Problem 1.272

A smooth uniform rod $A B$ of mass $M$ and length $l$ rotates freely with an angular velocity $\omega_{0}$ in a horizontal plane about a stationary vertical axis passing through its end $A$. A small sleeve of mass $m$ starts sliding along the rod from the point $A$. Find the velocity $v^{\prime}$ of the sleeve relative to the rod at the moment it reaches its other end $B$.