3.6: Force and Rate of Change of Momentum
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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:
\[\dot{\textbf{P}}_{i} =\sum_i \textbf{F}_{i} + \sum_i\sum_j \textbf{F}_{ij} \qquad (j \neq i ) \nonumber \]
\[\textbf{F} + \frac{1}{2}\sum_i\sum_j \textbf{F}_{ij} + \frac{1}{2}\sum_j\sum_i \textbf{F}_{ij} \nonumber \]
\[ \textbf{F} + \frac{1}{2}\sum_i\sum_j \textbf{F}_{ji}+ \textbf{F}_{ij} \label{eq:3.6.2} \]
But, by Newton’s third law of motion, \(\textbf{F}_{ji}+ \textbf{F}_{ij} = 0\), so the theorem is proved.
If the sum of the external forces on a system is zero, the linear momentum is constant. (Law of Conservation of Linear Momentum.)