11.5: Many Particles

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The generalization from two particles to many particles is quite easy in principle. If a subscripted i indicates the value of a quantity for the ith particle, then the center of mass is given by

\[\mathbf{R}_{c m}=\frac{1}{M_{\text {total }}} \sum_{i} M_{i} \mathbf{r}_{i}\label{11.19}\]

where

\[M_{\text {total }}=\sum_{i} M_{i}\label{11.20}\]

Furthermore, if we define \(\mathbf{r}_{\mathrm{i}}^{\prime}=\mathbf{r}_{\mathrm{i}}-\mathbf{R}_{\mathrm{cm}}\), etc., then the kinetic energy is just

\[K_{\text {total }}=M_{\text {total }} V_{\mathrm{cm}}^{2} / 2+\sum_{i} M_{i} v_{i}^{\prime 2} / 2\label{11.21}\]

and the angular momentum is

\[\mathbf{L}_{\text {total }}=M_{\text {total }} \mathbf{R}_{c m} \times \mathbf{V}_{c m}+\sum_{i} M_{i} \mathbf{r}_{i}^{\prime} \times \mathbf{v}_{i}^{\prime}\label{11.22}\]

In other words, both the kinetic energy and the angular momentum can be separated into two parts: one part is related to the overall motion of the system and the other is due to motions of system components relative to the center of mass, just as for the case of the dumbbell.