Revisiting the sliding ladder

Abstract

A uniform ladder leaning against a wall, or sliding down against a wall, is a familiar theme in introductory mechanics and calculus courses, made popular in recent years by several papers pertaining to its various intriguing aspects and important limitations. This configuration is particularly well-known in the context of static equilibrium where the friction exerted by the floor keeps the ladder from sliding [1]. The situation involving the ladder sliding down instead of being static is a staple topic of calculus courses, although usually only from a geometric point of view. The physics of the sliding ladder was addressed in [2, 3, 4], none of which took friction into consideration. In the present work we study in detail a situation involving a ladder sliding between a rough wall and a frictionless floor. This is a situation where in spite of the presence of friction the ladder is still destined to slide down, no matter how large the friction coefficient between the wall and the ladder is. As we will establish in this paper, in spite of the friction present, the ladder must always be speeding up while sliding down, until it breaks off the wall at some critical angle.