For *anything* hanging in stable equilibrium, if I push on
it *anywhere*
-- the energy is unchanged to first order, for a push in
*any* direction ... by definition of equilibrium.
-- the energy goes up to second order, for a push in
*any* direction ... by definition of stability.
Remapping the energy to the "height of the CM" is an amusing
twist. At this point (and not otherwise) we need to know that
the cable is non-stretchy.