A ladder of mass 34.4 kg and length 3.36 m is leaning against a wall at an angle θ. The coefficient of static friction between ladder and floor is 0.326; assume that the friction force between ladder and wall is zero. What is the maximum value that θ can have before the ladder starts slipping?
To find the maximum value of θ before the ladder starts slipping, we need to analyze the forces acting on the ladder.
Let's consider the forces in play:
1. Weight of the ladder (W): This force is acting vertically downwards from the center of mass of the ladder and is equal to the mass of the ladder multiplied by the acceleration due to gravity (W = m * g).
2. Normal force (N): This is the force exerted by the floor on the ladder perpendicular to the surface of the floor. It counteracts the weight of the ladder and acts vertically upwards.
3. Friction force (F): This is the force opposing the motion and acts parallel to the floor's surface.
The ladder starts slipping when the friction force reaches its maximum value, which is given by the equation:
F = μs * N
where μs is the coefficient of static friction between the ladder and the floor.
To analyze the forces acting on the ladder, we need to decompose the weight of the ladder into two components:
1. A component parallel to the surface of the floor (W_parallel): W_parallel = W * sin(θ)
2. A component perpendicular to the surface of the floor (W_perpendicular): W_perpendicular = W * cos(θ)
Next, let's balance the forces in the horizontal (x-axis) direction:
Friction force (F) = W_parallel
Therefore, we have:
μs * N = W * sin(θ)
Now, let's balance the forces in the vertical (y-axis) direction:
Normal force (N) = W_perpendicular
Now, substitute the value of N in the equation for the friction force:
μs * (W * cos(θ)) = W * sin(θ)
Simplifying the equation:
μs * cos(θ) = sin(θ)
Dividing both sides of the equation by cos(θ):
μs = tan(θ)
To find the maximum value of θ, we need to find the maximum value of the tangent function. This occurs when the tangent of an angle is equal to the coefficient of static friction (μs).
Therefore, θ = arctan(μs)
Now, let's substitute the value of μs (0.326) into the equation:
θ = arctan(0.326)
Using a calculator, we find:
θ ≈ 18.92 degrees
So, the maximum value of θ before the ladder starts slipping is approximately 18.92 degrees.