A uniform rod is kept vertically at a point O on a smooth...

Since the horizontal surface is smooth, there is no horizontal force acting on the rod.

The initial horizontal velocity of the center of mass is zero. Therefore, the horizontal position of the center of mass remains unchanged during the fall.

Initially, the rod is placed vertically at point $O$, so its center of mass lies exactly above $O$.

When the rod falls and lies completely on the horizontal surface, its center of mass will be at point $O$.

Since the rod is uniform, its center of mass is located at its midpoint, which is at a distance $l/2$ from either end.

Thus, the lower end of the rod will be at a distance $l/2$ from $O$.