Chapter 12, Problem 89GP

A uniform ladder of mass m and length ℓ leans at an angle θ against a wall, Fig. 12–101. The coefficients of static friction between ladder-ground and ladder-wall are μ_G and μ_W, respectively. The ladder will be on the verge of slipping when both the static friction forces due to the ground and due to the wall lake on their maximum values. (a) Show that the ladder will be stable if θ ≥ θ_min, where the minimum angle θ_min is given by

$\tan {\theta }_{\min }=\frac{1}{2{\mu }_{\text{G}}}(1-{\mu }_{\text{G}}{\mu }_{\text{W}}).$

(b) “Leaning ladder problems” are often analyzed under the seemingly unrealistic assumption that the wall is frictionless (see Example 12–6). You wish to investigate the magnitude of error introduced by modeling the wall as frictionless, if in reality it is frictional. Using the relation found in part (a), calculate the true value of θ_min for a frictional wall, taking μ_G = μ_W = 0.40. Then, determine the approximate value of θ_min for the “friction-less wall” model by taking μ_G = 0.40 and μ_W = 0. Finally, determine the percent deviation of the approximate value of θ_min from its true value.

FIGURE 12–101 Problem 95.