Four identical rods each of mass `m` and length `L` are joined to form a square. Find the `M.I.` about
(a) the `x`-axis,
(b) the `y`-axis,
(c) the `z`-axis,
(d) about `1-1`,
(e) an axis passing through one corner and perpendicular to the plane.

Explanation

(a) About the `x`-axis
Rods (`2`) and (`4`) each `=(mL^(2))/(12)`
Rods (`1`) and (`3`) each ` = [0+m((L)/(2))^(2)]=(mL^(2))/(4)`
`I_(x)=I_(2)+I_(4)+I_(1)+I_(3)`
` =(mL^(2))/(12)xx2+(mL^(2))/(4)xx(2)/(3)mL^(2)`
(b) `I_(y)=I_(x)=(2)/(3)mL^(2)` (due to symmetry)
(c) `I_(z)=I_(x)+I_(y)=(4mL^(2))/(2)` (`bot^(ar) ` axes theorem)
(d) `I_(1-1)=I_(1)+(I_(2)+I_(4))+I_(3)`
`= 0+(mL^(2))/(3)xx2+{0+mL^(2)}=(5)/(3)mL^(2)`
(e) An axis passing through `A` and `bot^(ar)` to the plane
`I_(A)=(I_(3)+I_(4))+(I_(1)+I_(2))`
`= (mL^(2))/(3)xx2+{(mL^(2))/(12)+m((sqrt5L)/(2))^(2)}xx2`
`= (10)/(3)mL^(2)`
OR
`O` is the center of mass system of rods
`I_(A)=(I_(z))_(c.m.)+4m((L)/(sqrt2))^(2)`
`= (4)/(3)mL^(2)+2mL^(2)=(10)/(3)mL^(2)`
For continuous or Distributed Mass
`I=int r^(2) dm`
`dm`: mass element
`r: bot^(ar)` distance of element from axis