How to Prove the Moment of Inertia for a Spinning Cone?

[Warmoth]

I am supposed to prove the moment of inertia about a spinning cone through the diameter. but I am supposed to do it using single integration and triple integration. I think I did it right in the triple integration but I really don't know what needs integrating with the single. the paper gave me my professors guess of 3/10MR^2 but when I did the triple integration it gave me 3/5 so I don't even know if I did it right. just a guiding hand would be appreciated. thanks a lot... maybe just a link to something to help with moments of inertia in general would do the trick because I have several... one of them is a sphere with varying density as the radius increases and I have no idea how that's to be done. I do appreciate the help and look forward to doing my best to contribute where I can. Adios.

 

[vsage]

Is the cone hollow? Is it of uniform density? I assume you mean spinning along an axis that goes through the center of the base and out the apex at the top. You can break up the cone into an infinite amount of uniform disks stacked along the axis (or rings depending on which the problem is).

For the sphere, you're going to need to integrate not only along the axis but out from the axis as well. You could derive the sphere using single integration by using the fact that the moment of inertia of a hollow sphere is 2/3mr^2

Edit: Welcome to PF!

 

[Warmoth]

sorry, I should have been more specific. the cone is of uniform density and the axis of rotation is through the apex and the center of the base. I think I found the way for triple integration, but the single is a little different. I think I am to find the area of anyone of the right triangles and just integrate from 0 to 2 pi. that would work right.

and for the sphere of varying density... it is most dense in the center and tapers off. my professor said to use spherical coordinates. but I don't know even what to integrate.

 

[vsage]

The triple integration shouldn't be much harder than the single integration for the cone (but a little unnecessary because it pretty much disregards symmetry). Show us what you have for each and maybe we can give you pointers.

I'm not sure why spherical coordinates would make this problem easier really. Anyway post what you have so far because I'm not sure what you understand and what you don't :)