A steel rod 3 cm in diameter and 80 cm long has an allowable elongation not to exceed 1.5 mm, find the allowable load in kN.

To determine the allowable load for the steel rod, we can use the following relationship based on **Hooke's Law** and the formula for **elongation** in a linear elastic material:

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

Where:
- \(\Delta L\) = elongation of the rod (1.5 mm = \(1.5 \times 10^{-3}\) m)
- \(P\) = applied load (which we are solving for)
- \(L\) = length of the rod (80 cm = 0.8 m)
- \(A\) = cross-sectional area of the rod (circular, with diameter 3 cm)
- \(E\) = Young's modulus for steel (typically around \(E = 210 \, \text{GPa} = 210 \times 10^9 \, \text{Pa}\))

### Step 1: Calculate the Cross-Sectional Area of the Rod
The area of a circle is given by:

\[
A = \frac{\pi d^2}{4}
\]

Where \(d = 3 \, \text{cm} = 0.03 \, \text{m}\).

\[
A = \frac{\pi (0.03)^2}{4} = 7.06858 \times 10^{-4} \, \text{m}^2
\]

### Step 2: Rearrange the Elongation Formula to Solve for Load \(P\)

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

Substituting the known values:

\[
P = \frac{(1.5 \times 10^{-3}) \cdot (7.06858 \times 10^{-4}) \cdot (210 \times 10^9)}{0.8}
\]

Let's calculate the load \(P\).The allowable load for the steel rod is approximately **278.33 kN**.

Would you like more details on the steps, or do you have any additional questions?

Here are some related questions to explore:

1. How would the allowable load change if the length of the rod were doubled?
2. What would happen to the allowable load if the diameter of the rod increased to 4 cm?
3. How does Young's modulus affect the allowable elongation?
4. What is the relationship between stress and strain in materials like steel?
5. How would the calculation change if the material were not steel, but aluminum?

**Tip:** Always check material properties like Young's modulus for the specific material being used, as they can vary significantly between different materials and alloys.

Calculate the allowable load for a steel rod with a diameter of 3 cm and length of 80 cm, given that the elongation must not exceed 1.5 mm. This problem involves applying Hooke's Law and the formula for elongation in a linear elastic material, covering concepts of cross-sectional area and Young's modulus. Suitable for college-level engineering students.

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