Two thin uniform rods `A` (`M.L`) and `B` (`3M,3L`) are joined as shown. Find the `M.I.` about an axis passing through the center of mass of system of rods and perpendicular to the length.

Explanation

First, locate the center of mass taking origin at `O`.

`x_(c.m)=(Mxx(L)/(2)+3Mxx(L+(3L)/(2)))/(M+3M)=2L`
`A` is `c.m.` is system of rods

`I_(A)=I_(1)+I_(2)=[(ML^(2))/(12)+M(2L-(L)/(2))^(2)]+[(3m(3L^(2)))/(12)+3m((5L)/(2)-2L)^(2)]`
`ML^(2)[{(1)/(12)+(9)/(4)}+{(27)/(12)+(3)/(4)}]=(16ML^(2))/(3)`