A smooth light horizontal \( \operatorname{rod} A B \) can rotate about a vertical axis passing ...

A smooth light horizontal \( \operatorname{rod} A B \) can rotate about a vertical axis passing through its end \( A \). The rod is fitted with a small sleeve of mass \( m \) attached to the end \( A \) by a weightless spring of length \( I_{0} \) and stiffness \( x \). What work must be performed to slowly get this system going and reaching the angular velocity \( \omega \) ?