From a complete ring of mass `M` and radius `R`, a `30^@`sector is removed. The moment of inertia of the incomplete ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is
,

Options

A. `(1)/(12) M (6R^(2) + L^(2))`
B. `(1)/(12) M (18 R^(2) + L^(2))`
C. `(1)/(3) M (6R^(2) + L^(2) + 3LR)`
D. None of these

Answer

C

Explanation

MOI of system `I= I_("ring") + I_("rod")`
`= MR^(2) + [(ML^(2))/(12) + M ((L)/(2) + R)^(2)]`
`= MR^(2) + (ML^(2))/(12) + (ML^(2))/(4) + 2M (L)/(2) R + MR^(2)`
`=2MR^(2) + (1)/(3) ML^(2) + MLR = (1)/(3) M [6R^(2) + L^(2) + 3LR]`