Solved Problem on Thermal Expansion

The average coefficient of linear expansion of iron is equal to 0.0000117 °C⁻¹. How much does it increase the temperature of an iron block so that its volume increases from 1%?

Problem data:

Coefficient of linear expansion of iron: α = 0.0000117 °C⁻¹;
Change in volume: ΔV = 1%.

Problem diagram:

Figure 1

Solution

Volume change will be 1%

\[ \begin{gather} \Delta V=1V_{0}\\ \Delta V=\frac{1}{100}V_{0}\\ \Delta V=0.01V_{0} \end{gather} \]

The problem gives the coefficient of linear expansion of the material, and for the calculation of the volume increase, we need the coefficient of volumetric expansion.

\[ \bbox[#99CCFF,10px] {\gamma =3\alpha} \]

\[ \begin{gather} \gamma =3\times 0.0000117\\ \gamma =0.0000351\\ \gamma=3.51\times 10^{-5}\;\text{°C}^{-1} \end{gather} \]

The final volume is given by

\[ \bbox[#99CCFF,10px] {\Delta V=\gamma V_{0}\Delta t} \]

substituting the problem data, we find the temperature variation

\[ \begin{gather} 0.01\cancel{V_{0}}=3.51\times 10^{-5}\cancel{V_{0}}\Delta t\\ 0.01=3.51\times 10^{-5}\Delta t\\ \Delta t=\frac{0.01}{3.51\times 10^{-5}} \end{gather} \]

\[ \bbox[#FFCCCC,10px] {\Delta t\approx 285\;\text{°C}} \]

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