General Mechanics/Many Particles

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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

\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

\mathbf{r}^*_n=\mathbf{r}_n-\mathbf{R}

The kinetic energy splits into

T=\frac{1}{2}M V^2 + \frac{1}{2} \sum_n m_n {v^*_n}^2

and the angular momentum into

\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.