An unwinder for rolls of paper web material includes: a support frame; a spindle cylinder, rotationally associated with the frame, supporting a parent roll with a longitudinal axis; unwinder to unwind a paper web from the parent roll; actuator keeping the unwindier contacting the parent roll; a command and control unit; first motor rotating the spindle cylinder; two support and drive rollers; units to measure instantaneous radius of the parent roll, rotational speed of the spindle cylinder, quantity of web unrolled from the parent roll per unit time, and weight unloaded from the parent roll onto the support and drive rollers; second motor synchronously rotating the support and drive rollers; and third motor moving the spindle cylinder vertically. The control unit receives and processes values detected by the measurement unit, causing the motor to maintain the weight force unloaded onto the rollers from the roll as the diameter varies.