The linear mass density (i.e. Mass per unit length) of a rod of length `L` is given by `rho=rho_(0)(x)/(L)`, where `rho_(0)` is constant and `x` is the distance from one end `A`. Find the `M.I.` about an axis passing through `A` and perpendicular to length of rod. Express your answer in terms of mass of rod `M` and length `L`.

Explanation

`dm=rhodx=(rho_(0)x)/(L)dx`
`I=int x^(2) dm = (rho_(o))/(L) int_(0)^(L) x^(3) dx`
`= (rho_(0))/(L)|(x^(4))/(4)|_(0)^(L)=(rho_(0)L^(3))/(4)`
Mass of rod
` M=int dm=(rho_(0))/(L) int_(0)^(L) x dx`
` =(rho_(0))/(L)|(x^(2))/(2)|_(0)^(L)=(rho_(0)L)/(2)`
`rho_(0)=(2M)/(L)`
`I=(rho_(0)L^(3))/(4)=(2M)/(L)xx(L^(3))/(4)=(ML^(2))/(2)`