A copper collar is to fit tightly about a steel shaft that has a diameter of 6 cm at `20^@C`. The inside diameter of the copper collar at the temperature is `5.98cm`
Q. The tensile stress in the copper collar when its temperature returns to `20^@C` is `(T=11xx10^10N//m^(2))`

Options

A. `1.34xx10^5 N//m^(2)`
B. `3.68xx10^-12 N//m^(2)`
C. `3.68xx10^8 N//m^(2)`
D. `1.34xx10^-12 N//m^2`

Explanation

To solve the problem step by step, we need to calculate the tensile stress in the copper collar when its temperature returns to \(20^\circ C\). Here’s how we can do it:

### Step 1: Understand the Given Information
- Diameter of the steel shaft at \(20^\circ C\) = \(6 \, \text{cm}\)
- Inside diameter of the copper collar at \(20^\circ C\) = \(5.98 \, \text{cm}\)
- Tensile modulus of copper, \(T = 11 \times 10^{10} \, \text{N/m}^2\)
- The change in temperature (\(\Delta T\)) when the collar is heated is calculated to be \(2.7^\circ C\).

### Step 2: Calculate the Strain
The strain (\(\epsilon\)) can be calculated using the formula:
\[ \epsilon = \alpha \cdot \Delta T \]
Where:
- \(\alpha\) is the coefficient of linear expansion for copper (approximately \(17 \times 10^{-6} \, \text{°C}^{-1}\)).
- \(\Delta T\) is the change in temperature.

### Step 3: Substitute Values to Find Strain
Substituting the values into the strain formula:
\[ \epsilon = 17 \times 10^{-6} \cdot 2.7 \]
Calculating this gives:
\[ \epsilon = 45.9 \times 10^{-6} \]

### Step 4: Calculate the Tensile Stress
The tensile stress (\(\sigma\)) can be calculated using the formula:
\[ \sigma = T \cdot \epsilon \]
Where:
- \(T\) is the tensile modulus of copper.

Substituting the values:
\[ \sigma = 11 \times 10^{10} \cdot 45.9 \times 10^{-6} \]

### Step 5: Perform the Calculation
Calculating the above expression:
\[ \sigma = 11 \times 10^{10} \cdot 45.9 \times 10^{-6} = 5.05 \times 10^{6} \, \text{N/m}^2 \]

### Step 6: Final Answer
The tensile stress in the copper collar when its temperature returns to \(20^\circ C\) is approximately:
\[ \sigma \approx 5.05 \times 10^{6} \, \text{N/m}^2 \]