I have difficulties in understanding, in the rotation of a rigid body, the properties of the component of the angular momentum vector →L which is perpendicular to the fixed axis of rotation z. I will call this component →Ln. Suppose that the angular velocity Ω is constant in direction but can vary in magnitude.
|→Ln,i|=miriRiΩcosθi⟹|→Ln|=Ω∑miriRicosθi
Can I therefore say that |→Ln|∝|→Ω| (1)?
If so, suppose to apply a torque perpendicular to the z axis and parallel to →Ln, so that the magnitude of this vector increases. Follows from (1) that there should be an angular acceleration →α, although we are in the absence of a torque with an axial component.
This would go against the fact that Iz→α=→Mz (Where Iz is the moment of inertia with respect to the z axis and Mz is the axial component of the exerted torque).
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