Solved Problem on Thermal Expansion

In the figure, the platform P is horizontal because it is supported by bases A and B of equal linear expansion coefficients, respectively, α_A and α_B. Determine the ratio of the L_A and L_B lengths of the bars so that the platform P remains horizontal at any temperature.

Problem Data:

Linear expansion coefficient of base A: α_A;
Linear expansion coefficient of base B: α_B.

Problem diagram:

Figure 1

Solution

In order to platform P to remain horizontally regardless of temperature, we must impose the condition that the expansions ΔL of the bars are equal (Figure 1)

\[ \begin{gather} \Delta L_{A}=\Delta L_{B} \tag{I} \end{gather} \]

The Linear Expansion Equation is given by

\[ \bbox[#99CCFF,10px] {\Delta L=L\alpha \Delta t} \]

Writing the expansion equations for each of the bars

\[ \begin{gather} \Delta L_{A}=L_{A}\alpha_{A}\Delta t\\[10pt] \Delta L_{B}=L_{B}\alpha_{B}\Delta t \end{gather} \]

Applying the condition given by (I)

\[ \begin{gather} L_{A}\alpha _{A}\Delta t=L_{B}\alpha _{B}\Delta t\\[5pt] \frac{L_{A}}{L_{B}}=\frac{\alpha _{B}\cancel{\Delta t}}{\alpha _{A}\cancel{\Delta t}} \end{gather} \]

\[ \bbox[#FFCCCC,10px] {\frac{L_{A}}{L_{B}}=\frac{\alpha _{B}}{\alpha _{A}}} \]

The ratio of lengths is proportional to the inverse of expansion coefficients, that is, the shorter base must expand proportionally more than the larger base.

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