The diameter of a brass rod is 4 mm and Young's modulus of brass is `9 xx 10^(10) N//m^(2)`. The force required to stretch by 0.1 % of its length is

Options

A. `360 pi N`
B. `36 pi N`
C. `144 pi xx 10^(3)N`
D. `36 pi xx 10^(5) N`

Explanation

To find the force required to stretch a brass rod by 0.1% of its length, we will follow these steps:

### Step 1: Understand the given parameters
- Diameter of the brass rod, \( d = 4 \, \text{mm} = 4 \times 10^{-3} \, \text{m} \)
- Young's modulus of brass, \( Y = 9 \times 10^{10} \, \text{N/m}^2 \)
- The stretch (change in length), \( \Delta L = 0.1\% \, \text{of} \, L = \frac{0.1}{100} \times L = \frac{L}{1000} \)

### Step 2: Calculate the strain
Strain is defined as the change in length divided by the original length:
\[ \text{Strain} = \frac{\Delta L}{L} = \frac{L/1000}{L} = \frac{1}{1000} \]

### Step 3: Calculate the stress using Young's modulus
Young's modulus is defined as the ratio of stress to strain:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
Rearranging gives us:
\[ \text{Stress} = Y \times \text{Strain} = 9 \times 10^{10} \, \text{N/m}^2 \times \frac{1}{1000} = 9 \times 10^{7} \, \text{N/m}^2 \]

### Step 4: Calculate the area of the cross-section of the rod
The area \( A \) of the cross-section of the rod can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Where the radius \( r \) is half of the diameter:
\[ r = \frac{d}{2} = \frac{4 \times 10^{-3}}{2} = 2 \times 10^{-3} \, \text{m} \]
Thus,
\[ A = \pi (2 \times 10^{-3})^2 = \pi \times 4 \times 10^{-6} \, \text{m}^2 = 4\pi \times 10^{-6} \, \text{m}^2 \]

### Step 5: Calculate the force required
The force \( F \) can be calculated using the formula:
\[ F = \text{Stress} \times A \]
Substituting the values we have:
\[ F = 9 \times 10^{7} \, \text{N/m}^2 \times 4\pi \times 10^{-6} \, \text{m}^2 \]
Calculating this gives:
\[ F = 36\pi \times 10^{1} \, \text{N} \approx 113.097 \, \text{N} \]

### Final Answer
The force required to stretch the brass rod by 0.1% of its length is approximately \( 113.097 \, \text{N} \).

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