A rubber rope of length `8 m` is hung from the ceiling of a room. What is the increase in length of rope due to its own weight? (Given: Young's modulus of elasticity of rubber `= 5 xx 10^(6) N//m` and density of rubber `=1.5xx10^(3)kg//m^(3)`. Take `g=10ms^(-12))`

Options

A. `1.5mm`
B. `6mm`
C. `24mm`
D. `96mm`

Explanation

To find the increase in length of a rubber rope due to its own weight, we can use the formula for elongation due to a uniform load:

\[ \Delta L = \frac{WL}{2AE} \]

where:
- \( W \) is the weight of the rope,
- \( L \) is the original length of the rope,
- \( A \) is the cross-sectional area of the rope,
- \( E \) is the Young's modulus of elasticity of the material.

### Step 1: Calculate the weight of the rope (W)

The weight \( W \) of the rope can be calculated using the formula:

\[ W = mg \]

where:
- \( m \) is the mass of the rope,
- \( g \) is the acceleration due to gravity.

To find the mass \( m \) of the rope, we use the formula:

\[ m = \text{Volume} \times \text{Density} \]

The volume of the rope can be expressed as:

\[ \text{Volume} = \text{Length} \times \text{Cross-sectional Area} = L \times A \]

Thus, we have:

\[ m = L \times A \times \text{Density} \]

Substituting this into the weight equation gives:

\[ W = (L \times A \times \text{Density}) \times g \]

### Step 2: Substitute values into the weight equation

Given:
- Length \( L = 8 \, \text{m} \)
- Density \( = 1.5 \times 10^3 \, \text{kg/m}^3 \)
- \( g = 10 \, \text{m/s}^2 \)

So,

\[ W = (8 \, \text{m}) \times A \times (1.5 \times 10^3 \, \text{kg/m}^3) \times (10 \, \text{m/s}^2) \]

\[ W = 12000A \, \text{N} \]

### Step 3: Substitute W into the elongation formula

Now, substituting \( W \) into the elongation formula:

\[ \Delta L = \frac{12000A \cdot L}{2AE} \]

### Step 4: Simplify the elongation formula

The area \( A \) cancels out:

\[ \Delta L = \frac{12000 \cdot 8}{2E} \]

### Step 5: Substitute the Young's modulus value

Given \( E = 5 \times 10^6 \, \text{N/m}^2 \):

\[ \Delta L = \frac{12000 \cdot 8}{2 \cdot (5 \times 10^6)} \]

### Step 6: Calculate the change in length

Calculating the above expression:

\[ \Delta L = \frac{96000}{10 \times 10^6} = \frac{96000}{10^7} = 9.6 \times 10^{-3} \, \text{m} = 9.6 \, \text{mm} \]

### Final Answer

The increase in length of the rubber rope due to its own weight is approximately:

\[ \Delta L \approx 9.6 \, \text{mm} \]