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To determine the elongation of a cylindrical steel alloy specimen when a load is applied, we need to use the formula derived from Hooke's Law and the definition of stress and strain in materials. The relevant formula is:

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

where:
- \( \Delta L \) is the elongation,
- \( F \) is the applied force,
- \( L \) is the original length of the specimen,
- \( A \) is the cross-sectional area,
- \( E \) is the Young's modulus of the material.

### Given dataTo determine the elongation of a cylindrical steel alloy specimen when a load is applied, we need to use the formula derived from Hooke's Law and the definition of stress and strain in materials. The relevant formula is:

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

where:
- \( \Delta L \) is the elongation,
- \( F \) is the applied force,
- \( L \) is the original length of the specimen,
- \( A \) is the cross-sectional area,
- \( E \) is the Young's modulus of the material.

### Given 
- Diameter (\(d\)) = 10.0 mm = 0.01 m
- Original length (\(L\)) = 75 mm = 0.075 m
- Applied load (\(F\)) = 20,000 N
- Young's modulus (\(E\)) for steel alloy (approximately) = 200 GPa = \( 200 \times 10^9 \) Pa

### Steps to Calculate Elongation:

1. **Calculate the cross-sectional area (\(A\)) of the cylindrical specimen**:
\[ A = \pi \left(\frac{d}{2}\right)^2 \]
\[ A = \pi \left(\frac{0.01}{2}\right)^2 \]
\[ A = \pi \left(0.005\right)^2 \]
\[ A = \pi \times 25 \times 10^{-6} \]
\[ A \approx 78.54 \times 10^{-6} \, \text{m}^2 \]

2. **Substitute the values into the elongation formula**:
\[ \Delta L = \frac{20,000 \times 0.075}{78.54 \times 10^{-6} \times 200 \times 10^9} \]
\[ \Delta L = \frac{1,500}{15.708 \times 10^3} \]
\[ \Delta L \approx 9.54 \times 10^{-5} \, \text{m} \]
\[ \Delta L \approx 0.0954 \, \text{mm} \]

Thus, the elongation of the cylindrical specimen is approximately \( 0.0954 \, \text{mm} \).

Would you like more details or have any questions on this calculation?

Here are five questions you might ask next:
1. How does Young's modulus affect the elongation of a material?
2. Can we use the same formula for materials other than steel?
3. What happens to the elongation if the diameter of the specimen is doubled?
4. How is the stress-strain relationship used in engineering applications?
5. What is the significance of the cross-sectional area in material deformation?

**Tip:** Always ensure that units are consistent when performing calculations involving physical quantities.

Learn how to calculate the elongation of a cylindrical steel alloy specimen under applied load using Hooke's Law and Young's modulus. Given data include diameter, original length, applied load, and Young's modulus. Detailed step-by-step solution provided.

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