a climber with a weight of 590 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 = 35˚ and φ = 35˚. 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 find the coefficient of static friction between the climber's shoes and the wall, we can use the condition that her feet are on the verge of sliding. The force of static friction is given by the equation:
fs = μs * N,
where fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force.
The normal force can be calculated as the component of the climber's weight perpendicular to the wall:
N = W * cos(θ),
where W is the weight of the climber.
The force of static friction can be found as the component of the climber's weight parallel to the wall:
fs = W * sin(θ).
Therefore, we can rewrite the equation for the force of static friction as:
W * sin(θ) = μs * W * cos(θ).
We can now solve for the coefficient of static friction:
μs = tan(θ).
Given that θ = 35˚, we can calculate the coefficient of static friction as:
μs = tan(35˚).
Using a calculator, we find:
μs ≈ 0.700
Therefore, the coefficient of static friction between the climber's shoes and the wall is approximately 0.700.
To find the coefficient of static friction between the climber's shoes and the wall, we need to analyze the forces acting on the climber.
Let's break down the forces:
1. Weight of the climber (590 N): This force acts vertically downwards, through the center of mass of the climber.
2. Force exerted by the rope: The rope supports the climber and prevents her from falling. This force can be broken down into two components:
- Vertical component: This component counters the weight of the climber and ensures she stays on the wall.
- Horizontal component: This component acts parallel to the wall and provides the necessary friction to prevent her feet from sliding.
The angle θ is the angle between the vertical component of the rope's force and the weight of the climber. Similarly, the angle φ is the angle between the horizontal component of the rope's force and the weight of the climber.
Since her feet are on the verge of sliding, we know that the maximum static friction force is acting on them. The formula for the maximum static friction force (F_friction) is given by:
F_friction = coefficient_of_static_friction * normal_force
In this case, the normal force is equal to the vertical component of the rope's force.
To calculate the normal force (N), we can use trigonometry:
N = vertical_component_of_rope_force = rope_force * cos(θ)
Now we can substitute the values into the formula for the maximum static friction force:
F_friction = coefficient_of_static_friction * N
Since the horizontal component of the rope's force is equal to the maximum static friction force, we have:
horizontal_component_of_rope_force = F_friction = coefficient_of_static_friction * N
Now, we can solve for the coefficient of static friction:
coefficient_of_static_friction = F_friction / N = horizontal_component_of_rope_force / (rope_force * cos(θ))
Given that rope_force = 590 N, θ = 35˚, and φ = 35˚, we can substitute these values into the equation to find the coefficient of static friction.
[radius (perpendicular) * sin35º] / [radius perpendicular*cos 35º]
-the radius cancel and your left with....
sin35º / cos35º
=.70 is the coefficient of static friction