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Chapter 7: Problem 895

The Coefficient of linear expansion of brass \& steel are \(\alpha_{1} \&\) \(\alpha_{2}\) If we take a brass rod of length \(\ell_{1} \&\) steel rod of length \(\ell_{2}\) at \(0^{\circ} \mathrm{C}\), their difference in length \(\left(\ell_{2}-\ell_{1}\right)\) will remain the same at a temperature if \(\ldots \ldots \ldots \ldots\) (A) \(\alpha_{1} \ell_{2}=\alpha_{2} \ell_{1}\) (B) \(\ell_{2} \alpha_{1}=\alpha_{2} \ell_{1}\) (C) \(a_{2}^{2} \ell_{1}=a_{2}^{2} \varepsilon_{2}\) (D) \(\alpha_{1} \ell_{1}=\alpha_{2} \mathcal{C}_{2}\)

Short Answer

Expert verified
The short answer is: \(\alpha_{1} \ell_{2}=\alpha_{2} \ell_{1}\) (Option A).

Step by step solution

01

Write the formula for linear expansion

The formula for linear expansion is: \[ Δ\ell = α \cdot \ell_0 \cdot ΔT \] Where \(Δ\ell\) is the change in length, \(α\) is the coefficient of linear expansion, \(\ell_0\) is the initial length, and \(ΔT\) is the change in temperature.
02

Write the equations for both rods.

We need to write the equations representing the change in length for both brass and steel rods at the same temperature change \(ΔT\): 1. Brass rod: \[ Δ\ell_1 = \alpha_1 \cdot \ell_{1_0} \cdot ΔT \] 2. Steel rod: \[ Δ\ell_2 = \alpha_2 \cdot \ell_{2_0} \cdot ΔT \]
03

Difference in length condition

We want the difference in length \(\left(\ell_{2}-\ell_{1}\right)\) to remain the same, so we can write: \[ \ell_{2_0} + Δ\ell_2 - (\ell_{1_0} + Δ\ell_1) = \ell_{2_0} - \ell_{1_0} \]
04

Substitute the equations from step 2 into the equation from step 3

We will now substitute our equations for \(Δ\ell_1\) and \(Δ\ell_2\) into the equation from step 3: \[\ell_{2_0} + \alpha_2 \cdot \ell_{2_0} \cdot ΔT - (\ell_{1_0} + \alpha_1 \cdot \ell_{1_0} \cdot ΔT) = \ell_{2_0} - \ell_{1_0} \]
05

Simplify the equation

Simplify the equation from step 4 by canceling out the terms \(\ell_{2_0}\) and \(-\ell_{1_0}\) on both sides: \[ \alpha_2 \cdot \ell_{2_0} \cdot ΔT - \alpha_1 \cdot \ell_{1_0} \cdot ΔT = 0 \]
06

Find the condition

In this step, we will find the condition under which the difference in length remains the same. Since \(ΔT\) is non-zero, we have: \[ \alpha_2 \cdot \ell_{2_0} = \alpha_1 \cdot \ell_{1_0} \] Comparing this result with the given options, we see that the correct answer is: (A) \(\alpha_{1} \ell_{2}=\alpha_{2} \ell_{1}\)

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