Deriving moment of inertia for a thin spherical shell

[dowjonez]

Hi

i have to give a presentation on an example of the defining equation for the moment of inertia of a thin spherical shell. I have to follow the example in my book "elements of Newtonian mechanics". I get most of it but there are a couple steps that the book skips that i cannot. I was wondering if anyone could better explain what's happening to me.


There is a thin spherical shell of mass M and radius R which is symetrically identical in the x, y and z coordinate system.

Ix = Iy = Iz

now Ix = integral (y^2 + z^2)dM i don't get this step.

R^2 = x^2 but i don't get the geometry of why x^2 = y^2 + z^2




Iy = integral (z^2 + x ^2)dM etc


now it says Itotal = 1/3(Ix + Iy + iz)

where does the 1/3 come from. Is it just taking the average or does it have to do with the parrallel axis theorem?

 

[Päällikkö]

I'm not sure if you have figures in your text, but I can't see them.Supposing I understood the situation correctly, the answer to your problem(s) would be the Pythagorean theorem.

 

[dowjonez]

yeah i understand that now. I still don't get why the total inertia is 1/3(Ix + Iy + iz) though