Angular Momentum: Is L = Iw True for Extended Bodies?

[asdf60]

Is angular momentum defined as L = Iw for extended bodies?

Take for example two masses attached to two ends of a massless stick. The stick is rotated at a constant frequency around the cm, except, at an angle with the stick, so the basically the stick revolves a cone. It's not hard to see that L does not point in the same direction as w, and torque must be applied to make it continue to rotate in the same fashion.

Why is this the case? Is it the (ficticious) centripedal force that puts a torque on the system?

Now, if we replaced the stick-mass object with, say a pencil, then it seems to me the analysis would be about the same, with the angular momentum NOT pointing in the same direction as w, so L = Iw is NOT true.

So then I'm confused as to what L = Iw even means and when it makes sense.

 

[Galileo]

In general, I is a tensor, known as the inertia tensor and the equation becomes:
$$\vec L = \mathbf{I} \cdot \vec \omega$$
or
$$L_k=\sum_{i=1}^3 w_l I_{kl}$$
as is (probably) explained in any classical mechanics textbook.