I'm looking for some references on a specific moment of inertia, for radial motions of a spherical body. In my calculations, I got this integral : \begin{equation}\tag{1} \bar{I} = \int r^2 \, dm = \int_{\mathcal{V}} \rho(r) \, r^2 \, d^3 x, \end{equation} where $\rho$ is the matter density and $r^2 = x^2 + y^2 + z^2$ defines the usual radial coordinate (the coordinates origin is located at the center of the spherical body). For an uniform mass distribution, this integral is easy to do : \begin{equation}\tag{2} \bar{I} = \frac{3}{5} \; M R^2. \end{equation} Please, don't confuse this with the well known moment of inertia of the sphere, around some rotation axis. This is about radial motions, and not rotation.
I never saw this in any books on mechanics.
Notice that expression (1) above is also half the trace of the inertia tensor : \begin{equation}\tag{3} I_{ij} = \int_{\mathcal{V}} (r^2 \, \delta_{ij} - x_i \, x_j) \, \rho \; d^3 x, \end{equation} Then we have this : \begin{equation}\tag{4} \bar{I} \equiv \frac{1}{2} \; I_{kk}. \end{equation}
I'm not sure the "radial inertia moment" defined by (1) (if it have a proper interpretation) is getting the proper factor.
Any thoughts on this ?