Wire Frame Cube Moment

Classical Mechanics Level 4

A cubic wire frame consists of 12 segments interconnecting 8 vertices. The vertices are located at the following points:

\[\begin{align} (x,y,z) = &(-1,1,1),(-1,-1,1),(1,-1,1),(1,1,1),\\&(-1,1,-1),(-1,-1,-1),(1,-1,-1),(1,1,-1). \end{align}\]

The wire frame has a mass \(M\) which is uniformly distributed over its constituent line segments. The object's moment of inertia with respect to an axis perpendicular to the \(xy\)-plane and passing through the point \((x,y) = (0,0)\) can be expressed as \(\dfrac{a}{b} M\), where \(a\) and \(b\) are coprime positive integers.

Determine \(a + b\).


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