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 mN, with total mass M. Then the centre of mass is
Note that the summation convention only applies to numbers indexing axes, not to n which indexes particles.
We again define
The kinetic energy splits into
and the angular momentum into
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.