a climber with a weight of 441 N is held by a belay rope connected to her climbing harness and belay device; the force of the rope on her has a line of action through her center of mass. The indicated angles are θ = 45˚ and φ = 25˚. If her feet are on the verge of sliding on the vertical wall, what is the coefficient of static friction between her climbing shoes and the wall?
To determine the coefficient of static friction, we first need to analyze the forces acting on the climber. Let's break down the forces in the vertical and horizontal directions.
Vertical Forces:
1. Weight (mg): The climber's weight is given as 441 N, acting downward.
Horizontal Forces:
1. Normal force (N): The force exerted by the wall perpendicular to it, acting upward.
2. Friction force (Ff): The force opposing the tendency of the climber's feet to slide, acting parallel to the wall.
Since the climber is on the verge of sliding, the friction force is at its maximum value and is equal to the maximum static friction force, given by:
Ff(max) = µs * N
where µs is the coefficient of static friction.
Now, let's determine the normal force and its components in the vertical and horizontal directions.
Vertical Component of Normal Force:
Nv = N * cos(θ)
Horizontal Component of Normal Force:
Nh = N * sin(θ)
With the vertical and horizontal components of the normal force, we can now determine the forces acting in the vertical and horizontal directions.
Vertical Forces:
Sum of forces in the vertical direction = 0
Nv - mg = 0
Nv = mg
Horizontal Forces:
Sum of forces in the horizontal direction = 0
Nh - Ff(max) = 0
Nh = Ff(max)
Now, let's substitute the known values and solve for the coefficient of static friction:
Nv = mg
N * cos(θ) = mg
N = mg / cos(θ)
Nh = Ff(max)
N * sin(θ) = µs * N
µs = Nh / N
µs = (Ff(max)) / (mg / cos(θ))
Substituting the given values:
µs = (Nh) / (mg / cos(θ))
µs = (N * sin(θ)) / (mg / cos(θ))
µs = (441 N * sin(θ)) / (441 N / cos(θ))
Now, simply substitute the given value of θ = 45° into the equation:
µs = (441 N * sin(45°)) / (441 N / cos(45°))
Simplify the expression:
µs = sin(45°) / cos(45°)
µs = tan(45°)
Using trigonometric properties, the coefficient of static friction between her climbing shoes and the wall is equal to the tangent of the angle θ, which is:
µs = tan(45°)
Therefore, the coefficient of static friction is 1.