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.

 

[charng]

I can confirm that the equation L = Iw is not always true for extended bodies. This equation is derived from the definition of angular momentum, which is the product of the moment of inertia (I) and the angular velocity (w). However, this definition assumes that the object has a fixed axis of rotation and that all of its mass is concentrated at that axis.

In the example given, the extended body (the stick) does not have a fixed axis of rotation, as it is rotating around a cone. Additionally, the mass is not concentrated at a single point, as it is spread out along the length of the stick. In this case, the equation L = Iw would not accurately describe the angular momentum of the system.

The reason for this is that the angular momentum of an extended body is not solely determined by its moment of inertia and angular velocity. Other factors such as the distribution of mass and the direction of rotation also play a role. In the case of the stick, the mass is not evenly distributed and the direction of rotation is constantly changing, which affects the angular momentum.

In summary, the equation L = Iw is a simplified version of the definition of angular momentum and is only applicable in certain situations, such as a point mass rotating around a fixed axis. For extended bodies with varying mass distribution and rotation, this equation may not accurately describe the angular momentum of the system.