Savchenko Problem 3.3.7
Physics · electromagnetism
Problem
A smooth uniform rope of length l is held in a vertical elbow of a bent pipe so that its lower end touches the horizontal part of the pipe. The rope is released. In what time will it be completely in the horizontal elbow? Neglect the friction. How will this time change if a part of the rope was already in the horizontal knee at the beginning?Intuitive explanation
This is a problem about variable-mass dynamics — as the rope slides around the corner, the portion accelerating downward keeps shrinking while the horizontal portion grows. The key insight is that you can't use constant acceleration; instead, track how the rope's configuration changes moment by moment. At any instant, only the hanging part experiences gravitational force, so the acceleration depends on what fraction of the rope remains vertical. Set up the equation of motion carefully by considering the mass and length ratios, then integrate to find the total time.
Intuitive explanation
This is a problem about variable-mass dynamics — as the rope slides around the corner, the portion accelerating downward keeps shrinking while the horizontal portion grows. The key insight is that you can't use constant acceleration; instead, track how the rope's configuration changes moment by moment. At any instant, only the hanging part experiences gravitational force, so the acceleration depends on what fraction of the rope remains vertical. Set up the equation of motion carefully by considering the mass and length ratios, then integrate to find the total time.