The th particle of a system of particles has mass and, at time , position vector with respect to an origin . It experiences an external force , and also an internal force due to the th particle (for each ), where is parallel to and Newton's third law applies.
(i) Show that the position of the centre of mass, , satisfies
where is the total mass of the system and is the sum of the external forces.
(ii) Show that the total angular momentum of the system about the origin, , satisfies
where is the total moment about the origin of the external forces.
(iii) Show that can be expressed in the form
where is the velocity of the centre of mass, is the position vector of the th particle relative to the centre of mass, and is the velocity of the th particle relative to the centre of mass.
(iv) In the case when the internal forces are derived from a potential , where , and there are no external forces, show that
where is the total kinetic energy of the system.