Moment of inertia of spherical shell

In summary, the conversation revolved around a presentation on the defining equation for the moment of inertia of a thin spherical shell. The book's example was followed, but there were some skipped steps that the person did not understand. The speaker asked for further explanation, particularly about the geometry involved in the equation. The answer was provided, which involved the use of the Pythagorean theorem. The conversation also touched on the origin of the 1/3 factor in the total inertia equation.
  • #1
dowjonez
22
0
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?
 
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  • #2
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.
 
  • #3
yeah i understand that now. I still don't get why the total inertia is 1/3(Ix + Iy + iz) though
 

FAQ: Moment of inertia of spherical shell

What is the formula for calculating the moment of inertia of a spherical shell?

The formula for calculating the moment of inertia of a spherical shell is I = (2/3)mr², where m is the mass of the shell and r is the radius of the shell.

How does the moment of inertia of a spherical shell differ from that of a solid sphere?

The moment of inertia of a spherical shell is less than that of a solid sphere with the same mass and radius. This is because the mass of a solid sphere is distributed throughout its entire volume, while the mass of a spherical shell is only concentrated at its outer surface.

How does the moment of inertia of a spherical shell change with respect to its radius?

The moment of inertia of a spherical shell increases as the radius increases. This is because the farther the mass is from the axis of rotation, the greater its contribution to the moment of inertia.

Can the moment of inertia of a spherical shell be negative?

No, the moment of inertia of a spherical shell cannot be negative. It is a physical property that represents the resistance of an object to changes in its rotational motion, and therefore cannot have a negative value.

How is the moment of inertia of a spherical shell affected by its mass?

The moment of inertia of a spherical shell is directly proportional to its mass. This means that as the mass of the shell increases, so does its moment of inertia.

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