A uniform ladder of mass m and length leans at an angle0

Chapter 12, Problem 12.95

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A uniform ladder of mass m and length leans at an angle0 against a wall, Fig. 12-101. The coefficients of static frictionbetween ladder-ground and ladder-wall are fxG and/itw, respectively. The ladder will be on the verge of slippingwhen both the static friction forces due to the ground anddue to the wall take on their maximum values, (a) Show thatthe ladder will be stable if 0 > 0mjn, where the minimum0min is given bytan dmm = ^ ( 1 -MGMw).(b) Leaning ladder problems are often analyzed under theseemingly unrealistic assumption that the wall is frictionless(see Example 12-6). You wish to investigate the magnitudeof error introduced by modeling the wall as frictionless, if inreality it is frictional. Using the relation found in part (a),calculate the true value of 0^ for a frictional wall,taking /jlq = piw = 0.40. Then, determine theapproximate value of 0min for the frictionlesswall model by taking /j lg = 0.40and /aw = 0. Finally, determine thepercent deviation of the approxi-value.FIGURE 12-101 95.