I'm studying the next problem what I found in a book.
A stick of length $l$ and mass $M$, initially upright on a frictionless table, starts falling. The problem is to find the speed of the center of mass as a function of the angle $\theta$ from the vertical.
I have analyzed the solution using energy methods, and I understand the problem almost completely, but I still do not understand why the normal force does not exert work.
Part of the solution that I found is below:
The key lies in realizing that because there are no horizontal forces, the center of mass must fall straight down. Since we must find velocity as a function of position, it is natural to apply energy methods.
The sketch shows the stick after it has rotated through angle $\theta$ and the center of mass has fallen distance $y$.
The initial energy is
The kinetic energy at a later time is
and the corresponding potential energy is
Because there are no dissipative forces, mechanical energy is conserved and $K+U=K_0+U_0=Mgl/2$. Hence
$$\frac12M\dot y^2+\frac12l_0\dot\theta^2+Mg\left(\frac12-y\right)=Mg\frac l2$$
We can eliminate $\dot\theta$ by using the constraint equation. The sketch shows that
$$\dot y=\frac l2\sin\theta\;\dot\theta$$
Question: Why does the normal force not do any work?