/*! 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 36 Two rods of length \(L_{1}\) and... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 36

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
The correct relationship between the lengths of the rods and the coefficients of linear expansion is given by option (B).

Step by step solution

01

Set up the equations for the change in length

Firstly, set up an equation for the change in length of each rod through thermal expansion using the formula \( \Delta L = 伪L\Delta T \). \n- For Rod 1: \( \Delta L_{1} = 伪_{1}L_{1}\Delta T \)\n- For Rod 2: \( \Delta L_{2} = 伪_{2}L_{2}\Delta T \)
02

Formulate the conditions given in the problem

The problem states that the difference in lengths of the two rods is independent of the temperature changes. Hence, we equate the changes - \n\( \Delta L_{1} = \Delta L_{2}\) \nSubstituting the equations from Step 1, we get - \n\( 伪_{1}L_{1}\Delta T = 伪_{2}L_{2}\Delta T \)
03

Simplify the equation

Since \( \Delta T \) is a common factor on both sides, it can be cancelled out. This leaves us with - \n\( 伪_{1}L_{1} = 伪_{2}L_{2} \)
04

Identify the matching option

Rewrite the resulting equation in the form of options given. Rearranging the equation for \( L_{1} / L_{2} \) yields - \n\( L_{1} / L_{2} = 伪_{2} / 伪_{1} \), which corresponds to option (B)

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Most popular questions from this chapter

We have a jar \(A\) filled with gas characterized by parameters \(P, V\), and \(T\) and another jar \(B\) filled with gas with parameters \(2 P, V / 4\), and \(2 T\), where the symbols have their usual meanings. The ratio of the number of molecules of jar \(A\) to those of jar \(B\) is (A) \(1: 1\) (B) \(1: 2\) (C) \(2: 1\) (D) \(4: 1\)

Cooking gas containers are loaded on to a truck moving with uniform speed. The temperature of the gas molecules inside the containers will (A) increase. (B) decrease. (C) remain same. (D) decrease for some, whereas increase for others.

If the degree of freedom of a gas molecule is \(f\), then the ratio of two specific heats \(C_{p} / C_{v}\) is given by (A) \(\frac{2}{f}+1\) (B) \(1-\frac{2}{f}\) (C) \(1+\frac{1}{f}\) (D) \(1-\frac{1}{f^{2}}\)

Mark the correct options (A) A system \(X\) is in thermal equilibrium with \(Y\) but not with \(Z\). The system \(Y\) and \(Z\) may be in thermal equilibrium with each other. (B) A system \(X\) is in thermal equilibrium with \(Y\) but not with \(Z\). The system \(Y\) and \(Z\) are not in thermal equilibrium with each other. (C) A system \(X\) is neither in thermal equilibrium with \(Y\) nor with \(Z\). The systems \(Y\) and \(Z\) must be in thermal equilibrium with each other. (D) A system \(X\) is neither in thermal equilibrium with \(Y\) nor with \(Z\). The systems \(Y\) and \(Z\) may be in thermal equilibrium with each other.

The ratio of coefficients of cubical expansion and linear expansion is (A) \(1: 1\) (B) \(3: 1\) (C) \(2: 1\) (D) None of these
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