Principle of Linear Impulse and Momentum for a System of Particles

In the context of a system of particles moving relative to an inertial frame of reference, the equation of motion is a crucial tool for understanding the dynamics of the system. This equation, which accounts for external forces acting on each particle, plays a fundamental role in describing the system's behavior.

Notably, internal forces between particles, occurring in equal and opposite collinear pairs, cancel out and are not part of the equation of motion. This exclusion simplifies the analysis, focusing on the impact of external forces on the system.

The principle of linear impulse and momentum for the system of particles emerges upon integrating the equation of motion and substituting the limits. According to this principle, the combined initial linear momenta of the particles, along with the impulses of all external forces from the initial to the final time, equals the final linear momenta of the system.

To extend the applicability of this principle, we delve into the equation governing the system's center of mass. By differentiating it, a relationship between the total linear momentum of the particles and the linear momentum of the center of mass is established. This crucial relationship is then incorporated back into the linear impulse and momentum equation, resulting in a modified equation that aptly applies to the broader context of a system of particles composing a rigid body. This nuanced approach enhances our understanding of the dynamic interactions within such systems.