Imagine a rod with a solid sphere attached to both ends. The rod it spinning around its midpoint. The distance from the sphere to the middle of the rod is $d$, the radius of the spheres is $r$, the mass of the spheres is m, and the mass of the rod is neglibible.
I thought that moment of inertia of the system would be the moment of inertia of all the points, which would be $I_{\mathrm{sphere}}+I_{\mathrm{sphere}}=md^2+md^2=2md^2$. But apparently, the moment of inertia is $2(md^2+\frac{2}{5}mr^2)$.
The moment of inertia of a sphere rotating around its own axis is $\frac{2}{5}mr^2$, so it seems that in this system, the moment of inertia of a solid sphere rotating around its axis is added to the total moment of inertia. That should mean that the spheres are spinning, but how are they spinning? Note that this is the situation of the Cavendish Experiment.
