Pulley and block with massive rope, sliding friction, and fluid drag

Abstract Newton’s second laws for translations and rotations are combined to describe the motion of a box sliding down an inclined plane that is tethered by a rope having a uniform linear mass density to a pulley. In addition, the box is subject to quadratic drag from the surrounding gas or liquid through which it moves. A variety of special cases are treated, including an anchor and chain dropped vertically into water, an untethered box descending a ramp in the presence of kinetic friction and air resistance, and a drag-free Atwood-type machine having a massive rope and pulley connected to a single block sliding along a rough incline. The purpose of the article is pedagogical, demonstrating how to apply Newton’s second law to these kinds of variable-mass problems involving both rotations and translations. The level of presentation is suitable for an introductory calculus-based course in physics.