A ring of mass `M` and radius `R` lies in `x-y` plane with its centre at origin as shown. The mass distribution of ring is non uniform such that, at any point `P` on the ring, the mass per unit length is given by `lamda = lamda_0 cos^2 theta` (where `lamda_0` is a positive constant). Then the moment of inertia of the ring about z-axis is :
.

Options

A. `MR^2`
B. `(1)/(2) MR^2`
C. `(1)/(2) (M)/(lamda_0) R`
D. `(1)/(pi) (M)/(lamda_0) R`

Answer

A

Explanation

(a) Divide the ring into infinitely small lengths of mass `dm_i`. Even through mass distribution is non-uniform, each mass `dm_i` is at same distance `R` from origin.
:. `MI` of ring about z-axis is
=`dm_ R^2 + dm_2 R^2 + …+ dm_n R^2 = MR^2`.