A chain of mass m and length l is placed on a table with one-sixth of it hanging freely from the table edge. The amount of work. done to pull the chain on the table is

Options

A. `mgl//4`
B. `mgl//6`
C. `mgl//72`
D. `mgl//36`

Answer

To solve the problem of how much work is done to pull the chain onto the table, we can follow these steps:

### Step 1: Understand the Setup
We have a chain of mass \( m \) and length \( l \) placed on a table, with \( \frac{1}{6} \) of it hanging over the edge. This means that the length of the hanging part of the chain is:
\[ l_h = \frac{l}{6} \]

### Step 2: Determine the Mass of the Hanging Part
The mass of the hanging part of the chain can be calculated as:
\[ m_h = \frac{m}{6} \]

### Step 3: Find the Center of Mass of the Hanging Part
The center of mass of the hanging part, which is \( \frac{l}{6} \) long, is located at half of its length from the edge of the table. Therefore, the distance from the edge of the table to the center of mass is:
\[ d_{cm} = \frac{1}{2} \left( \frac{l}{6} \right) = \frac{l}{12} \]

### Step 4: Calculate the Work Done Against Gravity
When pulling the chain onto the table, we need to lift the center of mass of the hanging part up to the level of the table. The work done \( W \) against gravity can be calculated using the formula:
\[ W = \text{Force} \times \text{Distance} \]
The force acting on the hanging mass is its weight, which is given by:
\[ F = m_h \cdot g = \frac{m}{6} \cdot g \]
The distance through which this mass is lifted is \( \frac{l}{12} \). Therefore, the work done is:
\[ W = \left( \frac{m}{6} \cdot g \right) \cdot \left( \frac{l}{12} \right) \]

### Step 5: Simplify the Expression
Now we can simplify the expression for work done:
\[ W = \frac{m \cdot g \cdot l}{6 \cdot 12} = \frac{m \cdot g \cdot l}{72} \]

### Final Answer
Thus, the amount of work done to pull the chain onto the table is:
\[ W = \frac{m \cdot g \cdot l}{72} \]

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