Moment of Inertia in Cylindrical Coordinates

AI Thread Summary
The discussion focuses on calculating the mass moment of inertia about the y-axis for an object with a density of 4 slugs/ft³. The relevant equation for the moment of inertia is I_y = ∫(x² + z²) dm, with dm defined as ρ dV. Participants debate the appropriate coordinate system, with one suggesting polar coordinates while another insists on using cylindrical coordinates for simplicity. The latter proposes slicing the object into vertical cylindrical shells, which allows for a single integral over r to find the volume and moment of inertia. The conversation emphasizes the importance of selecting the correct coordinate system for solving the problem efficiently.
mathmannn
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Homework Statement



What is the mass moment of inertia about the y-axis of the object shown if the density is 4 slugs/ft3?


Homework Equations



I_y = \int (x^2 + z^2) dm
dm = \rho dV

The Attempt at a Solution


Not really too sure how to set this one up, my prof said it should be done in polar coordinates. I'm pretty sure that it will come out to be a triple integral but it's been a while since I've done those and could use a little help setting this up.
 

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hi mathmannn! :smile:
mathmannn said:
… my prof said it should be done in polar coordinates. I'm pretty sure that it will come out to be a triple integral …

no, that's crazy , it should obviously be cylindrical coordinates

slice the region into vertical cylindrical shells of thickness dr (finding the volume and moment of inertia of each shell is elementary), and then you only have a single integral, over r :wink:
 

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