Find the work done by winding up a hanging cable of length 100 ft and weight density 5 lb/ft.

To find the work done in winding up a hanging cable of length 100 ft with a weight density of 5 lb/ft, we can follow these steps:

### Step 1: Understand the Problem
- **Length of cable** = 100 ft
- **Weight density** = 5 lb/ft

We need to calculate the work required to wind up the cable. The force required to lift any segment of the cable depends on how much of the cable is still hanging.

### Step 2: Define the Variables
Let \( x \) represent the distance (in feet) from the top of the cable to the segment being lifted. Thus, the weight of the remaining cable below \( x \) is proportional to the length still hanging, which is \( (100 - x) \) feet.

- The weight of the cable segment below \( x \) is \( 5 \times (100 - x) \) lb.

### Step 3: Set Up the Work Integral
Work is calculated as \( W = \int F \, dx \), where \( F \) is the force required to lift each infinitesimal segment of the cable. The force here is the weight of the cable that is still hanging at any height \( x \).

So, the work done in lifting the cable is:
\[
W = \int_0^{100} 5(100 - x) \, dx
\]

### Step 4: Evaluate the Integral
Now we evaluate the integral:
\[
W = 5 \int_0^{100} (100 - x) \, dx
\]
\[
W = 5 \left[ 100x - \frac{x^2}{2} \right]_0^{100}
\]
Substituting the limits of integration:
\[
W = 5 \left[ 100(100) - \frac{100^2}{2} \right]
\]
\[
W = 5 \left[ 10000 - 5000 \right]
\]
\[
W = 5 \times 5000 = 25000 \, \text{ft-lb}
\]

### Final Answer:
The total work required to wind up the cable is **25,000 ft-lb**.

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Do you want further details or have any questions?

Here are 5 related questions to expand on this topic:
1. How does the work change if the weight density of the cable is increased?
2. How would the work calculation differ if the cable had a non-uniform weight distribution?
3. What would be the work done if only half the cable was wound up?
4. How is the concept of work applied in other contexts, such as lifting objects with varying forces?
5. What is the physical meaning of work done in this context compared to other energy-related terms?

**Tip:** Work is the transfer of energy, and it depends not only on the force applied but also on the distance over which the force acts.

Learn how to calculate the work done in winding up a hanging cable with length 100 ft and a weight density of 5 lb/ft. This problem involves calculus concepts such as integration and the work-energy theorem. Suitable for advanced high school and college students.

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