## Moment of Inertia of Spherical Rings

Posted by diynovice on November 30, 2014

I recently needed to use the moment of inertia (MOI) of a spherical ring. Conveniently, the awesome Wolfram MathWorld website has the information I needed. An image of their spherical ring and MOI equation is below.

*Figure 1: Diagram and MOI as displayed on MathWorld’s site in 2012 (of course, I added the ‘wrong’ overlay to prevent it’s use)*

However, after doing a simple thought-check, something seemed off. The moment of inertia of any axis of a solid sphere is *( ^{2}/_{5})MR^{2}*. This means in figure 1, all three MOI terms should go to

*(*when

^{2}/_{5})MR^{2}*L*equals

*2R*.

*MOI(x)* and *MOI(y)* do go to *( ^{2}/_{5})MR^{2}*, however,

*MOI(z)*goes to

*(*. Now, I’m second guessing myself as there are not many errors on MathWorld. But, if in the

^{8}/_{5})MR^{2}*MOI(z)*expression, the (+) was changed to a (-) then the

*MOI(z)*would become

*(*. To confirm this is correct, lets derive

^{2}/_{5})MR^{2}*MOI(z)*.

The general expression for the moment of inertia of any 3D object through its centroid is given in equation (1).

(1)

Of course, (*x,y,z*) are coordinates and *ρ* is the material density. I am only interested in the z-axis expression, so the general form breaks down into (2).

When looking at this problem, it is easier to find the MOI of a ‘solid’ spherical ring, and subtract the hole. These MOI’s are denoted* I _{z,b}* and

*I*. The limits of integration are very similar to a normal sphere, except the integration in the z-direction only occurs for

_{z,c}*±*. These limits are:

^{1}/_{2}LEvaluating equation (2) gives the *MOI(z)* of a spheric ring with a solid ‘hole’.

The z-axis MOI of a cylinder is well known as:

where *r* is the hole radius and *m* is the cylinder mass. These can be represented as *r ^{2} = R^{2} – ¼L^{2}* and

*m = ρπr*With this,

^{2}L.*I*is expanded to:

_{z,c}Taking the solid Spherical Ring MOI and subtracting the cylindrical hole MOI gives:

Equation (11) is the solution, however, to get *MOI(z)* into MathWorld’s notation, the density must be removed.

So, the MathWorld matrix should appear as:

Being the dork that I am, I have contacted MathWorld multiple times with this derivation, however, the error still hasn’t been fixed. I am sure there are many more important things they have to do. In the meantime, hopefully this post will help the slim few who are looking for this information.

References:

Weisstein, Eric W. “Spherical Ring.” From *MathWorld*–A Wolfram Web Resource. http://mathworld.wolfram.com/SphericalRing.html

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