To solve this problem, we need to calculate the total work done in raising the climber, harness, and cable. The work $W$ is given by the formula:

$W = F_{\text{total}} \cdot d$ where:

• $F_{\text{total}}$ is the total force due to the weight of the climber, harness, and the cable.
• $d$ is the distance over which the climber is raised (13 meters).

Step 1: Calculate the force due to the climber's weight

The climber weighs 58 kg, and the force due to the climber's weight is given by: $F_{\text{climber}} = m_{\text{climber}} \cdot g = 58 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 568.4 \, \text{N}$

Step 2: Calculate the force due to the cable's weight

The weight of the cable increases as more of it is lifted. The cable weighs 0.6 kg per meter, and as the climber is raised, the length of the cable that is being lifted changes.

To account for this, we will treat the cable as a continuously distributed load and calculate the average weight of the cable over the distance of 13 meters.

The total length of the cable being lifted at the start is 13 meters. At the top, the length of the cable being lifted is 0 meters. So, the average length of the cable being lifted is:

$\text{average length of cable} = \frac{13 + 0}{2} = 6.5 \, \text{meters}$

Now, we calculate the weight of this average length of cable: $F_{\text{cable}} = m_{\text{cable}} \cdot g = (6.5 \, \text{m} \cdot 0.6 \, \text{kg/m}) \cdot 9.8 \, \text{m/s}^2 = 38.22 \, \text{N}$

Step 3: Total force to be lifted

The total force that needs to be lifted is the sum of the force due to the climber and the average force due to the cable: $F_{\text{total}} = F_{\text{climber}} + F_{\text{cable}} = 568.4 \, \text{N} + 38.22 \, \text{N} = 606.62 \, \text{N}$

Step 4: Calculate the work done

The work done is the total force multiplied by the distance (13 meters): $W = F_{\text{total}} \cdot d = 606.62 \, \text{N} \cdot 13 \, \text{m} = 7,886.06 \, \text{J}$

Final Answer:

The total work done in raising the climber, harness, and cable to the top of the rock face is approximately 7,886 Joules.

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Would you like more details on any step, or do you have further questions? Here are some related questions for you:

1. How does gravity affect the total work done in lifting an object?
2. What if the distance the climber needed to be lifted was different? How would that change the work done?
3. How would the problem change if the cable's weight per meter was heavier?
4. How do we calculate work when the force is not constant (as in the cable's case)?
5. What is the difference between average force and instantaneous force?

Tip: When dealing with continuously changing forces (like the cable), using the average value simplifies calculations!