A bicycle chain of length `1.6 m` and of mass `1 kg` is lying on a horizontal floor. If `g = 10 ms^(-2)`, the work done in lifting it with one end touching the floor and the other end `1.6 m` above the floor is

Options

A. `10J`
B. `3.2 J`
C. `8J`
D. `16J`

Explanation

To solve the problem of calculating the work done in lifting a bicycle chain from a horizontal position to a vertical position, we can follow these steps:

### Step 1: Understand the Problem
We have a bicycle chain of length \( L = 1.6 \, \text{m} \) and mass \( m = 1 \, \text{kg} \). The chain is initially lying flat on the floor. We need to lift one end of the chain to a height of \( 1.6 \, \text{m} \) while the other end remains on the floor.

### Step 2: Identify the Center of Mass
For a uniform chain, the center of mass is located at the midpoint. Therefore, the center of mass of the chain is at:
\[ \text{Height of center of mass} = \frac{L}{2} = \frac{1.6 \, \text{m}}{2} = 0.8 \, \text{m} \]

### Step 3: Calculate the Change in Height
When we lift the chain, the center of mass moves from an initial height of \( 0 \, \text{m} \) (when the chain is horizontal) to a final height of \( 0.8 \, \text{m} \) (when the chain is vertical). Thus, the change in height (\( \Delta h \)) is:
\[ \Delta h = 0.8 \, \text{m} - 0 \, \text{m} = 0.8 \, \text{m} \]

### Step 4: Calculate the Work Done
The work done against gravity to lift the chain can be calculated using the formula for gravitational potential energy:
\[ \text{Work Done} = m \cdot g \cdot \Delta h \]
Where:
- \( m = 1 \, \text{kg} \) (mass of the chain)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( \Delta h = 0.8 \, \text{m} \) (change in height)

Substituting the values:
\[ \text{Work Done} = 1 \, \text{kg} \cdot 10 \, \text{m/s}^2 \cdot 0.8 \, \text{m} = 8 \, \text{J} \]

### Conclusion
The work done in lifting the bicycle chain from a horizontal position to a vertical position is \( 8 \, \text{J} \).

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