A uniform chain of length `2m` is kept on a table such that a length of `60 cm` hangas freely from the adge of the table . The table . The total mass of the chain ia `4 kg` What is the work done in pulling the entire the chain the on the table ?

Options

A. ` 12 J`
B. ` 3.6 J`
C. `7.2 J`
D. `1200 J`

Explanation

To solve the problem of finding the work done in pulling the entire chain onto the table, we can follow these steps:

### Step 1: Understand the Problem
We have a uniform chain of length 2m (200 cm) with a total mass of 4 kg. A portion of the chain, specifically 60 cm (0.6 m), is hanging off the edge of the table. We need to calculate the work done to pull the entire chain onto the table.

### Step 2: Determine the Mass of the Hanging Portion
The mass of the chain per unit length can be calculated as follows:
- Total mass of the chain = 4 kg
- Total length of the chain = 2 m = 200 cm

The mass per meter of the chain is:
\[ \text{Mass per meter} = \frac{\text{Total mass}}{\text{Total length}} = \frac{4 \text{ kg}}{2 \text{ m}} = 2 \text{ kg/m} \]

Now, we can find the mass of the hanging portion (60 cm):
\[ \text{Mass of hanging portion} = \text{Mass per meter} \times \text{Length of hanging portion} = 2 \text{ kg/m} \times 0.6 \text{ m} = 1.2 \text{ kg} \]

### Step 3: Calculate the Height to Raise the Center of Mass
The center of mass of the hanging portion is located at its midpoint. Since the hanging length is 60 cm, the center of mass is at:
\[ \text{Height of center of mass} = \frac{60 \text{ cm}}{2} = 30 \text{ cm} = 0.3 \text{ m} \]

### Step 4: Calculate the Work Done
The work done in raising the mass is equal to the change in potential energy. The formula for potential energy (PE) is:
\[ \text{PE} = mgh \]
where:
- \( m \) = mass of the hanging portion = 1.2 kg
- \( g \) = acceleration due to gravity = 10 m/s² (approximate value)
- \( h \) = height the center of mass is raised = 0.3 m

Now, substituting the values:
\[ \text{Work Done} = mgh = 1.2 \text{ kg} \times 10 \text{ m/s}^2 \times 0.3 \text{ m} \]
\[ \text{Work Done} = 1.2 \times 10 \times 0.3 = 3.6 \text{ joules} \]

### Conclusion
The work done in pulling the entire chain onto the table is **3.6 joules**.

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