Moment of inertia about a cone

In summary, a spinning cone has an unknown moment of inertia. The student is trying to find the moment of inertia using single and triple integration, but is having difficulty. The professor gave the student a guess of 3/10MR^2, but the student's triple integration results in 3/5. The student is hoping for a link or guide that would help with moments of inertia in general.
  • #1
Warmoth
3
0
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.
 
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  • #2
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!
 
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  • #3
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.
 
  • #4
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 :)
 

1. What is the moment of inertia about a cone?

The moment of inertia about a cone is a measure of the resistance of the cone to change its rotational motion. It is a property that depends on the mass distribution of the cone and the axis of rotation.

2. How is the moment of inertia about a cone calculated?

The moment of inertia about a cone can be calculated using the formula I = (3/10)MR^2, where M is the mass of the cone and R is the radius of its base.

3. How does the moment of inertia about a cone compare to other shapes?

The moment of inertia about a cone is smaller than that of a solid cylinder with the same mass and radius. It is also smaller than that of a solid sphere with the same mass and radius.

4. What is the significance of the moment of inertia about a cone?

The moment of inertia about a cone is important in understanding the rotational motion of objects with a cone-like shape. It is also used in engineering and design to determine the stability and strength of rotating objects.

5. How can the moment of inertia about a cone be changed?

The moment of inertia about a cone can be changed by altering the mass distribution of the cone or by changing the axis of rotation. It can also be changed by changing the shape of the cone, such as making it taller or wider.

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