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

Explanation

A point mass is a hypothetical point particle which has non-zero mass and no size or shape. To find the center of mass for a collection of n point masses, say, `m._(1),m_(2),m_(3),m_(3).....m_(m)` we have to first choose an origin and an appropriate coordinate system as shown in Figure. Let `x_(1),x_(2),x_(3),....x_(n)` be the X-coordinates of the positions of these point masses in the X direction from the origin.
The equation for the X coordinate of the center of ass is
`x_(CM)=(summ_(i)x_(xi))/(sum m_(i))`
where, `sum mi_(i)` is the total mass M of all the
particles, `(sum m_(i)=M)`. Hence,
`x_(CM)=(sum m_(i)x_(i))/(sum M)`
Similarly, we can also find y and coordinates of the center of mass for these distributed point masses as indicated in figure.
`y_(CM)=(sum m_(i)y_(i))/(sumM)`
`z_(CM)=(sum m_(i)z_(i))/(sumM)`
Hence, the position of center of mass of these point masses in a Cartesian coordinate system is `(x_(CM), y_(CM) and z_(CM))`. In general, the position of center of mass can be written in a vector form as,
`vec r_(CM)= (sum m_(i) vecr_(i))/(M)`
where `vecr_(CM)=x_(CM) hatj+y_(CM)hatj+z_(CM)hatk` is is the position vector of the distributed point mass and `vecr_(i)=x_(i)hati+y_(i)hatj+z_(i)hatk` is the position vector of the distributed mass, where `hati, hatj and hatk` are the unit vectors along X, Y and Z-axis respectively.