Derive an expression for center of mass for distributed point masses.

Explanation

(i) Consider the point masses `m_(1), m_(2), m_(3), ... m_(n)` whose position coordinates from orgin O along X-direction are `x_(1), x_(2), x_(3),...x_(n)` as shown in figure.
(ii) The equation for the X coordinate of the center of mass is,
`x_(CM)=(sum m_(i)x_(i))/(sum_(m_(i))`
where `sum m_(i)=M`, is the total mass of all the particles.
(iii) Hence `x_(CM)= (sum m_(i)x_(i))/(M)`
(iv) Similarly, the Y and Z coordinates of center of mass can be written as,.
`y_(CM)=(sum m_(i)y_(i))/(M), z_(CM)=(sum m_(i)z_(i))/(M)`
(v) The position of the center of mass of these masses is `(X_(CM), Y_(CM), Z_(CM))`. In general, the position of center of mass in vector form can be expressed as,
`vecr_(CM)=(sum m_(i)vecr_(i))/(M)`
where `vecr_(CM)=x_(CM)hati_y_(CM)hatj+z_(CM)hatk` is the position vector of the center of mass and `vecr_(i)=x_(i)hati+y_(i)hatj+z_(i)hatk` is the position vector of the distributed point mass.