How Can You Determine the Principal Moments of Inertia for a Cone?

[Zenshin]

Hello again! About my last problem (cone principal moments of inertia around top vertice), I´ve found the main moments of inertia, but I did so by integrating the moments of inertia dI o a disk rotaing about it´s diameter (1/4 ML^2, by table), and found the correct answer. But I have done this by the way we did it in Basic Physics 1... in Classical Mechanics I, I think there´s a way in which you use "proportion factors" and find the moments of inertia like Ixx =M(Ky x Lambda-y + Kz x Lambda-z), but i could apply this method...Still, I haven´t figured out how to find a h/a (height / base radius) of that cone in which every axis passing through the vertice v is a principal axis... Somehow, I think it´s a eigenvalue and eigenvector problem of the Inertia tensor... Oh, and by the way, does anyone know how to find the moment of inertia of that disk, about it´s diameter?? I got the value from a table, since I could not find the answer myself...(tried something similar to the spherical moment of inertia by integrating a cascade of disks...didn´t work as well with a cascade of bars for the disk though...)


Thanks in advance!

 

[Nate810]

To find the moment of inertia of the disk about its diameter, you need to first calculate the total mass of the disk. This can be done by dividing the mass of the disk by its area. Once you have the total mass of the disk, you can use the formula I = mr^2, where 'I' is the moment of inertia, 'm' is the mass, and 'r' is the radius of the disk. From this, you can calculate the moment of inertia for the disk about its diameter. For the cone, you can use the same technique as before to calculate the moment of inertia. You will need to divide the mass of the cone by its surface area and then use the formula I = mr^2, where 'I' is the moment of inertia, 'm' is the mass, and 'r' is the radius of the cone. This will give you the moment of inertia for the cone about its vertice.

 

[wais]

It's great that you were able to find the main moments of inertia by integrating the moments of inertia of a disk rotating about its diameter. This is a valid method and can be used to solve many rigid body dynamics problems. However, as you mentioned, there is another method using "proportion factors" which can also be used to find the moments of inertia. This method involves using the parallel axis theorem and the perpendicular axis theorem to find the moments of inertia about different axes.

To find the moments of inertia of the cone about its principal axes, you can use the eigenvalue and eigenvector approach. This involves finding the eigenvalues and eigenvectors of the inertia tensor, which represents the moments of inertia of the cone. The eigenvalues represent the principal moments of inertia and the corresponding eigenvectors represent the principal axes. By finding the eigenvalues and eigenvectors, you can then use the formula you mentioned to find the moments of inertia about the principal axes.

As for finding the moment of inertia of a disk about its diameter, you can use the parallel axis theorem. This theorem states that the moment of inertia of a body about an axis parallel to its centroidal axis is equal to the moment of inertia about the centroidal axis plus the product of the mass of the body and the square of the distance between the two axes. In this case, the centroidal axis is the axis passing through the center of the disk and the parallel axis is the axis passing through its diameter.

I hope this helps and good luck with your studies!