A 10ft chain weighs 25lb and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that it's level with the upper end.

To solve the problem of finding the work done in lifting the lower end of the chain to the ceiling, we can approach it using the concept of work done by a variable force.

### Step 1: Understand the Problem
- **Total length of the chain:** \(10\) feet
- **Total weight of the chain:** \(25\) pounds
- **Objective:** Find the work done in lifting the lower end of the chain to the ceiling so that it's level with the upper end.

### Step 2: Set Up the Problem
Let's consider an infinitesimal segment of the chain at a distance \(x\) feet from the lower end of the chain. The weight of this small segment of the chain is proportional to its length.

- **Weight per unit length:** 
  \[
  \text{Weight per foot} = \frac{25 \text{ lb}}{10 \text{ ft}} = 2.5 \text{ lb/ft}
  \]

- **Weight of an infinitesimal segment \(dx\):**
  \[
  dW = 2.5 \, dx \text{ lb}
  \]

### Step 3: Calculate the Work for Each Segment
The work \(dW\) required to lift this infinitesimal segment to the ceiling (which is \(10 - x\) feet above) is:
\[
dW = \text{force} \times \text{distance} = 2.5 \, dx \times (10 - x)
\]

### Step 4: Integrate to Find the Total Work
We integrate this expression over the length of the chain from \(x = 0\) (the lower end) to \(x = 10\) feet (the upper end):
\[
W = \int_0^{10} 2.5 \times (10 - x) \, dx
\]

### Step 5: Evaluate the Integral
\[
W = 2.5 \int_0^{10} (10 - x) \, dx
\]
\[
W = 2.5 \left[10x - \frac{x^2}{2}\right]_0^{10}
\]
Substituting the limits:
\[
W = 2.5 \left[\left(10 \times 10 - \frac{10^2}{2}\right) - \left(10 \times 0 - \frac{0^2}{2}\right)\right]
\]
\[
W = 2.5 \left[100 - 50\right]
\]
\[
W = 2.5 \times 50 = 125 \text{ foot-pounds}
\]

### Conclusion
The work done in lifting the lower end of the chain to the ceiling so that it's level with the upper end is **125 foot-pounds**.

Would you like more details or have any questions?

### Related Questions:
1. How would the work change if the chain were twice as long?
2. What would be the work done if only half the chain was lifted to the ceiling?
3. How does the weight distribution along the chain affect the work done?
4. If the chain's weight per unit length was not uniform, how would that affect the calculation?
5. How can this problem be extended to calculate work in a three-dimensional setting?

**Tip:** When solving work problems, always ensure that you consider the variable force along the path of movement.

Learn how to calculate the work done in lifting the lower end of a 10ft chain weighing 25lb to the ceiling. This problem involves using the work formula and integration to find the total work done. Suitable for high school students learning about work and energy in physics.

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