Three particles each of mass `m` are placed on the vertices of an equilateral triangle of length `L`. Find the `M.I`.
(`a`) about an axis passing through
(`i`) one particle and perpendicular to the plane
(`ii`) mid-point of the side of triangle and `bot^(ar)` to the plane
(`iii`) center of triangle and `bot^(ar)` to the plane
(`b`) (`i`) an axis coinciding with side of triangle
(`ii`) about median of triangle

Explanation

(a) (i) An axis passing through `A` and `bot^(ar)` to the plane
`I=mxx0+mL^(2)+mL^(2)=2mL^(2)`
(ii) An axis passing through `D` and `bot^(ar)` to the plane
`I=m((L)/(2))^(2)+m((L)/(2))^(2)+m((sqrt3L)/(2))^(2)=(5)/(4)mL^(2)`
(iii) An axis passing through `O` (center of triangle) and `bot^(ar)` to the plane
`r=(L//2)/(cos30^(@))=(L)/(sqrt3)`
`I=mr^(2)xx3=m((l)/(sqrt3))^(2)xx3=mL^(2)`
(b) (i)
`I=mxx0+mxx0+m((sqrt3L)/(2))^(2)=(3mL^(2))/(4)`
(ii)
`I=mxx0+m((L)/(2))^(2)+m((L)/(2))^(2)=(mL^(2))/(2)`