The `KE` and moment of inertia about the given end point of a rod of mass `m` and length `l` and cross sectional area `A` which is rotating with `omega=sqrt(g/l)` as shown in figure. Will be [density of the rod varies as `rho=rho_(0)(1+x/l),x` is the distance measured from `O`)n ]

Options

A. `KE=7/36 mgl`
B. `I=(7rho_(0)Al^(3))/36`
C. `KE=7/39mgl`
D. `I=(11rho_(0)Al^(3))/36`

Answer

A, B

Explanation

`I=int_(0)^(L)1/3d mx^(2)`
`=int_(0)^(l)1/3 rho_(0)(1+x/l)(Adx)x^(2)=1/3rho_(0)Aint_(0)^(l)[x^(2)dx+(x^(3))/l dx]`
`=1/3rho_(0)A[(l^(3))/3+1/lxx(l^(4))/4]=(7rho_(0)Al^(3))/36`