The lap (W) to be fed to the combing head (1, 2, 3, 5, 6) is wound off a lap roll (7). The roll (7) has a winding tube (8) on which the lap is wound. The device has supports (10, 12) to hold the roll (7) rotatably in such a way that its axis is parallel to a drive roller (9) and its periphery contacts the drive roller (9). During unwinding the drive roller (9) and at least one component of the supports (10, 12) are mutually movable in relation to the strip (W) in such a way that, despite the gradual reduction in the weight of the roll (7) and as a function of this reduction, the contact force between the roll (7) and the drive roller (9) is compensated. Owing to the compensated contact force, the periphery of the roll (7) always bears against the contact point with approximately the same force. This prevents the quantity of lap wound off the roll (7) per revolution of the drive roller (9) from becoming gradually smaller during the unwinding process.