Center-of-mass Theorem

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In the case of a system of particles $P_i$, each with mass $m_i$ that are located in space with coordinates $\mathbf{r}_i$, the coordinates $\mathbf{R}$ of the center of mass satisfy the condition \[ \sum_{i=1}^n m_i(\mathbf{r}_i - \mathbf{R}) = 0.\] Solve this equation for R to obtain the formula \[\mathbf{R} = \frac{1}{M} \sum_{i=1}^n m_i \mathbf{r}_i,\] where $M$ is the sum of the masses of all of the particles.

If the mass distribution is continuous with the density $\rho(\mathbf{R})$ within a volume $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass $\mathbf{R}$ is zero, that is \[\int_V \rho(\mathbf{r})(\mathbf{r}-\mathbf{R})dV = 0.\] Solve this equation for the coordinates R to obtain \[\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV,\] where $M$ is the total mass in the volume.