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?

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 μₛ = 0.45 . What is the minimum angle the beams can make with the floor and still not fall?

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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.

(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 θₘᵢₙ for a frictional wall, taking μ_G = μ_W = 0.40 . Then, determine the approximate value of θₘᵢₙ 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 θₘᵢₙ from its true value.

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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).

(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 μₛ = 0.45 . What is the minimum angle the beams can make with the floor and still not fall?

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