A uriform chain hangs over the edge of a horizontal platform. A machine does 1.0 J of work in pulling the rest of the chain onto the platform. The chain has a mass of 2.0 kg and a length of 3.0 m. What length was initially hanging over the edge ? On the Moon, the gravitational acceleration is 1/6 of `9.8 m//s^(2)`.

Answer

To solve the problem, we need to determine the length of the chain that was initially hanging over the edge of the platform. We know the following:

- Total mass of the chain (M) = 2.0 kg
- Total length of the chain (L) = 3.0 m
- Work done (W) = 1.0 J
- Gravitational acceleration on the Moon (g) = \( \frac{9.8}{6} \, \text{m/s}^2 \)

### Step-by-Step Solution:

1. **Determine the mass per unit length of the chain:**
\[ \text{Mass per unit length} = \frac{M}{L} = \frac{2.0 \, \text{kg}}{3.0 \, \text{m}} = \frac{2}{3} \, \text{kg/m} \]

2. **Let \( x \) be the length of the chain that is initially hanging:**
- The mass of the hanging portion of the chain (denoted as \( M' \)) can be calculated as:
\[ M' = \text{Mass per unit length} \times x = \left(\frac{2}{3}\right) x \, \text{kg} \]

3. **Calculate the weight of the hanging portion:**
- The weight (force due to gravity) of the hanging portion is given by:
\[ F = M' \cdot g = \left(\frac{2}{3} x\right) \cdot \left(\frac{9.8}{6}\right) = \frac{2 \cdot 9.8}{18} x = \frac{9.8}{9} x \, \text{N} \]

4. **Calculate the distance the center of mass of the hanging portion is lifted:**
- The center of mass of the hanging portion is located at a distance of \( \frac{x}{2} \) from the edge of the platform.

5. **Calculate the work done to lift the hanging portion:**
- The work done \( W \) is given by:
\[ W = F \cdot d \]
where \( d = \frac{x}{2} \). Therefore:
\[ W = \left(\frac{9.8}{9} x\right) \cdot \left(\frac{x}{2}\right) = \frac{9.8}{18} x^2 \]

6. **Set the work done equal to the given work:**
\[ 1.0 = \frac{9.8}{18} x^2 \]

7. **Solve for \( x^2 \):**
\[ x^2 = \frac{1.0 \cdot 18}{9.8} = \frac{18}{9.8} \approx 1.8367 \]

8. **Calculate \( x \):**
\[ x = \sqrt{1.8367} \approx 1.354 \, \text{m} \]

### Final Answer:
The length of the chain that was initially hanging over the edge is approximately **1.35 m**.