A circular disc of radius b has a hole of radius a at its centre (see figure) if the mass per unit area of the disc varies as `((sigma_(0))/(r))` then the radius of gyration of the disc about its axis passing through the centre is

Options

A. `(a+b)/2`
B. `sqrt((a^(2) + b^(2) + ab)/2)`
C. `sqrt((a^(2) + b^(2) + ab)/3)`
D. `(a+b)/3`

Answer

C

Explanation

`I = int dm r^(2) rArr int_(a)^(b) (sigma_(0)/r 2pi r dr)r^(2) rArr I=2 sigma_(0)pi (b^(2)-a^(2))/3`

`rarr I=mk^(2) rArr 2sigma_(0)pi (b^(3) -a^(3))/3 = 2sigma_(0) pi(b-a)K^(2)`
`rArr k= sqrt((a^(2) + b^(2) + c^(2))/3`