Acceleration of Centre of Mass

In exactly the same way, the ACCELERATION of the centre of mass of a system of particles (with accelerations $\vec{a_1}, \vec{a_2}, \ldots, \vec{a_n}$) is the mass-weighted average of the individual accelerations: $\vec{a}_{cm}=\frac{\sum_{i=1}^n m_i\vec{a_i}}{M}$, again with x, y, z components obtainable similarly, and the integral form $\vec{a}_{cm}=\frac{\int \vec{a}\,dm}{M}$ for a continuous distribution. This result is what ultimately justifies treating an extended system's overall translational motion using a single equivalent point mass at the centre of mass: since internal (mutual) forces between the particles of a system always cancel out in pairs (Newton's third law), only the NET EXTERNAL …