Undergrad Is angular momentum perpendicular to fixed axis of rotation constant?

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Angular momentum perpendicular to a fixed axis of rotation is considered constant when torques acting on the system are ignored, assuming sufficient restraints. However, when a rod is hinged and allowed to rotate, the direction of its angular momentum can change at various points during its motion. The discussion raises the question of whether the torques acting perpendicular to the fixed axis also lie within the plane that is perpendicular to that axis. This highlights the complexity of angular momentum behavior in dynamic systems. Understanding these principles is crucial for analyzing rotational motion accurately.
TahirMaqbool
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TL;DR
Does the component of angular momentum perpendicular to the fixed axis of rotation change in direction or magnitude?
So my book states torques perpendicular to the fixed axis of rotation tend to tilt the axis , however we assume sufficient restraints exist so these torques are simply ignored.
It follows that angular momentum perpendicular to axis remians constant.
(See image )

My question is that if a rod is hinged at one of its ends and allows to rotate, wouldn't angular momentum perpendicular to axis change in direction at each point?
See image below.
 

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Are those “torques perpendicular to the fixed axis of rotation” also contained in the plane that is “perpendicular to the fixed axis of rotation”?
 

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