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 find the work done in winding 50 cm of the chain around the peg, we can follow these steps:

### Step 1: Understand the system
We have a heavy chain of length 1 m and weight 20 kg hanging vertically, with one end attached to a peg and the other end carrying a block of mass 10 kg. When we wind 50 cm of the chain around the peg, we need to calculate the work done against gravity.

### Step 2: Determine the initial height of the center of mass
Initially, the entire chain is hanging down, and the block is at the end of the chain. The center of mass of the chain is located at its midpoint, which is 0.5 m from the peg. The block is also at the end of the chain, which is initially at a height of 1 m from the peg.

### Step 3: Calculate the new height after winding
When we wind 50 cm of the chain around the peg, the height of the chain reduces by 0.5 m. Therefore, the new height of the center of mass of the chain becomes:
- Initial height of the center of mass of the chain = 0.5 m
- New height after winding = 0.5 m - 0.5 m = 0 m

The block, which was initially at 1 m, will now be at:
- Initial height of the block = 1 m
- New height of the block after winding = 1 m - 0.5 m = 0.5 m

### Step 4: Calculate the change in height
The change in height for the center of mass of the chain is:
- Change in height of chain = 0 m - 0.5 m = -0.5 m (it goes down)
The change in height for the block is:
- Change in height of block = 0.5 m - 1 m = -0.5 m (it also goes down)

### Step 5: Calculate the total weight
The total weight (force due to gravity) acting on the system is the sum of the weights of the chain and the block:
- Weight of the chain = 20 kg × g
- Weight of the block = 10 kg × g
- Total weight = (20 kg + 10 kg) × g = 30 kg × g

### Step 6: Calculate the work done
The work done (W) against gravity when winding the chain can be calculated using the formula:
\[ W = \text{Total weight} \times \text{Change in height} \]
Since both the chain and the block are moving down, we consider the absolute value of the change in height:
\[ W = 30g \times 0.5 \]

Substituting \( g \approx 9.8 \, \text{m/s}^2 \):
\[ W = 30 \times 9.8 \times 0.5 \]
\[ W = 30 \times 4.9 \]
\[ W = 147 \, \text{J} \]

### Final Answer:
The work done in winding 50 cm of the chain around the peg is **147 Joules**.

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