A weight of 200 kg is suspended by vertical wire of length `600.5 cm`. The area of cross-section of wire is `1mm^(2)`. When the load is removed, the wire contracts by `0.5 cm`. The Young's modulus of the material of wire will be

Options

A. `2.35xx10^(12)N//m^(2)`
B. `1.35 xx 10^(10)N//m^(2)`
C. `13.5 xx 10^(11)N//m^(2)`
D. `23.5 xx 10^(9)N//m^(2)`

Answer

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

### Step 1: Identify the given data
- Mass of the weight (m) = 200 kg
- Length of the wire (L) = 600.5 cm = 6.005 m (convert cm to m)
- Area of cross-section (A) = 1 mm² = 1 x 10⁻⁶ m² (convert mm² to m²)
- Change in length (ΔL) = 0.5 cm = 0.005 m (convert cm to m)

### Step 2: Calculate the force (F) applied on the wire
The force applied due to the weight is given by:
\[ F = m \cdot g \]
Where \( g \) (acceleration due to gravity) is approximately \( 9.8 \, \text{m/s}^2 \).

Calculating the force:
\[ F = 200 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 1960 \, \text{N} \]

### Step 3: Use the Young's modulus formula
The formula for Young's modulus (Y) is:
\[ Y = \frac{F \cdot L}{A \cdot \Delta L} \]

### Step 4: Substitute the values into the Young's modulus formula
Now substituting the known values into the formula:
- \( F = 1960 \, \text{N} \)
- \( L = 6.005 \, \text{m} \)
- \( A = 1 \times 10^{-6} \, \text{m}^2 \)
- \( \Delta L = 0.005 \, \text{m} \)

So,
\[ Y = \frac{1960 \, \text{N} \cdot 6.005 \, \text{m}}{1 \times 10^{-6} \, \text{m}^2 \cdot 0.005 \, \text{m}} \]

### Step 5: Calculate Young's modulus
Calculating the numerator:
\[ 1960 \cdot 6.005 = 11763.8 \, \text{N m} \]

Calculating the denominator:
\[ 1 \times 10^{-6} \cdot 0.005 = 5 \times 10^{-9} \, \text{m}^3 \]

Now, substituting these into the Young's modulus formula:
\[ Y = \frac{11763.8}{5 \times 10^{-9}} \]

Calculating:
\[ Y = 2.35276 \times 10^{12} \, \text{N/m}^2 \]

### Step 6: Final result
Thus, the Young's modulus of the material of the wire is approximately:
\[ Y \approx 2.35 \times 10^{12} \, \text{N/m}^2 \]

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