I Rotation about two axes and angular momentum

AI Thread Summary
The discussion focuses on the changes in angular momentum along the z-axis for a body with an initial angular velocity, considering fixed axes in inertial space. It highlights confusion regarding the contribution to the change in Lz due to rotation about the y-axis, specifically how Lx remains constant in magnitude while affecting Lz. The participants question the accuracy of the assumed motion, noting that the body does not simply rotate around the y-axis while maintaining its spin. There is speculation about whether Lz can be expected to remain zero or exhibit periodic behavior. Overall, the conversation emphasizes the complexities of angular momentum conservation in multi-axis rotations.
Kashmir
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IMG_20210709_103319.JPG

I've a body having initial angular velocity at ## t=0 ## as shown. The axis shown are fixed in inertial space and initially match with the principal axis. I want to find the infinitesimal change at ##t+\Delta t## in the angular momentum along the ##z## axis.

I've seen the following approach which I don't understand:
One contribution to change in ##L_z## is due to rotation about y axis. This causes ##L_x## to rotate and hence a component ##-L_x \Delta{_y}## appears.
IMG_20210709_105348.JPG

How do we know that ##Lx## will remain constant in magnitude? Also the actual motion won't be as is shown, in which the body simply goes around the y-axis while maintaining it's spin ##L_x##

A similar method is used here by Kleppner and Kolenkow here
IMG_20210709_112436.JPG
 
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May we expect Lz=0? At least periodic it seems.
 
anuttarasammyak said:
May we expect Lz=0? At least periodic it seems.
Initially?
 
Yes, and I assume ##\mathbf{L}=(L_x,L_y,0)## is conserved in later time.
 
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