Show that a system of particle moving under the influence of an external force, moves as if the force is applied as its centre of mass.

Explanation

Consider `vec(r_(1)),vec(r_(2)),vec(r_(3))......vec(r_(n))` be the position vectors of masses `m_(1),m_(2),m_(3),…..m_(n)` of n particle system. According to Defination of centre of mass.
`vecR=(m_(1)vec(r_(1))+m_(2)vec(r_(2))+......+m_(n)vec(r_(n)))/(m_(1)+m_(2)+......+m_(n))`
`vecR=(m_(1)vec(r_(1))+m_(2)vec(r_(2))+......+m_(n)vec(r_(n)))/(M)` `[m_(1)+m_(2)+......+m_(n)=M="mess of the body"]`
`MvecR=m_(1)vec(r_(1))+m_(2)vec(r_(2))+....+m_(n)vec(r_(n))`
Differentiating the two sides of the equation with respect to time we get
`M(vec(dR))/(dt)=m_(1)(vec(dr)_(1))/(dt)+m_(2)(vec(dr)_(2))/(dt)+......+m_(n)(vec(dr)_(n))/(dt)`
`MvecR=m_(1)vec(r_(1))+m_(2)vec(r_(2))+......+m_(n)vec(r_(n))`
`Mvecv=m_(1)vec(v_(1))+m_(2)vec(v_(2))+......+m_(n)vec(v_(n))`
Differentiating the above equation w.r.t. time, we obtain
`M(dvecv)/(dt)=m_(1)(dvec(v_(1)))/(dt)+m_(2)(dvec(v_(2)))/(dt)+......+m_(n)(dvec(v_(n)))/(dt)orMvecA=m_(1)vec(a_(1))+m_(2)vec(a_(2))+......+m_(n)vec(a_(n))`
But `m_(1)vec(a_(1))=vec(F_(1)),m_(2)vec(a_(2))=vec(F_(2)),.........m_(n)vec(a_(1))=vec(F_(n)),MvecA=vec(F_(1))+vec(F_(2))+......+vec(F_(n))=vec(F_(ext))`
Where `F_(ext)` represents the sum of all external forces acting on the particles of the system.
This equation states that the centre of mass of a system of particles moves as if all the mass of the system was concentrated at the centre of mass and all the external forces were applied at that point.