Peculiarities in the gravitational field of a filamentary ring Available to Purchase

In all textbooks cited before, all mentions are that in a uniform gravitational field, the center of mass and center of gravity coincide. While this practice is defensible from the view of practicality, we feel that pedagogically is questionable: an exceptional case (point particle or perfectly spherical objects) is taught. Only Ref. 6, among the already cited textbooks, treats non-uniform mass distribution, and non-spherical bodies explicitly in a subsection.

An even simpler approach would be to first substitute $r⊥=0$ into the expression of the gravitational potential and obtain $V(z)=−GM/R2+z2$ and then differentiate this expression twice and solve the $∂z2V(z)=0$ equation for $z∗$⁠.

For the sake of completeness we provide here the exact expression for the period $T=(4/ω) (1/2(1−2k2)) [2E(k2)−K(k2)+Π(2k2|k2)]$ in terms of the complete elliptic integrals of the first, second, and third kind, K, E, and Π, respectively. Their definitions can be found in Byrd and Friedman.²³ The power series expansion of this exact expression, up to the second order, agrees with that of given in the text, $T≈(2π/ω)(1+94k2)$⁠, but derived from a hand-waving argument.