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.
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Options

A. `(ML^2)/(12)`
B. `(ML^2)/(54)`
C. `(ML^2)/(162)`
D. `(ML^2)/(108)`

Answer

B

Explanation

(b) Mass of each side `= (M)/(3)`
`x = (1)/(3)[(L)/(3) cos 30^@] = (L//3)/(2 sqrt(3))`
moment of interia of one side about `O` :
`I = [(1)/(12) (M)/(3) ((L)/(3))^2 + (M)/(3) x^2] = (ML^2)/(162)`
moment of interia of whole triangle about
`O : 3 I = (ML^2)/(54)`.
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