This has been causing me some pain.
An 89.03 kg person is half way up a uniform ladder of length 3.453 m and mass 22.75 kg that is leaning against a wall. The angle between the ladder and the wall is θ. The coefficient of static friction between the ladder and the floor is 0.2561. Assume that the friction force between the ladder and the wall is zero. What is the maximum value of the angle θ between the ladder and the wall before the ladder starts slipping?
To find the maximum value of the angle θ before the ladder starts slipping, we need to analyze the forces acting on the ladder.
Let's consider the forces acting on the ladder:
1. Weight force (mg): Since the person has a mass of 89.03 kg and is halfway up the ladder, the weight force acting on the ladder is m * g = 89.03 kg * 9.8 m/s^2.
2. Normal force (N): The normal force is the force exerted by the floor on the ladder and is equal and opposite to the weight force.
3. Friction force (F_friction): The friction force opposes the tendency of the ladder to slip along the floor.
To determine the maximum angle θ before the ladder starts slipping, we need to analyze the forces along the ladder:
1. Horizontal forces: The horizontal forces acting on the ladder are due to the friction force F_friction.
2. Vertical forces: The vertical forces acting on the ladder are due to the weight force mg and the normal force N.
For the ladder to remain static (not slip), the static friction force must be equal to or greater than the horizontal component of the weight force.
The horizontal component of the weight force can be found using trigonometry:
Horizontal component = weight force * sin(θ)
So we have:
F_friction ≥ weight force * sin(θ)
The maximum value of the static friction force can be found using the coefficient of static friction (μ) and the normal force (N):
F_friction ≤ μ * N
Since the normal force is equal and opposite to the weight force, N = weight force:
F_friction ≤ μ * weight force
Combining the above inequalities, we get:
μ * weight force ≥ weight force * sin(θ)
Cancelling the weight force from both sides:
μ ≥ sin(θ)
To find the maximum value of θ, we just need to take the inverse sine (arcsin) of μ:
θ = arcsin(μ)
Plugging in the given coefficient of static friction μ = 0.2561 into the equation, we can calculate the maximum angle θ before the ladder starts slipping:
θ = arcsin(0.2561)
Using a scientific calculator or trigonometric table, we find that θ ≈ 15.06 degrees. Therefore, the maximum value of the angle θ between the ladder and the wall before the ladder starts slipping is approximately 15.06 degrees.