The linear mass density i.e. mass per unit length of a rod of length `L` is given by `rho = rho_(0)(1 + (x)/(L))`, where `rho_(0)` is constant , `x` distance from the left end. Find the total mass of rod and locate `c.m.` from the left end.

Explanation

Take a small length dx at a distance x from O.
Mass of element
`dm=rhodx`
Total mass of rod
`m=intdm=intrhodx=int_(0)^(L)intrho_(0)(1+(x)/(L))dx`
`=p_(0)|x+(x^(2))/(2L)|_(0)^(L)=rho_(0)(L+(L)/(2))=(3rho_(0)L)/(2)=(3rho_(0)L)/(2)`
`x_(c.m.)=(intxdm)/(intdm)=(int_(0)^(L)xrho_(0)(1+(x)/(L))dx)/((3rho_(0)L)/(2))`
`=(int_(0)^(L)(x+(x^(2))/(L))dx)/((3L)/(2))=(|(x^(2))/(2)+(x^(3))/(2L)|_(0)^(L))/((3L)/(2))`
`=((L^(2))/(2)+(L^(3))/(3L))/((3L)/(2))=((5L^(2))/(6))/((3L)/(2))=(5L)/(9)`