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Chapter 17: Problem 71

Prove the equation \(\beta=3 \alpha\) (Section 17.3 ) by considering a cube of side \(s\) and therefore volume \(V=s^{3}\) that undergoes a small temperature change \(d T\) and corresponding length and volume changes \(d s\) and \(d V\)

Short Answer

Expert verified
Given that the linear thermal expansion is represented by \(d s=\alpha s d T\), the change in volume for a cube (volumetric expansion) can be found as \(d V = 3s^{2} ds\). Upon comparing the latter with the expression for volumetric expansion \(d V=\beta V d T\), it is evident that the equation \(\beta=3\alpha\) holds.

Step by step solution

01

Calculate the change in length

Recall the formula for linear expansion: \(d s=\alpha s d T\). This formula basically says the change in length \(d s\) is equal to the linear thermal expansion coefficient \(\alpha\), multiplied by the original length \(s\) and the change in temperature \(d T\)
02

Calculate the change in volume

The volume of a cube is given by \(s^{3}\). If the side \(s\) has a small increment \(d s\), the new volume is \((s + ds)^{3}\). Expanding this, we have \( V + 3s^{2} ds + 3s d s^{2} + ds^{3}\). However, because \(d s\) is very small, \(d s^{2}\) and \(d s^{3}\) are practically zero and can be ignored. Therefore, \(d V = V + 3s^{2} ds = s^{3} + 3s^{2} ds\). Subtracting the original volume \(V = s^{3}\), we find \(d V = 3s^{2} ds\)
03

Establish relationship between α and β

The formula for volumetric expansion is \(d V = \beta V d T\). From previous step, we found that \(d V = 3s^{2} ds = 3s^{2}\alpha s d T = 3 \alpha V d T\). Comparing both expressions for \(d V\), we see that \(\beta = 3\alpha\)

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