Question

Two rods of different materials having coefficients of thermal expansions \(\alpha_{1}, \alpha_{2}\) and Young's module \(\mathrm{y}_{1}, \mathrm{y}_{2}\) respectively are fixed between two rigid walls. The rods are heated such that they undergo the same increase in temperature. There is no bending of the rod. if \(\alpha_{1}: \alpha_{2}=2: 3\) the thermal stresses developed in the rod are equal provided \(\mathrm{y}_{1}: \mathrm{y}_{2}\) equals. (A) \(3: 2\) (B) \(2: 3\) (C) \(4: 9\) (D) \(1: 1\)

Short answer

The correct answer is (A) \(3 : 2\).

Step-by-step solution

Write the thermal stress formula for both rods

Since both rods develop the same stress and undergo the same temperature change, we can write the thermal stress formula for each rod as follows: Thermal stress in rod 1 = \(Y_1 \alpha_1 (\Delta T)\) Thermal stress in rod 2 = \(Y_2 \alpha_2 (\Delta T)\) As both thermal stresses are equal, we have: \(Y_1 \alpha_1 (\Delta T)= Y_2 \alpha_2 (\Delta T)\)

Use the provided ratio of \(\alpha_1\) to \(\alpha_2\)

It is given that \(\alpha_1 : \alpha_2 = 2 : 3\). Therefore, we can write \(\alpha_1 = 2x\) and \(\alpha_2 = 3x\), where \(x\) is a constant. Now, we can rewrite the equation from Step 1 using these values: \(Y_1 (2x) (\Delta T) = Y_2 (3x) (\Delta T)\)

Solve for the ratio of \(Y_1\) to \(Y_2\)

First, divide both sides of the equation by \((\Delta T) x\): \(Y_1 (2) = Y_2 (3)\) Next, divide both sides by 2 to isolate \(Y_1\): \(Y_1 = Y_2 \frac{3}{2}\) This gives us the ratio of \(Y_1\) to \(Y_2\) as \(3 : 2\). So, the correct answer is (A) \(3 : 2\).

Key concepts

Thermal Expansion

Thermal expansion is when materials expand when they heat up. Every material has a specific coefficient of thermal expansion, which is a measure of how it expands with a temperature increase. This coefficient is denoted by \( \alpha \) and is different for different materials.
When you heat a rod, for example, it gets slightly longer. If two rods with different materials are heated equally and fixed at their ends, their expansion will cause stress because they can't expand freely.
In the problem, two rods have different coefficients of thermal expansion, \( \alpha_1 \) and \( \alpha_2 \), and are held between two rigid walls. Because their ability to expand differs, the walls cause thermal stress in them when their temperatures rise.

Young's Modulus

Young's modulus measures a material's stiffness. It tells us how much it will deform under stress. A higher Young's modulus means the material is stiffer and less likely to stretch.
It is denoted by \( Y \) in equations. In the given problem, each rod has a different Young's modulus, \( Y_1 \) and \( Y_2 \). These values determine how each rod reacts to the stress caused by thermal expansion.
When rods with different Young's moduli are heated and create the same stress, we use this property to determine how the materials react to the stress applied by fixing them in place while they expand.

Temperature Change

Temperature change is the difference between the final and initial temperatures of an object. In the problem, both rods experience the same temperature change, denoted as \( \Delta T \).
This uniform temperature change is essential as it directly influences the thermal expansion of the rods. The formula for thermal stress includes the temperature change, showing how it directly impacts stress.
Thus, knowing the magnitude of \( \Delta T \) helps us calculate how much the rods try to expand and thus the stress developed.

Stress Analysis

Stress analysis involves calculating the stress a material experiences when different forces, like temperature changes, are applied. In the context of thermal stress, stress analysis involves understanding how temperature changes cause internal forces within materials.
The thermal stress formula is given by: \( \text{Thermal Stress} = Y \alpha \Delta T \), where \( Y \) is Young's modulus, \( \alpha \) is the thermal expansion coefficient, and \( \Delta T \) is the temperature change.
By setting the stress in both rods equal and substituting the given ratios, we find the ratio of Young's moduli of the rods. This analysis helps us to determine the impacts of the temperature change on the materials.