Work done by constraints on rotating rigid bodies I am trying to understand why constraint forces do no work on extended, rotating bodies. For instance, consider the problem of a rigid rod falling on a frictionless surface (K&K 7.17)

There are no horizontal forces, so the work-energy theorem says that
$$\Delta K=\int_{y_0}^{y_1}\mathbf{F}\cdot dR+\int_{\theta_0}^{\theta_1}\tau\cdot d\theta$$
Now in non-rotating bodies, it's obvious that the normal force does no work, but clearly here the normal force is applying a force upward while the center of mass is moving in the same direction, doing negative work. At the same time, the contact point is exerting a torque on the rod, increasing the angular kinetic energy. It seems likely that these two contributions cancel each other, but I don't know how to prove or even intuit that this is true as a general rule. Why are we permitted to assume such a force is conservative?
 A: I think you're 'overthinking' a bit.
Your formula for $\Delta K$ counts the work twice.
As the CoG of the object has been lowered by $\frac{\ell}{2}$, the reduction in potential energy is simply:
$$W=\Delta K=\Delta U=mg\frac{\ell}{2}$$
We can also calculate it by the work done by the torque:
$$W=\int_{path}\tau \mathrm{d}\theta$$
$$W=\int_0^{\pi/2}mg\frac{\ell}{2}\sin\theta \mathrm{d}\theta$$
$$W=mg\frac{\ell}{2}\Big[-\cos\theta\Big]_0^{\pi/2}=mg\frac{\ell}{2}(0-(-1))$$
$$W=mg\frac{\ell}{2}$$
These are equivalent ways of calculating $W=\Delta K=\Delta U$. If the object was both free falling and rotating we would have to add these energies but here the object is rotating only, not translating too.
The normal force (in the right hand 'resting point') as such performs no work because the point doesn't move in the direction of the force.
A: I'm not exactly sure what you are aiming at with your question. We can strictly prove that the work done by constraint forces vanishes only for the case of particles (point systems), in which case the conclusion is near-trivial: These forces eliminate movement in the direction of the constraint, so they cannot perform work on the system. For extended systems, on the other hand, it is not always true that constraint forces do not perform any work. Wikipedia links to Goldstein's Classical Mechanics, 3rd Edition, page 16, for an example.
