Question

The stress that has to be applied to the ends of a steel wire of length 20 cm to keep its length constant, when its temperature is raised by \(100^\circ C\) is \(2.2 \times 10^x \, \text{Pa}\). The value of \(x\) is
(Given \( Y = 2 \times 10^{11} \, \text{Nm}^{-2}, \, \alpha = 1.1 \times 10^{-5} \, ^\circ C^{-1} \))

Hint:

If expansion is restricted, directly use: Stress = \( Y \alpha \Delta T \).

Answer 1

Step 1: Understanding the Concept:
Heating a material causes thermal expansion. If the material is constrained (ends are fixed), it cannot expand, and an internal force per unit area develops to oppose this expansion. This is known as thermal stress.
Step 2: Key Formula or Approach:
1. Thermal strain: \( \frac{\Delta L}{L} = \alpha \Delta T \).
2. Stress formula: \( \text{Stress} = Y \times \text{Strain} \).
3. Thermal Stress: \( \sigma = Y \alpha \Delta T \).
Step 3: Detailed Explanation:
We are given:
\( Y = 2 \times 10^{11} \text{ Nm}^{-2} \)
\( \alpha = 1.1 \times 10^{-5} \text{ C}^{-1} \)
\( \Delta T = 100^\circ \text{C} \)
Applying the thermal stress formula:
\[ \text{Stress} = (2 \times 10^{11}) \times (1.1 \times 10^{-5}) \times 100 \]
\[ \text{Stress} = 2.2 \times 10^{11-5+2} \]
\[ \text{Stress} = 2.2 \times 10^8 \text{ Pa} \]
Comparing this with the given form \( 2.2 \times 10^x \), we get:
\[ x = 8 \]
Step 4: Final Answer:
The value of x is 8.