The density of a linear 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 `x`-axis be along the length of the rod and origin at one of its ends. As rod is along `x`-axis, for all points on it `y` and `z` coordinates are zero.

Centre of mass will be on the rod. Now consider an element of rod of length `dx` at a distance `x` from the origin, then `dm=lambdadx=(A+Bx) dx`
`x_(CM)=(int_(0)^(L)xdm)/(int_(0)^(L)dm)=(int_(0)^(L)x(A+Bx)dx)/(int_(0)^(L)(A+Bx)dx)`
`x_(CM)=((AL^(2))/2+(BL^(3))/3)/(AL+(BL^(2))/2)=(3AL+2BL^(2))/(3(2A+BL))=(L(3A+2BL))/(3(2A+BL))`