A uniform chain of length L and mass m is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. If g is the acceleration due to gravity, then work required to pull the hanging part on the table will be :

Question

Accepted Answer

To determine the work required to pull the hanging part of the chain onto the table, we need to consider the gravitational potential energy of the hanging portion of the chain.

### Explanation for Option 4:

1. **Length of the Hanging Part:**
   - The chain has a total length $ L $, and one-third of it is hanging down. Therefore, the length of the hanging part is $ \frac{L}{3} $.

2. **Mass of the Hanging Part:**
   - The total mass of the chain is $ m $. Since mass is uniformly distributed, the mass of the hanging part is $ \frac{m}{3} $.

3. **Center of Mass of the Hanging Part:**
   - The center of mass of the hanging part is at its midpoint, which is at a distance of $ \frac{L}{6} $ from the edge of the table (half of the hanging length).

4. **Gravitational Potential Energy:**
   - The gravitational potential energy of the hanging part when it is hanging is given by:
     $$
     U = \text{mass} \times g \times \text{height of center of mass}
     $$
   - Substituting the values, we get:
     $$
     U = \left(\frac{m}{3}\right) \times g \times \left(\frac{L}{6}\right) = \frac{mgL}{18}
     $$

5. **Work Done:**
   - The work required to pull the hanging part onto the table is equal to the gravitational potential energy of the hanging part, which is $ \frac{mgL}{18} $.

Thus, the work required to pull the hanging part onto the table is indeed $\frac{MgL}{18}$, making Option 4 the correct answer.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $\frac{MgL}{6}$**
   - This option suggests a larger amount of work than is actually required. It incorrectly assumes a different distribution or calculation of potential energy.

2. **Option 2: $\frac{MgL}{9}$**
   - This option also overestimates the work required. It does not correctly account for the center of mass of the hanging part being at $\frac{L}{6}$.

3. **Option 3: $\frac{MgL}{12}$**
   - This option is closer but still overestimates the work required. It fails to accurately calculate the potential energy based on the correct center of mass position.

By following the correct calculation steps, we confirm that Option 4 is the only accurate choice, as it correctly represents the work needed to pull the hanging part of the chain onto the table.