newtonian mechanics - Why normal doesn't do any work in this case

Suppose the stick slides without friction along flat surface.

Why in this case N does not do any work? As i see stick not only slides in horizontal direction, but also falls in vertical direction, at least its COM falls.

EDIT: I confused by fact that object actually moves in N's direction. Its COM falls vertically.

Answer

Work is defined as $$W =\int F \cdot dx= \int F \cos {\theta}dx$$ This definition only holds for non rotating objects. When normal force is applied to the object sliding down, the displacement of the object occurs along the incline. So the angle between force applied and displacement is 90°.

Your question is explained because the work done is not because the object does not slide along the plane. You would also have to use concepts of rotational mechanics to calculate work. In rotational mechanics, $$ W = \int \tau d\theta +\int F \cdot dx$$ Where $\tau$ is torque about the axis through centre of mass. The work done by normal no longer zero as it generated torque.

Incidentally, the work fine would be zero if the object were to slide down a curved path as long as it slid along it.