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Posts Tagged ‘Spherical Ring’

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.

Spherical Ring - Figure1

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).

Spherical Ring - Equation1  (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).

Spherical Ring - Equation2  (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 Iz,b and Iz,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:

Spherical Ring - Equation3 (3)

Spherical Ring - Equation4  (4)

Spherical Ring - Equation5  (5)

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

Spherical Ring - Equation6  (6)

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

Spherical Ring - Equation7  (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, Iz,c is expanded to:

Spherical Ring - Equation8  (8)

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

Spherical Ring - Equation9  (9)

Spherical Ring - Equation10  (10)

Spherical Ring - Equation11  (11)

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

Spherical Ring - Equation12  (12)

Spherical Ring - Equation13  (13)

Spherical Ring - Equation14  (14)

Spherical Ring - Equation15  (15)

So, the MathWorld matrix should appear as:

Spherical Ring - Equation16

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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