How Do You Calculate the Moment of Inertia of a Cone?

AI Thread Summary
To calculate the moment of inertia of a uniform solid cone about its center axis, the relevant equation is I = integral(r^2 dm), where dm can be expressed as ρ dV. The user is uncertain about how to set up the integration, particularly whether to use cylindrical coordinates and how to define the limits of integration. The known answer for the moment of inertia is 3/10 MR^2, but the user seeks guidance on deriving this result. Suggestions include using cylindrical coordinates to define dV and determining appropriate integration limits. Understanding these concepts will aid in solving similar moment of inertia problems in the future.
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Homework Statement



Calculate the moment of inertia of a uniform solid cone about an axis through its center. The cone has mass M and altitude h. The radius of its circular base is R. (see attached photo)



Homework Equations


I know I need to somehow use the equation I= intergral r^2 dm
also, I have an equation from my proffessor, dm=rho dv I'm not sure if I need this though since its unifrom density so it doesn't seem like \rho should matter.

The Attempt at a Solution


I don't have a solution right now. I know the answer is 3/10 MR2 but I don't know how to get there. From class, since we did some examples, I think I need an equation that has a triple integral in it but I don't know what to integrate and to where. Do I need to get dm in terms of something like d\vartheta, dr, and dh?

Thanks for your help. Also, if you have any general suggestions on how to complete moment of inertia problems like this that would be great. I know we're going to be doing a lot of them.

Thanks again!
 
Last edited:
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Yes dm = \rhodV will be used. Have you tried setting up dV in cylindrical coordinates? Then try to look for your limits of integration.
 

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