A rod of length `l=2m` is maintained to rotate with a constant angular velocity `omega=1 rad//s` about vertical axis passing through one end (fig). There is a spring of spring constant `k=1 N//m` which just encloses rod inside it in natural length. One end of the spring is attached to axis of rotation. `S` is sleeve of mass `m=1kg` which can just fix on rod. All surfaces are smooth. With what minimum kinetic energy (in `J`) sleeve should be projected so that it enters on the rod without impulse and completely compresses the spring.

Explanation

For entering without jerk `v_(2)=lomega_(0)=2rad//s`
Using work energy theorem on sleeve after entering in the frame of rod.
`W_("spring")+W_("centrifugal")=DeltaK`
`-(1)/(2)kl^(2)-(1)/(2)momega^(2)l^(2)=0-(1)/(2)mv_(1)^(2)rArrv_(1)^(2)=8`
Now `K=(1)/(2)m(v_(1)^(2)+v_(2)^(2))=6`