A uniform wire 6m long weighing 0.04 kg elongates by 0.8 mm when stretched by a load of 1 kg . If its density is 8.9 gm `//` cc, find the Young's modulus of the material.

Explanation

To find the Young's modulus of the material of the wire, we will follow these steps:

### Step 1: Understand the formula for Young's Modulus
Young's modulus (Y) is defined as the ratio of stress to strain:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]

### Step 2: Calculate the Stress
Stress is defined as the restoring force per unit area. The restoring force (F) can be calculated using the weight of the load applied to the wire:
\[ F = mg \]
Where:
- \( m = 1 \, \text{kg} \) (mass of the load)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)

So,
\[ F = 1 \times 9.8 = 9.8 \, \text{N} \]

### Step 3: Calculate the Area of the Wire
To find the area (A) of the wire, we can use the density (ρ) and mass (M) of the wire:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
The volume (V) can be expressed as:
\[ V = A \times L_0 \]
Where \( L_0 = 6 \, \text{m} \) (original length of the wire).

Rearranging gives:
\[ A = \frac{M}{\rho \times L_0} \]
Given:
- \( M = 0.04 \, \text{kg} \)
- \( \rho = 8.9 \, \text{g/cm}^3 = 8900 \, \text{kg/m}^3 \) (converting to SI units)

Now substituting the values:
\[ A = \frac{0.04}{8900 \times 6} \]

### Step 4: Calculate the Strain
Strain is defined as the change in length (ΔL) divided by the original length (L_0):
\[ \text{Strain} = \frac{\Delta L}{L_0} \]
Given:
- \( \Delta L = 0.8 \, \text{mm} = 0.8 \times 10^{-3} \, \text{m} \)

So,
\[ \text{Strain} = \frac{0.8 \times 10^{-3}}{6} \]

### Step 5: Substitute values into Young's Modulus formula
Now we can substitute the values of stress and strain into the Young's modulus formula:
\[ Y = \frac{F/A}{\Delta L/L_0} \]

### Step 6: Calculate Young's Modulus
Substituting the values we have:
\[ Y = \frac{(9.8 \, \text{N}) / A}{(0.8 \times 10^{-3} \, \text{m}) / 6} \]

### Step 7: Final Calculation
After calculating the area and substituting it back, we will find the Young's modulus value.

### Final Answer
After performing the calculations, we find:
\[ Y \approx 9.8 \times 10^{10} \, \text{N/m}^2 \]