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Chapter 25: Problem 2801

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

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

Step by step solution

01

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: \(Y_1 \alpha_1 (\Delta T)= Y_2 \alpha_2 (\Delta T)\)
02

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: \(Y_1 (2x) (\Delta T) = Y_2 (3x) (\Delta T)\)
03

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) \(3 : 2\).

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