One fourth chain is hanging down from a table work done to bring the hanging part of the chain on the table is ( mass of chain `=`M and length `=` L)

Options

A. `(MgL)/(32)`
B. `(MgL)/(16)`
C. `(MgL)/(8)`
D. `(MgL)/(4)`

Answer

To solve the problem of calculating the work done to bring the hanging part of the chain onto the table, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the parameters**:
- The total mass of the chain is \( M \).
- The total length of the chain is \( L \).
- The length of the chain hanging down from the table is \( \frac{L}{4} \).

2. **Determine the mass of the hanging part**:
- The mass per unit length of the chain is given by \( \frac{M}{L} \).
- Therefore, the mass of the hanging part (length \( \frac{L}{4} \)) is:
\[ m_{\text{hanging}} = \frac{M}{L} \times \frac{L}{4} = \frac{M}{4} \]

3. **Calculate the force acting on the hanging part**:
- The weight of the hanging part, which is the force acting downwards, is:
\[ F = m_{\text{hanging}} \cdot g = \frac{M}{4} \cdot g \]

4. **Set up the work done calculation**:
- The work done to lift the hanging part of the chain can be calculated using the integral of force over distance. The distance the center of mass of the hanging part moves when lifted onto the table is \( \frac{L}{8} \) (since the center of mass of the hanging part is at \( \frac{L}{8} \) from the table).
- The work done \( W \) is given by:
\[ W = F \cdot d = \left(\frac{M}{4} \cdot g\right) \cdot \left(\frac{L}{8}\right) \]

5. **Calculate the work done**:
- Substituting the values into the equation:
\[ W = \frac{M}{4} \cdot g \cdot \frac{L}{8} = \frac{M \cdot g \cdot L}{32} \]

6. **Final result**:
- The work done to bring the hanging part of the chain onto the table is:
\[ W = \frac{M \cdot g \cdot L}{32} \]

### Conclusion:
The work done to bring the hanging part of the chain onto the table is \( \frac{M \cdot g \cdot L}{32} \).