`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 is 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

Answer

A, C, D

Explanation

At equilibrium forces balance `m(l+x)omega^(2)=Kximpliesx=(mlomega^(2))/(K-momega^(2))`
Also till this point by work energy theorem `underset(0)overset(x)intm(l+x)omega^(2)dx-(Kx^(2))/(2)=(1)/(2)mv^(2)`
`V=sqrt((ml^(2)omega^(4))/(K-momega^(2)))`
and at maximum elongation velocity becomes zero apply above concept of work & energy from start to zero velocity.