A roll-up tarp apparatus for a truck box having an open top. The roll-up tarp apparatus includes a roll bar, a flexible tarp secured to the truck box and the roll bar, a converter unit, a motive source and an arm operatively connected to the converter unit, the motive source and the roll bar. The roll bar is operatively connected to the truck box and rotatably movable between the first and second longitudinal sides of the truck box. The tarp is secured to the roll bar so that the tarp can roll up on the roll bar to open the truck box and unroll from the roll bar to cover or close the open top. The roll bar is preferably biased in an unroll direction or toward a closed position. In preferred embodiments, at least one biasing member is operatively connected to the roll bar or the arm.