A thin circular plate of mass `M` and radius `R` has its density varying as `p(r)=p_(0) r` with `p_(0)` as constant and `r` is the distance from its center. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is `I=a MR^(2)`. The value of the coefficient a is:

Options

A. `1/2`
B. `3/2`
C. `8/5`
D. `3/5`

Answer

C

Explanation

`I_(CM) = int dm r^(2) = int_(0)^( r) (2pi rdr)(rho_(0)r)r^(2) =(2pi rho_(0)r^(5))/5`, `M=int dm = int_(0)^(r )2pi r dr rho_(0)r=(2pi rho_(0)R^(3))/3`
Using parallel axis theorem `I=I_(CM) + MR^(2)`
`=2/5 pi rho_(0)R^(5) + 2/3pi rho_(0)R^(5) = 16/15 pi rho_(0) R^(5) = 3/2 xx 16/15 xx 2/3 pi rho_(0)R^(5) = 8/5 MR^(2)`