What will be the work done by external agent to slowly hang the lower end of the chain to the peg?

Options

A. `-(MgL)/(2)`
B. `(MgL)/(2)`
C. `-(MgL)/(4)`
D. `(MgL)/(4)`

Explanation

To find the work done by an external agent to slowly hang the lower end of a chain to a peg, we can follow these steps:

### Step 1: Understand the System
We have a chain of mass \( m \) and length \( L \). The chain is initially hanging vertically, and we need to hang the lower end of the chain to a peg, which will change the configuration of the chain.

### Step 2: Identify the Initial and Final States
- **Initial State**: The entire chain is hanging down. The center of mass of the chain is at a height of \( \frac{L}{2} \) from the ground.
- **Final State**: When the lower end of the chain is attached to the peg, the chain will form a "U" shape. The center of mass of the chain will be at a height of \( \frac{3L}{4} \) from the ground.

### Step 3: Calculate the Initial Potential Energy
The potential energy (PE) of the chain when it is fully hanging is given by:
\[ PE_{\text{initial}} = mgh_{\text{initial}} = mg\left(\frac{L}{2}\right) = \frac{mgL}{2} \]

### Step 4: Calculate the Final Potential Energy
When the lower end is hung to the peg, the center of mass of the chain is at a height of \( \frac{3L}{4} \):
\[ PE_{\text{final}} = mgh_{\text{final}} = mg\left(\frac{3L}{4}\right) = \frac{3mgL}{4} \]

### Step 5: Calculate the Change in Potential Energy
The work done by the external agent is equal to the change in potential energy:
\[ W = PE_{\text{final}} - PE_{\text{initial}} \]
Substituting the values:
\[ W = \frac{3mgL}{4} - \frac{mgL}{2} \]

### Step 6: Simplify the Expression
To simplify, we need a common denominator:
\[ \frac{mgL}{2} = \frac{2mgL}{4} \]
Thus,
\[ W = \frac{3mgL}{4} - \frac{2mgL}{4} = \frac{mgL}{4} \]

### Conclusion
The work done by the external agent to slowly hang the lower end of the chain to the peg is:
\[ \boxed{\frac{mgL}{4}} \]