A mountaineer is doing a vertical rappel. She is using a big boulder/stone as her anchor. The Mountain guides association advises that in this type of situation, the boulder should be bigger than a refrigerator. The boulder should be sitting on a surface that is horizontal rather than sloping. The aim for this problem is to approximate the coefficient of static friction between the boulder and the ledge is required if this setup is to hold the climber's body weight. For reference, granite blocks typically has a static coefficient of approximately 0.6. It is expected that our calculation will result in a value that's smaller than this. There is no friction where the rope goes over the lip of the cliff.

Question

A mountaineer is doing a vertical rappel. She is using a big boulder/stone as her anchor. The Mountain guides association advises that in this type of situation, the boulder should be bigger than a refrigerator. The boulder should be sitting on a surface that is horizontal rather than sloping. The aim for this problem is to approximate the coefficient of static friction between the boulder and the ledge is required if this setup is to hold the climber's body weight. For reference, granite blocks typically has a static coefficient of approximately 0.6. It is expected that our calculation will result in a value that's smaller than this. There is no friction where the rope goes over the lip of the cliff.

What is the minimum value of the static friction coefficient, express in terms of m-mass of the climber, V -volume of the boulder, d- density, and g

Answer 1

To determine the minimum value of the static friction coefficient required to hold the climber's body weight, we need to consider the forces acting on the system.

First, let's analyze the forces acting on the climber:
1. The weight of the climber (mg), where m is the mass of the climber and g is the acceleration due to gravity.

Next, let's consider the forces acting on the boulder:

1. The weight of the boulder (Vdg), where V is the volume of the boulder and d is its density.
2. The static friction force between the boulder and the ledge (Fs).

For the climber to descend safely, the force of static friction (Fs) must be greater than or equal to the weight of the climber (mg). Mathematically, we can express this as:

Fs ≥ mg

Now, we need to relate the friction force (Fs) to the normal force (N) acting on the boulder. The normal force is equal to the weight of the boulder (Vdg). The friction force can be expressed as the product of the static friction coefficient (μs) and the normal force (N). Therefore:

Fs = μs * N

Substituting the weight of the boulder (Vdg) for the normal force (N), we have:

Fs = μs * Vdg

Since we want to approximate the static friction coefficient (μs), we can rearrange the equation to solve for it:

μs = Fs / (Vdg)

Now, substituting the minimum value of the friction force (mg) for Fs, we can express the minimum value of the static friction coefficient as:

μs(min) = mg / (Vdg)

Therefore, the minimum value of the static friction coefficient is given by mg / (Vdg), where m is the mass of the climber, V is the volume of the boulder, d is the density of the boulder, and g is the acceleration due to gravity.