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Component of angular momentum perpendicular to rotation axis
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[QUOTE="Soren4, post: 5430337, member: 589474"] [h2]Homework Statement [/h2] Consider the rigid body in the picture, rotating about a fixed axis [itex]z[/itex] not passing through a principal axis of inertia, with an angular velocity [itex]\Omega[/itex] that can vary in magnitude but not in direction. Find the angular momentum vector and its component parallel to [itex]z[/itex] axis ([itex] \vec {L_z } [/itex]) and perpendicular to it ([itex] \vec {L_n } [/itex]). 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. [ATTACH=full]98388[/ATTACH] (I start by saying that I'm totally ok with the calculation of angular momentum and the properties of its component parallel to the [itex]z[/itex] axis ([itex] \vec {L_z } [/itex]). [U]My difficulties are in understanding what are the properties of [itex] \vec {L_n } [/itex].)[/U] [h2]Homework Equations[/h2] From the picture we have that [itex] | \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[/itex] (1) [h2]The Attempt at a Solution[/h2] Firstly, does it follow from (1) that [itex] | \vec {L_n} | \propto | \vec {\Omega} | [/itex] (2)? If this is true, suppose to apply a torque perpendicular to the [itex]z[/itex] axis and parallel to [itex]\vec {L_n} [/itex], so that the magnitude of this vector increases. Follows from (2) that there should be an angular acceleration [itex] \vec {\alpha} [/itex], although we are in the absence of a torque with an axial component. This would go against the fact that [itex] I_z \vec {\alpha} = \vec { M_z}[/itex] (Where [itex]I_z [/itex] is the moment of inertia with respect to the [itex]z[/itex] axis and [itex]M_z[/itex] is the [B]axial component[/B] of the exerted torque). How is this possible? [/QUOTE]
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Component of angular momentum perpendicular to rotation axis
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