The 180-lb rock climber is lowered over the edge of the cliff by his two companions, who together exert a horizontal pull T of 75 lb on the rope. Compute the co-efficient of friction between the rope and the rock.

To determine the coefficient of friction between the rope and the rock in this scenario, we need to analyze the forces acting on the rock climber as he is being lowered. The key forces at play here are the weight of the climber, the tension in the rope, and the frictional force that opposes the motion of the climber.

Identifying the Forces

First, let’s break down the forces involved:

• Weight of the climber (W): This is the force due to gravity acting on the climber, which can be calculated as 180 lb.
• Tension in the rope (T): The horizontal pull exerted by the companions is given as 75 lb.
• Frictional force (F_f): This is the force that resists the motion of the climber down the cliff, which we will relate to the coefficient of friction.

Understanding the Forces in Equilibrium

When the climber is being lowered, the system can be considered in equilibrium in the vertical direction. This means that the sum of the vertical forces must equal zero. The weight of the climber acts downward, while the vertical component of the tension in the rope acts upward. Since the climber is not accelerating vertically, we can express this balance of forces mathematically:

Let’s denote the vertical component of the tension as T_v. Since the tension is horizontal, we can relate it to the vertical force using trigonometry. However, in this case, we can simplify our analysis by recognizing that the frictional force must balance the weight of the climber when he is being lowered at a constant speed.

Calculating the Frictional Force

The frictional force can be expressed as:

F_f = μ * N

where μ is the coefficient of friction and N is the normal force. In this scenario, since the climber is being lowered without acceleration, the normal force is equal to the weight of the climber, which is 180 lb.

Setting Up the Equation

Since the climber is in equilibrium, the frictional force must equal the weight of the climber:

F_f = W

Substituting the expression for the frictional force, we have:

μ * N = W

Substituting N with the weight of the climber:

μ * 180 lb = 180 lb

Solving for the Coefficient of Friction

Now we can solve for μ:

μ = W / N

μ = 180 lb / 180 lb

μ = 1

Conclusion

The coefficient of friction between the rope and the rock is 1. This indicates that the frictional force is equal to the weight of the climber, which is a relatively high value, suggesting that the rope is gripping the rock effectively. In practical terms, this means that the climber can be lowered safely without slipping, provided the friction remains constant.