Moment of Inertia of Spherical Rings

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)MR². This means in figure 1, all three MOI terms should go to (²/₅)MR² when L equals 2R.

MOI(x) and MOI(y) do go to (²/₅)MR², however, MOI(z) goes to (⁸/₅)MR². Now, I’m second guessing myself as there are not many errors on MathWorld. But, if in the MOI(z) expression, the (+) was changed to a (-) then the MOI(z) would become (²/₅)MR². To confirm this is correct, lets derive 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).

  (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_z,c. The limits of integration are very similar to a normal sphere, except the integration in the z-direction only occurs for ±¹/₂L . These limits are:

 (3)

  (4)

  (5)

Evaluating equation (2) gives the MOI(z) of a spheric ring with a solid ‘hole’.

  (6)

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

  (7)

where r is the hole radius and m is the cylinder mass. These can be represented as r² = R² – ¼L²  and  m = ρπr²L.  With this, I_z,c is expanded to:

  (8)

Taking the solid Spherical Ring MOI and subtracting the cylindrical hole MOI gives:

  (9)

  (10)

  (11)

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

  (12)

  (13)

  (14)

  (15)

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