Component of angular momentum perpendicular to rotation axis

[Soren4]

Homework Statement

Consider the rigid body in the picture, rotating about a fixed axis $z$ not passing through a principal axis of inertia, with an angular velocity $\Omega$ that can vary in magnitude but not in direction. Find the angular momentum vector and its component parallel to $z$ axis ($\vec {L_z }$) and perpendicular to it ($\vec {L_n }$). How do this two vectors vary in time? And if a torque is exerted? Explain how is the angular acceleration related to the variation of the components of angular momentum.

(I start by saying that I'm totally ok with the calculation of angular momentum and the properties of its component parallel to the $z$ axis ($\vec {L_z }$). My difficulties are in understanding what are the properties of $\vec {L_n }$.)

Homework Equations

From the picture we have that $| \vec {L_ {n, i}} | = m_i r_i R_i \Omega cos \theta_i \implies | \vec {L_n} | = \Omega \sum m_i r_i R_i cos \theta_i$ (1)

The Attempt at a Solution

Firstly, does it follow from (1) that $| \vec {L_n} | \propto | \vec {\Omega} |$ (2)?

If this is true, suppose to apply a torque perpendicular to the $z$ axis and parallel to $\vec {L_n}$, so that the magnitude of this vector increases. Follows from (2) that there should be an angular acceleration $\vec {\alpha}$, although we are in the absence of a torque with an axial component.
This would go against the fact that $I_z \vec {\alpha} = \vec { M_z}$ (Where $I_z$ is the moment of inertia with respect to the $z$ axis and $M_z$ is the axial component of the exerted torque). How is this possible?

 

[nilesh_pat]

If (2) is not true, then what can be said about the variation of \vec {L_n} in time? Does it change if a torque is applied perpendicular to the z axis?Finally, if a torque is applied with an axial component, how is the angular acceleration related to the variation of the components of angular momentum?