Linear Momentum Conservation Theorem

up:: Linear Momentum

Let be the positions of particles , of mass , with Center of Mass and external forces . Then

That is, the sum of the external forces over a system of particles induces Newton’s Second Law to the equivalent particle , located at the center of mass and with mass equal to the system’s total mass.

Proof

By Newton’s Second Law, we have that, for each particle ¹,

Summing for all particles , we have that

The sum of all reciprocal forces in a system of particles is always zero ². Thus,

Using the Center of Mass definition, and supposing that , one can write the above as

Corollaries

A system’s total momentum is conserved if it has no net external forces: If the external forces sum to , then the system’s total momentum

is conserved.

Examples

Let a system of two particles with no external forces acting upon them. Then

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References

• LEMOS, Nivaldo A. Mecânica analítica. Editora Livraria da Física, 2007.

Footnotes

1. Denoting as the force the particle exerts upon particle . ↩

2. Due to Newton’s Third Law. One can think about it as summing all values of an antisymmetric matrix. ↩