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# 3.6: Force and Rate of Change of Momentum

Theorem:

The rate of change of the total momentum of a system of particles is equal to the sum of the external forces on the system.

Thus, consider a single particle. By Newton’s second law of motion, the rate of change of momentum of the particle is equal to the sum of the forces acting upon it:

$\dot{\textbf{P}}_{i} = \textbf{F}_{i} + \sum_i \textbf{F}_{ij} \qquad (j \neq i ) \label{eq:3.6.1}$

Now sum over all the particles:

$\dot{\textbf{P}}_{i} =\sum_i \textbf{F}_{i} + \sum_i\sum_j \textbf{F}_{ij} \qquad (j \neq i )$

$\textbf{F} + \frac{1}{2}\sum_i\sum_j \textbf{F}_{ij} + \frac{1}{2}\sum_j\sum_i \textbf{F}_{ij}$

$\textbf{F} + \frac{1}{2}\sum_i\sum_j \textbf{F}_{ji}+ \textbf{F}_{ij} \label{eq:3.6.2}$

But, by Newton’s third law of motion, $$\textbf{F}_{ji}+ \textbf{F}_{ij} = 0$$, so the theorem is proved.

Corollary:

If the sum of the external forces on a system is zero, the linear momentum is constant. (Law of Conservation of Linear Momentum.)