In Fig. 12-39, 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?

Question

In Fig. 12-39, 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?

Answer 1

To find the coefficient of static friction between the climber's climbing shoes and the wall, we need to consider the forces acting on the climber.

First, let's analyze the forces acting vertically. The weight of the climber (441 N) can be split into two components: one acting downward vertically and one acting perpendicular to the wall. The vertical component is given by:

Vertical component = weight * cos(θ)
= 441 N * cos(45°)
= 441 N * 0.707
= 311.25 N

This vertical component represents the force with which the climber pushes against the wall, trying to slide down.

Now, let's analyze the forces acting horizontally. The force of the rope, which acts through the center of mass, can be split into two components: one acting parallel to the wall and one acting perpendicular to the wall. The horizontal component is given by:

Horizontal component = weight * sin(θ)
= 441 N * sin(45°)
= 441 N * 0.707
= 311.25 N

This horizontal component represents the force with which the climber is pushing against the wall, trying to slide sideways.

Since the climber's feet are on the verge of sliding on the vertical wall, we know that the static friction force is equal in magnitude to the horizontal component of the force of the rope. Therefore, the maximum static friction force is 311.25 N.

The maximum static friction force can be calculated using the equation:

Maximum static friction force = coefficient of static friction * Normal force

Since the normal force is equal to the vertical component of the weight, we have:

Maximum static friction force = coefficient of static friction * 311.25 N

Now, we can solve for the coefficient of static friction:

Coefficient of static friction = (Maximum static friction force) / (Normal force)
= 311.25 N / 311.25 N
= 1

Therefore, the coefficient of static friction between the climber's climbing shoes and the wall is 1.