq. 5.7, we have (F _1e + F ₁₂)= d p ₁/ dt for m ₁, and (F _2e + F ₂₁)= d p ₂/ dt for m ₂. When we add these equations and use Newton's third law in the form F ₁₂ + F ₂₁ = 0, we find

(F ₁₂ + F ₂₁) = (p ₁+ p ₂)

This result can be extended to a system with any number of particles. The internal forces between the particles in the system cancel in pairs and Newton's second law takes the form

Newton's second law for a system of particles

FEXT= P (5.8)

where F _EXT = ∑ F _ie is the net external force on the system and P = ∑ p _i is the total linear momentum of the particles. From Eq. 5.8 we infer the condition for the applicability of the conservation of linear momentum:

If F _EXT = 0, then P = ∑ p _i = constant

If the net external force on a system is zero, the total linear momentum is constant.

If a net external force acts in the x and z directions, say, but not in the у direction. then the у component of linear momentum is still conserved.

The principle of the conservation of linear momentum is remarkably simple and general. It is valid for all types of interaction and may be applied to such diverse phenomena as collisions, explosions, radioactive decay, nuclear reac­tions, and the emission and absorption of light. It also helps us analyze such commonplace phenomena as the kick of a rifle and rocket propulsion.

The conservation of linear momentum may be applied as a first approximation even in situations in which the net external force is not zero. This is allowable if the internal forces, such as those in an explosion or in a collision, are much greater than the external force, for example, the force of gravity.

Дата добавления: 2014-01-05