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

$𝐑=\frac{\sum _n{m}_n{𝐫}_n}{M}$

Note that the summation convention only applies to numbers indexing axes, not to n which indexes particles.

We again define

${𝐫}_n^{*}={𝐫}_n-𝐑$

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

$𝐋=M𝐑\times 𝐕+\sum _n{m}_n{𝐫}_n^{*}\times {𝐯}_n^{*}$

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.