General Mechanics The uneven dumbbell consisted of just two particles. These results can be extended to cover systems of many particles, and continuous media. Suppose we have N particles, masses m1 to mN, with total mass M. Then the centre of mass is R = ∑ n m n r n M {\displaystyle \mathbf {R} ={\frac {\sum _{n}m_{n}\mathbf {r} _{n}}{M}}} Note that the summation convention only applies to numbers indexing axes, not to n which indexes particles. We again define r n ∗ = r n − R {\displaystyle \mathbf {r} _{n}^{*}=\mathbf {r} _{n}-\mathbf {R} } The kinetic energy splits into T = 1 2 M V 2 + 1 2 ∑ n m n v n ∗ 2 {\displaystyle T={\frac {1}{2}}MV^{2}+{\frac {1}{2}}\sum _{n}m_{n}{v_{n}^{*}}^{2}} and the angular momentum into L = M R × V + ∑ n m n r n ∗ × v n ∗ {\displaystyle \mathbf {L} =M\mathbf {R} \times \mathbf {V} +\sum _{n}m_{n}\mathbf {r} _{n}^{*}\times \mathbf {v} _{n}^{*}} It is not useful to go onto moments of inertia unless the system is approximately rigid but this is still a useful split, letting us separate the overall motion of the system from the internal motions of its part.
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