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 \(B\).