The density of a rod of length L varies as `rho=A+Bx` where x is the distance from the left end. Locate the centre of mass.

Explanation

Let the cross sectioN/Al area be `alpha`. The mass of an element of length dx located at a distance x away from the left end is `(A+Bx) alphadx`. The x coordilN/Ate of the centre of mass is given by
`X_(CM)=(intx dm)/(int dm)=(int_0^Lx(A+Bx)alpha dx)/(int_0^L(A+Bx)alphaxd)`
`=(AL^2/2+BL^3/3)/(AL+BL^2/2)=(3AL+2BL^2)/(3(2A+BL))`