Flying Circus of Physics

Mountain pull. A large mountain can slightly affect the direction of "down" as determined by a plumb line. Assume that we can model a mountain as a sphere of radius $\backslash ($ R$=3\mathrm{.}00\backslash$mathrm$\{$~km$\}\backslash )$ and density, $($mass per unit volume$)\backslash (2\mathrm{.}6\backslash$times $10^\{3\}\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ).$ Assume also that we hang a $0\mathrm{.}500$ m plumb line at a distance of $\backslash (3$ R $\backslash )$ from the sphere's center and such that the sphere pulls horizontally on the lower end. How far would the lower end move toward the sphere?