Write the expression of centre of mass of a system of .n. particles and derive the formula of force acting on its centre of mass.

Explanation

The centre of mass of a system of .n. particle
`vec(v)_(cm)orvecV=(m_(1)vec(v_(1))+m_(2)vec(v_(2))+....m_(n)vec(v_(n)))/(m_(1)+m_(2)+...m_(n))`
`therefore MvecV=m_(1)vec(v_(1))+m_(2)vec(v_(2))+...m_(n)vec(v_(n))...(1)`
Where `M=m_(1)_m_(2)+...,m_(n)` total mass of a system.
Assume that the value of masses does not change with time, then differentiate equation (1) w.r.t. time,
`M(dvecV)/(dt)=m_(1)(dvec(v_(1)))/(dt)+m_(2)(dvec(v_(2)))/(dt)+...m_(n)(dvec(v_(n)))/(dt)`
but `(dvec(V))/(dt)=vecA` acceleration of centre of mass
`dvec(v_(1))/(dt)=vec(a_(1))` acceleration of frist particle
`dvec(v_(2))/(dt)=vec(a_(2))` acceleration of second particle and
`dvec(v_(n))/(dt)=vec(a_(n))` acceleration of .n. particle
`therefore MvecA=m_(1)vec(a_(1))+m_(2)vec(a_(2))+...m_(n)vec(a_(n))...(2)`
`vec(a)_(cm)orvec(A)=(m_(1)vec(a_(1))+m_(2)vec(a_(2))+...m_(n)vec(a_(n)))/(m_(1)+m_(2)+....m_(n))`
is a formula for acceleration of centre of mass.
From Newton.s second law, `vec(F_(i))=m_(1)vec(a_(i))` but `Mvec(A)=vec(F_(1))+vec(F_(2))+...vec(F_(n))`
Hence, the vector sum of all the forces acting on the all particles of a system is equal to the product of total mass of system and the acceleration of its centre of mass.