Radial motions moment of inertia

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 ?