 Gravitational potential of a truncated inverse-radius sphere

The sphere $\rho =k^{\prime }/(2\pi Gr)$ for $r<r_0$ has $M(r)=k^{\prime }{r}^2/G$ and $\Phi (r)=-k^{\prime }(2r_0-r)$, with zero potential at infinity. Its finite central potential contrasts with the logarithmic central divergence of a truncated inverse-square density.