Plumb-line deflection from cylindrical gravity ($d\simeq 2GL\lambda /(rg)$)

A pendulum of length $L$ in vertical gravity $g$ is weakly attracted horizontally by an added long circular cylinder of line density $\lambda$. Outside it, $g_h=2G\lambda /r$ at axis distance $r$. Small-angle equilibrium gives horizontal deflection $d\simeq L{g}_h/g=2GL\lambda /(rg)$. Removing the cylinder's mass reverses the change. It is a before/after differential estimate; finite length, a nonhorizontal radial direction and mechanical motion require corrections.