To find Young's modulus of steel, we will use the formula:
\[
Y = \frac{\text{Stress}}{\text{Strain}}
\]
### Step 1: Calculate the Stress
Stress is defined as the load applied per unit area.
1. **Convert the load from kg to Newtons**:
\[
\text{Load (F)} = 12.7 \, \text{kg} \times g = 12.7 \, \text{kg} \times 9.81 \, \text{m/s}^2 \approx 124.5 \, \text{N}
\]
2. **Calculate the cross-sectional area (A) of the wire**:
The diameter of the wire is given as 0.075 cm, which we need to convert to meters:
\[
\text{Diameter} = 0.075 \, \text{cm} = 0.075 \times 10^{-2} \, \text{m} = 0.00075 \, \text{m}
\]
The radius (r) is half of the diameter:
\[
r = \frac{0.00075}{2} = 0.000375 \, \text{m}
\]
Now, calculate the area using the formula for the area of a circle \( A = \pi r^2 \):
\[
A = \pi (0.000375)^2 \approx 3.14 \times (0.000375)^2 \approx 4.42 \times 10^{-7} \, \text{m}^2
\]
3. **Calculate Stress**:
\[
\text{Stress} = \frac{F}{A} = \frac{124.5 \, \text{N}}{4.42 \times 10^{-7} \, \text{m}^2} \approx 2.82 \times 10^{8} \, \text{N/m}^2
\]
### Step 2: Calculate the Strain
Strain is defined as the change in length divided by the original length.
1. **Convert the elongation from cm to meters**:
\[
\text{Change in length} = 0.7 \, \text{cm} = 0.7 \times 10^{-2} \, \text{m} = 0.007 \, \text{m}
\]
2. **Calculate the original length in meters**:
\[
\text{Original length} = 5.08 \, \text{m}
\]
3. **Calculate Strain**:
\[
\text{Strain} = \frac{\text{Change in length}}{\text{Original length}} = \frac{0.007 \, \text{m}}{5.08 \, \text{m}} \approx 0.001376
\]
### Step 3: Calculate Young's Modulus
Now we can substitute the values of stress and strain into the formula for Young's modulus:
\[
Y = \frac{\text{Stress}}{\text{Strain}} = \frac{2.82 \times 10^{8} \, \text{N/m}^2}{0.001376} \approx 2.05 \times 10^{11} \, \text{N/m}^2
\]
### Final Answer
Thus, the Young's modulus of steel is approximately:
\[
Y \approx 2.05 \times 10^{11} \, \text{N/m}^2
\]
