A uniform chain of length `l` and mass m is placed on a smooth table with one-fourth of its length hanging over the edge. The work that has to be done to pull the whole chain back onto the table is :

Options

A. `1/4 m gl`
B. `1/8 m gl`
C. `1/16 m gl`
D. `1/32 mgl`

Explanation

To solve the problem of calculating the work done to pull the entire chain back onto the table, we will follow these steps:

### Step 1: Understand the setup
We have a uniform chain of length \( l \) and mass \( m \). One-fourth of the chain's length is hanging over the edge of a smooth table. This means that the length of the chain hanging off the table is \( \frac{l}{4} \), and the length on the table is \( \frac{3l}{4} \).

### Step 2: Determine the mass of the hanging part
Since the chain is uniform, the mass per unit length is given by:
\[ \text{Mass per unit length} = \frac{m}{l} \]
The mass of the hanging part (length \( \frac{l}{4} \)) is:
\[ m_h = \frac{m}{l} \cdot \frac{l}{4} = \frac{m}{4} \]

### Step 3: Find the center of mass of the hanging part
The center of mass of the hanging part is located at a distance of \( \frac{l}{8} \) below the table (since it hangs halfway down its length of \( \frac{l}{4} \)).

### Step 4: Calculate the change in potential energy
When the entire chain is pulled back onto the table, the potential energy of the hanging part will change. The initial potential energy \( U_i \) of the hanging mass (which is at a height of \( -\frac{l}{8} \)) is given by:
\[ U_i = m_h \cdot g \cdot h = \frac{m}{4} \cdot g \cdot \left(-\frac{l}{8}\right) = -\frac{mgl}{32} \]
When the chain is fully on the table, its potential energy \( U_f \) is zero (as we take the table level as the reference point).

### Step 5: Calculate the work done by the external agent
The work done \( W \) by the external agent to pull the chain back onto the table is equal to the change in potential energy:
\[ W = U_f - U_i = 0 - \left(-\frac{mgl}{32}\right) = \frac{mgl}{32} \]

### Conclusion
The work that has to be done to pull the whole chain back onto the table is:
\[ \boxed{\frac{mgl}{32}} \]

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