General Mechanics/Many Particles

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 m_N, 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

$T={\frac {1}{2}}MV^{2}+{\frac {1}{2}}\sum _{n}m_{n}{v_{n}^{*}}^{2}$

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.