`AB` is a light rigid rod. Which is rotating about a vertical axis passing through `A,A` spring of force constant `K` and natural length `l` is attached at `A` and its other end is attached to a small bead of mass `m`. The bead can slide without friction on the rod. At the initial moment the bead is at rest (w.rt. the rod) and the spring is unstreached Select correct option

Options

A. The maximum velocity attained by the bead w.r.t. the rod is given by `V_("max")=sqrt((momega^(4)l^(2))/(k-momega^(2)))`
B. The maximum velocity attained by the bead w.r.t the rod is given by `V_("max")=sqrt(((momega^(4)+K)/(momega^(2)-K))omega^(2)l^(2))`
C. The maximum extension in the spring in given by `X_("max")=(2momega^(2)l)/(K-momega^(2))`
D. The maximum value of contact force between the bead and the rod is greater than `mg`

Explanation

Velocity will be maximum at equilibrium position

`rArr momega^(2)(l+x)=kx`
`rArr x=(momega^(2)l)/(K-momega^(2))`
`(1)/(2)mV_("max")^(2)=underset(0)overset(x)intmomega^(2)(l+x)dx-(1)/(2)kx^(2)`
`rArrV_("max")=(2momega^(2)lx+momega^(2)x^(2)-kx^(2))/(m)`
`V_("max")^(2)=((momega^(2)l+momega^(2)(l+x)-kx)x)/(m)`
`rArrV_("max")^(2)=omega^(2)l=(momega^(4)l^(2))/(momega^(2)-K)`
`V_("max")=sqrt((momega^(4)l^(2))/(k-momega^(2)))`
For maximum extension
`underset(0)overset(x)intmomega^(2)(l+x)dx-(1)/(2)kx_("max")^(2)=0 rArr x_("max")=(2momega^(2)l)/(k-momega^(2))`