The linear density of a thin rod of length 1m lies as `lambda = (1+2x)`, where x is the distance from its one end. Find the distance of its center of mass from this end.

Explanation

Let us consider a small length dx of the rod shown in the figure.

Its mass `dm = lambda dx = (1+2x)dx`
`X_(cm)=(int x dm)/(int dm)=(int_(0)^(1)x(1+2x)dx)/(int_(0)^(1)(1+2x)dx)=([(x^(2))/(2)+(2x^(3))/(3)]_(0)^(1))/([x + x^(2)]_(0)^(1))=(7)/(12)m`