/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Two rods of length \(L_{1}\) and... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 26

Two rods of length \(L_{1}\) and \(L_{2}\) are made of materials whose coefficients of linear expansion are \(\alpha_{1}\) and \(\alpha_{2}\). If the difference between the two lengths is independent of temperature (A) \(\left(L_{1} / L_{2}\right)=\left(\alpha_{1} / \alpha_{2}\right)\) (B) \(\left(L_{1} / L_{2}\right)=\left(\alpha_{2} / \alpha_{1}\right)\) (C) \(L_{1}^{2} \alpha_{1}=L_{2}^{2} \alpha_{2}\) (D) \(\alpha_{1}^{2} L_{1}=\alpha_{2}^{2} L_{2}\)

Short Answer

Expert verified
(B) \(\left(\frac{L_{1}}{L_{2}}\right)=\left(\frac{\alpha_{2}}{\alpha_{1}}\right)\)

Step by step solution

01

Write the linear expansion equations for both rods

Let's write the linear expansion equations for both rods. The length change of a rod due to temperature change is given by: \(\Delta L = L \cdot \alpha \cdot \Delta T\) Where \(\Delta L\) is the change in length of the rod, \(L\) is the initial length, \(\alpha\) is the coefficient of linear expansion, and \(\Delta T\) is the change in temperature. For the first rod, we have: \(\Delta L_1 = L_1 \cdot \alpha_1 \cdot \Delta T\) For the second rod, we have: \(\Delta L_2 = L_2 \cdot \alpha_2 \cdot \Delta T\)
02

Determine the condition when difference between lengths is constant

Since we are looking for a condition where the difference in lengths remains constant, we can set up an equation as follows: \(\Delta L_1 - \Delta L_2 = C\) Where \(C\) is a constant value. Substituting the linear expansion equations from step 1 for \(\Delta L_1\) and \(\Delta L_2\), we get: \((L_1 \cdot \alpha_1 \cdot \Delta T) - (L_2 \cdot \alpha_2 \cdot \Delta T) = C\)
03

Simplify the equation and compare with the given options

Let's simplify the above equation: \(\Delta T(\alpha_1 L_1 - \alpha_2 L_2) = C\) Since \(\Delta T\) can change, the only way this equation can hold is if: \(\alpha_1 L_1 = \alpha_2 L_2\) Now let's divide both sides by \(L_1 \cdot \alpha_2\), we get: \(\frac{L_1}{L_2} = \frac{\alpha_2}{\alpha_1}\) Comparing this result with the given options, the correct answer is: (B) \(\left(L_{1} / L_{2}\right)=\left(\alpha_{2} / \alpha_{1}\right)\)

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Most popular questions from this chapter

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