A uniform ladder 5.0 m long and weighing 200 N is placed against a very smooth wall so that the lower end of the ladder is 3.0 m from the wall. If the coeficient of friction between the ladder and the ground is 0.50, determine the maximum distance that a 700 N man could climb along the length of the ladder before the ladder slips.
where did the 3x/5 come from
what about the friction?
To determine the maximum distance that a 700 N man could climb along the length of the ladder before the ladder slips, we need to analyze the forces acting on the ladder.
1. Weight of the ladder:
The weight of the ladder is given as 200 N. This force acts downward and can be represented by the symbol Wladder.
2. Normal force exerted by the ground:
The normal force exerted by the ground on the ladder counteracts the weight of the ladder. Since the ladder is on a smooth surface, there is no friction involved. Therefore, the normal force (Nground) is equal to the weight of the ladder (Wladder).
3. Frictional force between the ladder and the ground:
The coefficient of friction between the ladder and the ground is given as 0.50. Frictional force (Ffriction) can be calculated using the formula Ffriction = coefficient of friction × normal force.
4. Force exerted by the man:
The man exerts a force (Fman) in the upward direction, along the length of the ladder. This force opposes the downward forces acting on the ladder.
To prevent the ladder from slipping, the maximum force exerted by the man (Fman) should not exceed the maximum frictional force (Ffriction).
Now, let's calculate the maximum distance the man can climb before the ladder slips.
1. Calculate the normal force (Nground):
Since the ladder is on a smooth surface, the normal force is equal to the weight of the ladder.
Nground = Wladder = 200 N
2. Calculate the maximum frictional force (Ffriction):
Ffriction = coefficient of friction × normal force
Ffriction = 0.50 × 200 N = 100 N
3. Calculate the maximum force exerted by the man (Fman):
To calculate the maximum force that the man can exert, we need to consider both the weight of the man and the weight of the ladder.
Total weight = weight of the man + weight of the ladder
Total weight = 700 N + 200 N = 900 N
4. Calculate the maximum distance the man can climb:
To find the maximum distance, we need to equate the force exerted by the man (Fman) to the maximum frictional force (Ffriction).
Fman = Ffriction
Fman = 100 N (from step 2)
Since the force exerted by the man is equal to the maximum frictional force, this means the ladder is about to slip. Therefore, the maximum distance the man can climb is the distance from the ground to where the ladder starts to slip.
In this case, the distance is the difference between the length of the ladder (5.0 m) and the distance of the lower end of the ladder from the wall (3.0 m).
Maximum distance = 5.0 m - 3.0 m = 2.0 m
Therefore, the maximum distance that a 700 N man could climb along the length of the ladder before the ladder slips is 2.0 m.
3,4,5 triangle
Vertical forces
200 N down at middle, 1.5 m from foot of ladder
700 N down at (3x/5) from foot of ladder
900 N up at foot of ladder
Horizontal forces
F at top of ladder, 4 m above base
900 mu at base of ladder = F at top
take moments about
base of ladder
700 (3x/5) + 200(1.5) - 4F = 0
but F = .5*900 = 450
so
700(3x/5) +300 = 1800
420 x = 1500
x = 3.57