A muon scattering tomography system which uses extruded plastic scintillator bars for muon tracking and a dedicated muon spectrometer that measures scattering through steel slabs has been constructed and successfully tested. The atmospheric muon detection efficiency is measured to be 97% per plane on average and the average intrinsic hit resolution is 2.5 mm. In addition to creating a variety of three-dimensional images of objects of interest, a quantitative study has been carried out to investigate the impact of including muon momentum measurements when attempting to detect high-density, high-Z material. As expected, the addition of momentum information improves the performance of the system. For a fixed data-taking time of 60 s and a fixed false positive fraction, the probability to detect a target increases when momentum information is used. This is the first demonstration of the use of muon momentum information from dedicated spectrometer measurements in muon scattering tomography.