15. Rotational Equilibrium

A uniform ladder of mass m and length ℓ leans at an angle θ against a wall, Fig. 12–105. 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 take on their maximum values. “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 “frictionless 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.

A ladder of mass 20 kg (uniformly distributed) and length 6 m rests against a vertical wall while making an angle of Θ = 60° with the horizontal, as shown. A 50 kg girl climbs 2 m up the ladder. Calculate the magnitude of the total contact force at the bottom of the ladder (Remember:You will need to first calculate the magnitude of N,_BOT and f,_S).

A uniform ladder 5.0 m long rests against a frictionless, vertical wall with its lower end 3.0 m from the wall. The ladder weighs 160 N. The coefficient of static friction between the foot of the ladder and the ground is 0.40. A man weighing 740 N climbs slowly up the ladder. Start by drawing a free-body diagram of the ladder. What is the maximum friction force that the ground can exert on the ladder at its lower end?

A 3.0-m-long ladder, as shown in Figure 12.35, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.40. What is the minimum angle the ladder can make with the floor without slipping?

(III) Consider a ladder with a painter climbing up it (Fig. 12–71). The mass of the uniform ladder is 12.0 kg, and the mass of the painter is 55.0 kg. If the ladder begins to slip at its base when the painter’s feet are 70% of the way up the length of the ladder, what is the coefficient of static friction between the ladder and the floor? Assume the wall is frictionless.

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Two identical, uniform beams are symmetrically set up against each other (Fig. 12–95) on a floor with which they have a coefficient of friction μ_s = 0.45. What is the minimum angle the beams can make with the floor and still not fall?