Diameter-rod oscillation in cylindrical gravity (${\omega }^2=2\pi G\rho$)

A thin rod of length $2R$ and linear mass density $\lambda$ slides along a diameter of an infinite uniform source cylinder of radius $R$. Neglect the narrow hole's perturbation of the source field. If the rod centre is displaced by $x$, $F(x)=\lambda {\int }_{x-R}^{x+R}{g}_y(y)dy$. The endpoint derivative gives $F^{\prime }(0)=\lambda [g_y(R)-g_y(-R)]=-4\pi G\rho \lambda R$. Dividing the restoring coefficient by rod mass $2R\lambda$ gives the displayed small-oscillation frequency.