A machine for finishing a flat surface, for example a floor made of cement, concrete, or similar hardened material, includes a chassis supported by at least three wheels arranged to carry the weight of the chassis. A grinding tool, preferably a grinding wheel, is mounted to the chassis to rotate about a vertical rotational axis and to rest on the floor when grinding work is in progress. The grinding wheel is mounted to move freely in the vertical direction relative to the chassis such that the weight of the grinding wheel is carried by the floor, and the grinding pressure is readily adjustable for differing applications.