Question

A smooth chain of length 2 m is kept on the table such that its length of 60 cm hangs freely from the edge of the table . The total mass of the chain is 4 kg. The work done in pulling the entire chain on the table is. ( Take g = 10 m/s²)

• A. 3.6 J
• B. 12.9 J
• C. 2.0 J
• D. 6.3 J

Answer 1

The Correct Option is A

Problem: Given the following values:

• Length of the chain = 2 m
• Length of the hanging part = 60 cm = 0.6 m
• Total mass of the chain = 4 kg
• Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)

We are tasked with calculating the work done in pulling the entire chain onto the table. Let's proceed step by step.

Step 1: Calculate the potential energy of the hanging part of the chain.

The potential energy (PE) is given by:

\(PE = \text{mass} \times g \times \text{height}\)

The height of the hanging part of the chain is \( 0.6 \, \text{m} \). We first calculate the mass of the hanging part. Since the total mass of the chain is \( 4 \, \text{kg} \) and the hanging part is \( 0.6 \, \text{m} \) out of the total \( 2 \, \text{m} \), the mass of the hanging part is:

\(\text{mass of hanging part} = \frac{0.6}{2} \times 4 = 0.6 \, \text{kg}\)

Now, using this mass, we can calculate the potential energy of the hanging part:

\(PE_{\text{hanging}} = 0.6 \, \text{kg} \times 10 \, \text{m/s}^2 \times 0.6 \, \text{m} = 3.6 \, \text{J}\)

Step 2: Calculate the potential energy of the entire chain when it is on the table.

The mass of the entire chain is \( 4 \, \text{kg} \), and the height of the entire chain on the table is \( 0 \, \text{m} \), so:

\(PE_{\text{table}} = 4 \, \text{kg} \times 10 \, \text{m/s}^2 \times 0 \, \text{m} = 0 \, \text{J}\)

Step 3: Calculate the work done in pulling the entire chain onto the table.

The work done is the difference between the initial potential energy (PE_hanging) and the final potential energy (PE_table):

\(\text{Work done} = PE_{\text{hanging}} - PE_{\text{table}} = 3.6 \, \text{J} - 0 \, \text{J} = 3.6 \, \text{J}\)

Therefore, the work done in pulling the entire chain onto the table is: 3.6 J (option A).

Answer 2

Mass of the chain lying freely from the table is given by: 

\( \frac{M}{L_1} = 4 \, \text{kg} \times \frac{0.6}{1} = 1.2 \, \text{kg} \)

The distance of the center of mass of the chain from the table is:

\( \frac{2}{1} \times 0.6 \, \text{m} = 0.3 \, \text{m} \)

Thus, the work done in pulling the chain can be calculated using the formula \( W = mgh \):

\( W = 1.2 \, \text{kg} \times 10 \, \text{m/s}^2 \times 0.3 \, \text{m} = 3.6 \, \text{J} \)

Therefore, the work done in pulling the chain is 3.6 J.