A rod of mass `M kg` and length `L m` is bent in the from of an equilateral tringle as shown in Gig. The moment of inertia of the triangle about a vertical axis perpendicular to the plane of the triangle and passing through the centre (in units of `kg m^2`) is.
.

Options

A. `( ML ^(2))/( 12)`
B. `( ML ^(2))/( 54)`
C. `(ML ^(2))/( 162)`
D. `(ML ^(2))/( 10 8)`

Answer

B

Explanation

Moment of inertia of any one side about its perpendicular bisector is
`I = I _(cm) + Md ^(2)`
`I = [ (1)/(12) ((M )/( 3)) ( ( L )/(3)) ^(2) + ((M )/(3)) (((L )/( 3))/( 2 sqrt3)) ^(2) ]`
`I = ( ML ^(2))/( 162)`
Moment of Inertia of a triangle
`= 3 I = ( ML ^(2))/( 54)`