Question

Consider three rods of length $L_{1}, \, L_{2}$ and $L_{3}$ respectively joined in series. Each has same cross-sectional area with Young's moduli Y, 2Y and 3Y respectively and thermal coefficients of linear expansion $\alpha , \, 2\alpha $ and $3\alpha \, $ respectively. They are placed between two rigid fixed walls. The temperature of the whole system is increased and it is found that length of the middle rod does not change with temperature rise. Find the value of $\frac{9 L_{1}}{L_{3}}$ .

Answer 1

Since the middle rod doesn't expand at all, the strain in the rod will be
$strain=2\alpha \Delta T$
$stress=2Y\left(2 \alpha \Delta T\right)$
The stress is same in each rod
$4Y\alpha \Delta T=Y\frac{L_{1} \alpha \Delta T + x}{L_{1}}=3Y\frac{L_{3} \left(3 \alpha \right) \Delta T + x}{L_{3}}$
On solving for $x$ , we get
$x=3L_{1}\alpha \Delta T=\frac{5 L_{3} \alpha \Delta T}{3}$
$\Rightarrow \frac{9 L_{1}}{L_{3}}=5$