In my previous blog post I demonstrated how to use linear programming for solving the ladder against a wall problem. More specifically, the R code in that blog found the maximum position x a person could reach before the ladder starts sliding.
In this blog post I will go one step further. In particular, I will develop a solution strategy for finding the minimum angle θ such that a ladder doesn’t slip down. For simplicity, I will assume that no person is on the ladder, and that the ladder slips down due to its own weight (Q). Furthermore, I assume that both the wall and floor are rough surfaces. As a consequence, friction occurs between the top of the ladder and the vertical wall, and between the base of the ladder and the horizontal floor. This minimum angle problem is depicted in the following figure (with on the right-hand side a free body diagram).