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 fir 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.

Answer

6

Explanation

For entering without jerk `v_(2) = lomega_(0) = 2ms^(-1)`
Using work-energy theorem on the sleeve after entering in the frame of rod

`W_("spring") + W_("centrifugal") = DeltaK`
`-1/2 kl^(2) - 1/2 momega^(2)l^(2) = 0 - 1/2m_(1)^(2)`
`rArr v_(1)^(2) = 8`
Now, `K = 1/2 m (v_(1)^(2) + v_(2)^(2)) = 6`