A man begins to climb up a 12-ft ladder . The man has mass 80 kg, while
the ladder has mass 10 kg. The wall against which the ladder rests is very smooth,
which means that the tangential (vertical) component of force at the contact between
ladder and wall is negligible. The foot of the ladder is placed 6 ft from the wall. The
ladder, with the man’s weight on it, will slip if the tangential (horizontal) force at the
contact between ladder and ground exceeds 350 N. How far up the ladder can the
man safely climb?
To determine how far up the ladder the man can safely climb, we need to calculate the maximum value of the horizontal force that the ladder can sustain before slipping.
Let's start by analyzing the forces acting on the ladder. There are two main forces at play: the weight of the man and the ladder, and the contact force between the ladder and the ground.
1. Weight of the man and the ladder: The weight acts vertically downwards. The total weight is the sum of the man's weight and the ladder's weight:
Weight = (mass of man + mass of ladder) * acceleration due to gravity
Weight = (80 kg + 10 kg) * 9.8 m/s²
Weight = 90 kg * 9.8 m/s²
2. Contact force between the ladder and the ground: This force acts perpendicular to the ladder and opposes the weight. If the ladder is on the verge of slipping, the maximum contact force will be equal to the weight of the man and the ladder. We can calculate this force by subtracting the horizontal component of the weight from the weight itself.
Contact force = Weight - horizontal component of weight
To find the horizontal component of the weight, we need to consider the angle between the ladder and the ground. In this case, the ladder forms a right triangle with the ground, with the ladder length being the hypotenuse and the horizontal distance from the foot of the ladder to the wall as the base.
Using the Pythagorean theorem, we can find the length of the ladder:
Ladder length = √(base² + height²)
Ladder length = √(6 ft² + 12 ft²)
Ladder length = √(36 ft² + 144 ft²)
Ladder length = √180 ft²
Ladder length ≈ 13.42 ft
Now, let's find the angle θ between the ladder and the ground:
tan(θ) = opposite/adjacent
tan(θ) = height/base
θ = arctan(height/base)
θ = arctan(12 ft/6 ft)
θ ≈ 63.43°
The horizontal component of the weight (F_horizontal) can be calculated using trigonometry:
F_horizontal = Weight * sin(θ)
Finally, we can determine the maximum contact force:
Contact force = Weight - F_horizontal
If the maximum contact force exceeds 350 N, the ladder will slip. Otherwise, the ladder will be safe to climb.
By plugging in the values, you can calculate the maximum contact force and determine how far up the ladder the man can safely climb.