Equation of Motion: Center of Mass

The equation of motion for a single particle can be expanded to encompass a system of particles consisting of n particles. For any arbitrarily chosen particle within this system, the net force acting upon it is the aggregate of both internal and external forces. Extending this principle to all particles within the system results in the equation of motion for the entire assembly.

Internal forces between any pair of particles manifest as collinear pairs of equal magnitude but opposite directions, leading to their summation equating to zero. Now, introduce a center of mass G expressed in terms of position vectors of the various particles. Differentiating this expression twice concerning time yields the equation of motion relative to the center of mass of the entire system.

Consequently, the net external forces influencing the system of particles translate to the product of the system's overall mass and the acceleration of its center of mass. This comprehensive formulation captures the dynamics of a multi-particle system, considering both internal interactions and external influences. The center of mass concept provides a helpful perspective, simplifying the description of the system's motion in relation to its overall characteristics.