Calculate the force required to incrase the length of wire of cross-sectional area ` 10^(-6) m^(2)` by 50% if the Young's modulus of the material of the wire is `90 xx 10^(9) Pa`.

Explanation

To solve the problem of calculating the force required to increase the length of a wire by 50%, we can use the formula related to Young's modulus. Here’s a step-by-step solution:

### Step 1: Understand the given data
- Cross-sectional area (A) = \(10^{-6} \, m^2\)
- Young's modulus (Y) = \(90 \times 10^9 \, Pa\)
- Change in length (ΔL) = 50% of original length (L)

### Step 2: Express the change in length
Since the change in length is 50%, we can express it as:
\[ \Delta L = 0.5L \]
Thus, the ratio \(\frac{\Delta L}{L} = 0.5\).

### Step 3: Use the formula for Young's modulus
Young's modulus (Y) is defined as:
\[ Y = \frac{F/A}{\Delta L/L} \]
Where:
- F = Force applied
- A = Cross-sectional area
- \(\Delta L/L\) = Change in length per unit original length

Rearranging the formula to solve for force (F):
\[ F = Y \cdot A \cdot \frac{\Delta L}{L} \]

### Step 4: Substitute the values into the formula
Now we can substitute the known values into the formula:
\[ F = (90 \times 10^9 \, Pa) \cdot (10^{-6} \, m^2) \cdot (0.5) \]

### Step 5: Calculate the force
Calculating the force:
\[ F = 90 \times 10^9 \cdot 10^{-6} \cdot 0.5 \]
\[ F = 90 \times 0.5 \times 10^3 \]
\[ F = 45 \times 10^3 \]
\[ F = 45000 \, N \]

### Final Answer
The force required to increase the length of the wire by 50% is **45000 N**.

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