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?
You will have to write a system of equations. The ladder can have a normal to the wall force at the ladder, and forces at the base. Horizontal force will be friction.
Sum moments about the base, sum vertical and horizonal forces. You will find your solution in those three equations.
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 can be exerted by the wet floor on the ladder.
First, let's calculate the weight of the glass cleaner and the ladder:
Weight of the glass cleaner = mass × acceleration due to gravity
Weight of the glass cleaner = 95 kg × 9.8 m/s^2 = 931 N
Weight of the ladder = mass × acceleration due to gravity
Weight of the ladder = 22 kg × 9.8 m/s^2 = 216 N
Now, let's calculate the maximum static friction force:
Maximum static friction force = coefficient of static friction × normal force
The normal force on the ladder is equal to the weight of the glass cleaner plus the weight of the ladder:
Normal force = weight of the glass cleaner + weight of the ladder
Normal force = 931 N + 216 N = 1147 N
Now, we can calculate the maximum static friction force:
Maximum static friction force = 0.40 × 1147 N
Maximum static friction force = 459 N
To prevent the ladder from slipping, the force exerted by the window cleaner (climbing force) should not exceed the maximum static friction force.
The climbing force can be calculated using the formula:
Climbing force = weight of the window cleaner × vertical distance climbed / length of the ladder
Note that the vertical distance climbed is the height of the window cleaner above the floor, which we'll denote as 'h'. Given that the ladder is inclined at an angle, the vertical distance climbed is h = length of the ladder × sin(angle).
Substituting the values, the climbing force becomes:
Climbing force = 931 N × h / 10 m
Setting the climbing force equal to the maximum static friction force gives us:
931 N × h / 10 m = 459 N
Simplifying, we have:
h = 459 N × 10 m / 931 N
h ≈ 4.93 m
Therefore, the maximum length that the window cleaner can climb before the ladder slips is approximately 4.93 meters.