A method and apparatus for uniformly polishing the magnetic recording layer of a floppy disc by producing uniform relative velocities between the magnetic oxide material and the polishing medium across the floppy disc. The floppy disc is placed on a circular resilient pad disposed on a first spindle-mounted circular platen. A circular fibrous or abrasive polishing disc is disposed on a second spindle-mounted circular platen. The radius of the polishing disc is greater than the width of an annular region to be polished on the floppy disc, so that the relative velocities between points of the polishing disc and adjacent points of the floppy disc are uniform. Uniform polishing of the annular region results when the rotating polishing disc contacts the annular region of the rotating disc when the first and second platens are rotated in the same angular direction, at first and second predetermined angular velocities, respectively.