A heavy chain of length `1 m` and weight `20 kg` hangs vertically with one end attached to a peg and carries a block of mass `10 kg` at the other end. Find the work done in winding `50 cm` of chain round the peg.

Options

A. `85J`
B. `100J`
C. `120J`
D. `125J`

Explanation

To solve the problem of finding the work done in winding 50 cm of the chain around the peg, we can break it down into steps as follows:

### Step 1: Understand the system
We have a vertical chain of length 1 m and weight 20 kg hanging from a peg, with a block of mass 10 kg attached at the end. When we wind up 50 cm of the chain, we need to calculate how much work is done against gravity.

### Step 2: Identify the center of mass of the chain
The chain is uniform, so the center of mass of the entire chain is located at its midpoint. Since the chain is 1 m long, the center of mass is at 0.5 m from the top.

### Step 3: Calculate the change in height for the center of mass of the chain
When we wind up 50 cm (0.5 m) of the chain, the center of mass of the upper half of the chain (which is 0.5 m long) will rise by 25 cm (0.25 m), and the center of mass of the lower half of the chain (also 0.5 m long) will rise by 50 cm (0.5 m).

### Step 4: Calculate the work done on the block
The work done on the block of mass 10 kg that is raised by 50 cm is given by:
\[ W_{\text{block}} = m_{\text{block}} \cdot g \cdot h_{\text{block}} \]
Where:
- \( m_{\text{block}} = 10 \, \text{kg} \)
- \( g = 10 \, \text{m/s}^2 \) (approximate value of gravitational acceleration)
- \( h_{\text{block}} = 0.5 \, \text{m} \)

Calculating this gives:
\[ W_{\text{block}} = 10 \cdot 10 \cdot 0.5 = 50 \, \text{J} \]

### Step 5: Calculate the work done on the chain
Now we need to calculate the work done on the chain. The weight of the chain is 20 kg, and we can consider it in two parts: the upper half and the lower half.

1. **Upper half of the chain (0.5 m, mass = 10 kg)**:
The center of mass rises by 25 cm (0.25 m):
\[ W_{\text{upper}} = m_{\text{upper}} \cdot g \cdot h_{\text{upper}} \]
\[ W_{\text{upper}} = 10 \cdot 10 \cdot 0.25 = 25 \, \text{J} \]

2. **Lower half of the chain (0.5 m, mass = 10 kg)**:
The center of mass rises by 50 cm (0.5 m):
\[ W_{\text{lower}} = m_{\text{lower}} \cdot g \cdot h_{\text{lower}} \]
\[ W_{\text{lower}} = 10 \cdot 10 \cdot 0.5 = 50 \, \text{J} \]

### Step 6: Total work done
Now, we sum the work done on the block and the work done on both halves of the chain:
\[ W_{\text{total}} = W_{\text{block}} + W_{\text{upper}} + W_{\text{lower}} \]
\[ W_{\text{total}} = 50 + 25 + 50 = 125 \, \text{J} \]

### Conclusion
The total work done in winding 50 cm of the chain around the peg is **125 Joules**.