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

Explanation

Consider a system of particles of masses `m_(1),m_(2),……m_(n)` moves with velocity vectors `overline(v_(1)),overline(v_(2)),overline(v_(3))......,overline(v_(n))`
`therefore` Velocity of centre of mass
`overline(V)_(CM)=(1)/(M){m_(1)overline(v_(1))+m_(2)overline(v_(2))+...+m_(n)overline(v_(n))}`
`becausem_(1)+m_(2)+.......+m_(n)=M`
We know
`thereforeoverline(a)=(d)/(dt)(overline(V))`
`therefore` Acceleration of centre of mass
`a_(CM)=(d)/(dt)(overline(V)_(CM))`
`thereforea_(CM)=(1)/(M){m_(1)(d)/(dt)overline(v_(1))+m_(2)(d)/(Dt)overline(v_(2))+....+m_(n)(d)/(dt)overline(v_(n))}`
But Force (F) = ma , so total force on the body is
`F=Ma_(C.M)=m_(1)a_(1)+m_(2)a_(2)+m_(3)a_(3)+......+m_(n)a_(n)`
or Total Force `F=Ma_(C.M)=F_(1)+F_(2)+F_(3)+.....+F_(n)`
Hence ,total force on the bosy is the sum of forces on individual particles and it is equals to force on centre of mass of the body.