We introduce a new technique to model spherical geophysical fluid dynamics in the terrestrial laboratory. The local vertical projection of planetary vorticity, f, varies with latitude on a rotating spherical planet and allows an important class of waves in large-scale atmospheric and oceanic flows. These Rossby waves have been extensively studied in the laboratory for middle and polar latitudes. At the equator f changes sign where gravity is perpendicular to the planetary rotation. This geometry has made laboratory studies of geophysical fluid dynamics near the equator very limited. We use ferrofluid and static magnetic fields to generate nearly spherical geopotentials in a rotating laboratory experiment. This system is the laboratory analogue of those large-scale atmospheric and oceanic flows whose horizontal motions are governed by the Laplace tidal equations. As the rotation rate in such a system increases, waves are trapped to latitudes near the equator and the dynamics can be formulated on the equatorial β-plane. This transition from planetary modes to equatorially trapped modes as the rotation rate increases is observed in the experiments. The equatorial β-plane solutions of non-dispersive Kelvin waves propagating eastward and non-dispersive Rossby waves propagating westward at low frequency are observed in the limit of rotation fast compared to gravity wave speed.