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. `1/2MR^(2)`
C. `(MR)/(2lambda)`
D. `(MR)/(5lambda)`

Answer

A

Explanation

Divide the ring into infinitely small lengths of mass dm. even though mass distribution in non-uniform, each mass `dm_(1)` is at same distance `R` from origin
`:' MI` of ring about `z`-axis is
`=dm_(1)R^(2)+dm_(2)R^(2)+....dm_(n)R^(2)=MR^(2)`