The force required to stretch a steel wire `100m^(2)` a section to triple its length is `(Y= 2xx 10^(11)Nm^(-2))`

Options

A. `4 xx 10^(10)N`
B. `4 xx 10^(12)N`
C. `4 xx 10^(11)N`
D. `4 xx 10^(13)N`

Explanation

To solve the problem of finding the force required to stretch a steel wire with a cross-sectional area of \(100 \, m^2\) to triple its length, we can use the relationship between stress, strain, and Young's modulus. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- Young's modulus \(Y = 2 \times 10^{11} \, N/m^2\)
- Cross-sectional area \(A = 100 \, m^2\)
- The wire is to be stretched to triple its original length.

### Step 2: Define the Variables
Let:
- Original length of the wire = \(L\)
- Final length of the wire = \(3L\) (since it is to be tripled)
- Change in length \(\Delta L = 3L - L = 2L\)

### Step 3: Calculate the Strain
Strain is defined as the change in length divided by the original length:
\[ \text{Strain} = \frac{\Delta L}{L} = \frac{2L}{L} = 2 \]

### Step 4: Calculate the Stress
Stress is defined as the force per unit area:
\[ \text{Stress} = \frac{F}{A} \]

### Step 5: Relate Stress and Strain using Young's Modulus
Young's modulus \(Y\) is defined as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
From this, we can express force \(F\) as:
\[ F = Y \times \text{Strain} \times A \]

### Step 6: Substitute the Values
Substituting the known values into the equation:
\[ F = (2 \times 10^{11} \, N/m^2) \times (2) \times (100 \, m^2) \]

### Step 7: Calculate the Force
Calculating the force:
\[ F = 2 \times 10^{11} \times 2 \times 100 = 4 \times 10^{13} \, N \]

### Final Answer
The force required to stretch the steel wire to triple its length is:
\[ \boxed{4 \times 10^{13} \, N} \]

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