Velocity of end B of the rigid rod

To determine the velocity of end B of the rigid rod, we need to consider the motion of the rod and the constraints imposed by the vertical walls. Let's analyze the situation step by step.

1. Assumptions:
- The rod is rigid, meaning that its length does not change.
- The vertical walls are smooth, implying that there is no friction between the rod and the walls.
- We assume that the rod is in equilibrium, and there are no external forces acting on it.

2. Motion of the rod:
Since end A of the rod is moving with velocity u along the vertical direction, the entire rod will have a vertical velocity component. This means that the rod is rotating about a fixed point where it touches the vertical walls.

3. Constraint at end A:
Since end A is moving vertically, it will have a non-zero vertical velocity. However, the rod is in contact with the vertical walls at end A, and these walls impose a constraint on the motion of the rod. As a result, the vertical velocity of end A will be zero with respect to the walls.

4. Constraint at end B:
The constraint at end A affects the motion of the entire rod. As a consequence, end B of the rod must also have a vertical velocity of zero with respect to the vertical walls. This means that end B will also move vertically with velocity u, just like end A.

5. Conclusion:
In conclusion, the velocity of end B of the rigid rod will always remain constant and equal to the velocity u along the vertical direction. The constraints imposed by the smooth vertical walls ensure that both end A and end B move vertically with the same velocity, despite the rotation of the rod about the point of contact with the walls.