A glass cleaner of mass 95 kg places a 22 kg ladder against the wall.The ladder is 10 m
long and rests on a wet floor with a
coefficient of static friction equal to
0.40. What is the maximum length
that the window cleaner can climb
before the ladder slips?
To find the maximum length that the window cleaner can climb before the ladder slips, we need to determine the maximum static friction force that the wet floor can exert on the ladder. Once we determine this force, we can use it to calculate the maximum angle at which the ladder can rest before slipping.
1. Calculate the maximum static friction force:
- The coefficient of static friction between the ladder and the floor is given as 0.40.
- The weight of the ladder, W_ladder, can be calculated as mass multiplied by acceleration due to gravity: W_ladder = mass_ladder x g, where g = 9.8 m/s^2.
- W_ladder = 22 kg x 9.8 m/s^2 = 215.6 N (rounded to one decimal place)
- The maximum static friction force, F_friction, can be calculated as the coefficient of static friction (μ) multiplied by the normal force (N).
- N = weight of the ladder = 215.6 N (rounded to one decimal place)
- F_friction = μ x N = 0.40 x 215.6 N = 86.2 N (rounded to one decimal place)
2. Calculate the maximum angle of inclination:
- The tangent of the angle of inclination, θ, can be determined using the ratio of the maximum static friction force to the weight of the window cleaner (mass_cleaner x g).
- F_friction / W_cleaner = tan(θ)
- W_cleaner = mass_cleaner x g = 95 kg x 9.8 m/s^2 = 931 N (rounded to three significant figures)
- F_friction = 86.2 N (from the previous calculation)
- tan(θ) = 86.2 N / 931 N = 0.0926 (rounded to four decimal places)
- Taking the inverse tangent of 0.0926, we find:
- θ = tan^(-1)(0.0926) = 5.3 degrees (rounded to one decimal place)
3. Calculate the maximum length the window cleaner can climb:
- To find the maximum length, we need to calculate the height of the window, h, using the ladder length (10 m) and the angle of inclination (θ). The formula is:
- h = ladder_length x sin(θ)
- h = 10 m x sin(5.3 degrees) ≈ 0.92 m (rounded to two decimal places)
Therefore, the maximum length that the window cleaner can climb before the ladder slips is approximately 0.92 meters.