A drive mechanism for a wafer processing machine of the kind which holds a silicon wafer on a vacuum chuck and rotates the wafer for processing operations on the wafer periphery or upper surface incorporates a low-speed motor, a high-speed motor and an overrunning clutch. The low-speed motor is connected to a drive shaft for the chuck through the overrunning clutch and rotates the chuck and wafer at a relatively low and precisely-controlled speed while the edge of the wafer is contoured by a grinding wheel or while the upper surface is mechanically cleaned by a brush engaged with the upper surface of the wafer. The high-speed motor is directly connected to the drive shaft and, when energized for a rinsing and drying operation, rotates the chuck and wafer in the same direction of rotation as the low-speed motor but at high speed while the wafer is being rinsed and dried after the grinding or cleaning operation. During the grinding or cleaning operation the high-speed motor is energized toward rotation in a direction opposed to the direction of rotation produced by the low-speed motor. The biasing torque so produced insures against any slack or rebounding through the overrunning clutch which could result in skipping or chattering of the engagement of the grinding wheel with the periphery of the wafer during the contouring of the periphery of the wafer or which could result in loss of clamping contact between the chuck and the wafer.