We present a study of the macroscopic dynamics of a polariton condensate formed by nonresonant optical excitation in a quasi-one-dimensional ring-shaped microcavity. The presence of a gradient in the cavity photon energy creates a macroscopic trap for the polaritons in which a coherent condensate is formed which evolves into a single-mode condensate at late times. With time- and energy-resolved imaging we show the role of interactions in the motion of the condensate as it undergoes equilibration in the ring. These experiments also give a direct measurement of the polariton-polariton interaction strength above the condensation threshold. Our observations are compared to the open-dissipative one-dimensional Gross-Pitaevskii equation which shows excellent qualitative agreement.