Four identical spheres each of mass `M` and radius `R` are placed on the vertices of a square of edge `L`. Find the `M.I.` about an axis
(`a`) coinciding with the side of a square and
(`b`) coinciding with the diagonal of a square.

Explanation

(a)
`I_(1-1)=?`
For two spheres `A` and `B` , we have direct formulae and for `C` and `D`, we will use the parallel axes theorem
`I_(1-1)=I_(A)+I_(B)+I_(C)+I_(D)`
`M.I.` of `A` and `B` about `1-1=(2)/(5)MR^(2)`
`M.I.` of `C` and `D` about `1-1` is
`(I_(c.m))+Md^(2)=(2)/(5)MR^(2)+ML^(2)`
`I_(1-1)=(2)/(5)MR^(2)xx2+((2)/(5)MR^(2)+ML^(2))xx2`
`=(8)/(5)MR^(2)+2ML^(2)`
(b)
`I_(1-1)=I_(B)+I_(D)+I_(A)+I_(C)`
`=(2)/(5)MR^(2)xx2+[(2)/(5)MR^(2)+M((L)/(sqrt2))^(2)]xx2`
`=(8)/(5)MR^(2)+ML^(2)`