Question

Three identical roads each of mass `M` and `L` are joined to form an equilateral triangle. Find the `M.I.` about an axis perpendicular to the plane of triangle and passing through
(`a`) center of triangle
(`b`) mid-point of a rod and
(`c` ) one vertex of triangle.

Answer 1

(`a`) Axis is passing through `O` and `bot^(ar)` to the plane. `M.I.` of rod (`1`), about and axis passing through `D` and `bot^(ar)` to the length (`bot^(ar)` to the plane) `=(ML^(2))/(12)`

`M.I.` of rod (`1`), about axis through `O`
`I_(1)=(ML^(2))/(12)+Md^(2)=(ML^(2))/(12)+M((1)/(2sqrt3))^(2)=(ML^(2))/(6)`
`{In DeltaODC, tan30^(@)=(d)/(L//2)impliesd=(L)/(2sqrt3)}`
Similar anyalysis for two other rods
`I_(0)=3I_(1)=(ML^(2))/(2)`
(`b`)
Axis is passing through `D` and `bot^(ar)` to the plane
`I_(D)=I_(1)+I_(2)+I_(3)`
`I_(1)=(ML^(2))/(12)`
`I_(2)=I_(3)=(ML^(2))/(12)+M((L)/(2))^(2)=(ML^(2))/(3)`
`I_(D)=(ML^(2))/(12)+(ML^(2))/(3)xx2=(3)/(4)ML^(2)`
(If we join the mid-point of equilateral triangle of side `L`, we get another equilateral triangle of side `L//2`.)
(`c`)
Axis is passing through `A` and `bot^(ar)` to the plane. For rods (`2`) and (`3`), the axis through `A` is an axis passing through the end of rod and `bot^(ar)` to the length
`I_(2)=I_(3)=(ML^(2))/(3)`
For rod (`1`)
`I_(1)=(ML^(2))/(12)+M((sqrt3L)/(2))^(2)=(5ML^(2))/(6)`
`I_(A)=I_(1)+I_(2)+I_(3)=((5)/(6)+(1)/(3)+(1)/(3))=(9ML^(2))/(6)=(3ML^(2))/(2)`
Alternatively for parts (`b`) and (`c`)

`O` is the center of mass of system of rods
`I_(D)=I_(O)+3M((L)/(2sqrt3))^(2)=(ML^(2))/(2)+3M(L^(2))/(12)=(3ML^(2))/(4)`
`I_(A)=I_(O)+3M((L)/(sqrt3))^(2)=(ML^(2))/(2)+3M(L^(2))/(3)=(3ML^(2))/(2)`