How much force is required to produce an increase of 0.2 % in the length of a brass wire of diameter 0.6 mm
(Young's modulus for brass `=0.9 xx 10^(11) N//m^(2)`)

Options

A. Nearly 17 N
B. Nearly 34 N
C. Nearly 51 N
D. Nearly 68 N

Answer

To find the force required to produce a 0.2% increase in the length of a brass wire, we can use the formula related to Young's modulus:

\[ Y = \frac{F/A}{\Delta L/L_0} \]

Where:
- \( Y \) is Young's modulus,
- \( F \) is the force applied,
- \( A \) is the cross-sectional area of the wire,
- \( \Delta L \) is the change in length,
- \( L_0 \) is the original length.

### Step 1: Convert the percentage increase in length to a decimal.
Given that the increase in length is 0.2%, we convert this to a decimal:

\[ \Delta L = 0.2\% \times L_0 = 0.002 \times L_0 \]

### Step 2: Calculate the cross-sectional area of the wire.
The diameter of the wire is given as 0.6 mm. First, we convert this to meters:

\[ d = 0.6 \text{ mm} = 0.6 \times 10^{-3} \text{ m} = 0.0006 \text{ m} \]

The radius \( r \) is half of the diameter:

\[ r = \frac{d}{2} = \frac{0.0006}{2} = 0.0003 \text{ m} \]

The cross-sectional area \( A \) of the wire is given by the formula for the area of a circle:

\[ A = \pi r^2 = \pi (0.0003)^2 \approx 2.827 \times 10^{-7} \text{ m}^2 \]

### Step 3: Rearrange the Young's modulus formula to find the force \( F \).
From the Young's modulus formula, we can rearrange it to solve for \( F \):

\[ F = Y \cdot A \cdot \frac{\Delta L}{L_0} \]

### Step 4: Substitute the known values into the equation.
We know:
- \( Y = 0.9 \times 10^{11} \text{ N/m}^2 \)
- \( A \approx 2.827 \times 10^{-7} \text{ m}^2 \)
- \( \Delta L = 0.002 \times L_0 \)

Substituting these values into the equation gives:

\[ F = (0.9 \times 10^{11}) \cdot (2.827 \times 10^{-7}) \cdot \frac{0.002 \times L_0}{L_0} \]

The \( L_0 \) cancels out:

\[ F = (0.9 \times 10^{11}) \cdot (2.827 \times 10^{-7}) \cdot 0.002 \]

### Step 5: Calculate the force \( F \).
Now we can calculate:

\[ F = 0.9 \times 10^{11} \times 2.827 \times 10^{-7} \times 0.002 \]

Calculating this step-by-step:

1. \( 0.9 \times 0.002 = 0.0018 \)
2. \( 0.0018 \times 2.827 \approx 0.00509 \)
3. \( 0.00509 \times 10^{11} = 5.09 \times 10^{8} \text{ N} \)

Thus, the force required is approximately:

\[ F \approx 5.09 \times 10^{8} \text{ N} \]

### Final Answer:
The force required to produce a 0.2% increase in the length of the brass wire is approximately \( 5.09 \times 10^{8} \text{ N} \).