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 calculating the thermal stress on a rod of aluminum fixed between two rigid supports when the temperature is increased by \(10°C\), we can follow these steps:

### Step 1: Identify the given values
- Young's modulus (\(Y\)) = \(7 \times 10^{10} \, \text{Pa}\)
- Coefficient of linear expansion (\(\alpha\)) = \(2.4 \times 10^{-5} \, \text{K}^{-1}\)
- Change in temperature (\(\Delta T\)) = \(10°C\)

### Step 2: Calculate the strain produced in the rod
The strain (\(\epsilon\)) produced due to thermal expansion can be calculated using the formula:
\[ \epsilon = \alpha \Delta T \]
Substituting the values:
\[ \epsilon = (2.4 \times 10^{-5} \, \text{K}^{-1}) \times (10 \, \text{K}) = 2.4 \times 10^{-4} \]

### Step 3: Relate stress and strain using Young's modulus
The relationship between stress (\(\sigma\)), strain (\(\epsilon\)), and Young's modulus (\(Y\)) is given by:
\[ \sigma = Y \cdot \epsilon \]
Substituting the values:
\[ \sigma = (7 \times 10^{10} \, \text{Pa}) \times (2.4 \times 10^{-4}) \]

### Step 4: Calculate the thermal stress
Now, perform the multiplication:
\[ \sigma = 7 \times 2.4 \times 10^{10} \times 10^{-4} = 16.8 \times 10^{6} \, \text{Pa} \]
This can be approximated to:
\[ \sigma \approx 1.68 \times 10^{7} \, \text{Pa} \]

### Step 5: Final result
Thus, the thermal stress on the rod is approximately:
\[ \sigma \approx 1.68 \times 10^{7} \, \text{Pa} \]