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A point mass is constrained to move on a massless hoop of radius a fixed in a vertical plane that rotates about its vertical symmetry axis with constant angular speed [tex] \omega [/tex].
a. Obtain the Lagrange's equations of motion assuming that the only external forces arise from gravity.
Should I have separate KE components for the linear velocities as well as the angular velocity? I have this so far (with separate x,y and z velocity components written in spherical coordinates) [tex]T=\frac{m}{2} v^2 + I\omega^2[/tex]. I'm pretty sure that is correct, but I don't know what to use for the moment of inertia? Can I just use the moment of inertia for a spherical shell, or would I use that of a ring or something else entirely?
EDIT:
Since [tex]\omega[/tex] is the rate of change of the angle [tex]\theta[/tex] in spherical coordinates, could I set that equal to [tex]\frac{d}{dt}\theta[/tex] in the kinetic energy term? Or can I not ignore the moment of inertia like that?
a. Obtain the Lagrange's equations of motion assuming that the only external forces arise from gravity.
Should I have separate KE components for the linear velocities as well as the angular velocity? I have this so far (with separate x,y and z velocity components written in spherical coordinates) [tex]T=\frac{m}{2} v^2 + I\omega^2[/tex]. I'm pretty sure that is correct, but I don't know what to use for the moment of inertia? Can I just use the moment of inertia for a spherical shell, or would I use that of a ring or something else entirely?
EDIT:
Since [tex]\omega[/tex] is the rate of change of the angle [tex]\theta[/tex] in spherical coordinates, could I set that equal to [tex]\frac{d}{dt}\theta[/tex] in the kinetic energy term? Or can I not ignore the moment of inertia like that?
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