A rod of aluminium is fixed between two rigid support. If the temperature of rod is increased by `10°C`, then the thermal stress on the rod is (Take Y=`7×10^10 `Pa and `alpha`=2.4*`10^-5 K^-1`)

Options

A. `1.4×10^3` Pa
B. `1.7×10^7` Pa
C. `1.7×10^3` Pa
D. `2.4×10^7` Pa

Answer

To solve the problem of finding the thermal stress on a fixed aluminium rod when its temperature is increased by \(10^\circ C\), we will use the relationship between stress, Young's modulus, and thermal expansion.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Change in temperature, \(\Delta T = 10^\circ C\)
- Young's modulus, \(Y = 7 \times 10^{10} \, \text{Pa}\)
- Coefficient of linear expansion, \(\alpha = 2.4 \times 10^{-5} \, \text{K}^{-1}\)

2. **Understand the Concept:**
- When a rod is fixed at both ends and its temperature increases, it tries to expand. However, since it is fixed, this expansion causes thermal stress in the rod.
- The formula for thermal stress (\(\sigma\)) in a material that is constrained from expanding is given by:
\[ \sigma = Y \cdot \text{strain} \]
- The strain due to thermal expansion can be expressed as:
\[ \text{strain} = \alpha \cdot \Delta T \]

3. **Calculate the Strain:**
- Substitute the values of \(\alpha\) and \(\Delta T\) into the strain formula:
\[ \text{strain} = \alpha \cdot \Delta T = (2.4 \times 10^{-5} \, \text{K}^{-1}) \cdot (10 \, \text{K}) = 2.4 \times 10^{-4} \]

4. **Calculate the Thermal Stress:**
- Now, substitute the strain into the stress formula:
\[ \sigma = Y \cdot \text{strain} = (7 \times 10^{10} \, \text{Pa}) \cdot (2.4 \times 10^{-4}) \]
- Perform the multiplication:
\[ \sigma = 7 \times 2.4 \times 10^{10} \times 10^{-4} = 16.8 \times 10^{6} \, \text{Pa} \]

5. **Convert to Standard Form:**
- Convert \(16.8 \times 10^{6} \, \text{Pa}\) to standard form:
\[ \sigma = 1.68 \times 10^{7} \, \text{Pa} \]

6. **Final Answer:**
- Thus, the thermal stress on the rod is:
\[ \sigma \approx 1.7 \times 10^{7} \, \text{Pa} \]