Heat and Thermodynamics 3 Question 12
12. A composite rod is made by joining a copper rod, end to end, with a second rod of different material but of the same cross-section. At $25^{\circ} C$, the composite rod is $1 m$ in length, of which the length of the copper rod is $30 cm$. At $125^{\circ} C$ the length of the composite rod increases by $1.91 mm$.
When the composite rod is not allowed to expand by holding it between two rigid walls, it is found that the length of the two constituents do not change with the rise of temperature. Find the Young’s modulus and the coefficient of linear expansion of the second rod. (Given, Coefficient of linear expansion of copper $=1.7 \times 10^{-5}$ per ${ }^{\circ} C$, Young’s modulus of copper $=1.3 \times 10^{11} N / m^{2}$ )
(1979)
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Solution:
- $Q \propto A T^{4}$ and $\lambda _m T=$ constant
Hence,
$$ \begin{aligned} & Q \propto \frac{A}{\left(\lambda _m\right)^{4}} \\ & \text { or } \quad Q \propto \frac{r^{2}}{\left(\lambda _m\right)^{4}} \\ & Q _A: Q _B: Q _C=\frac{(2)^{2}}{(3)^{4}}: \frac{(4)^{2}}{(4)^{4}}: \frac{(6)^{2}}{(5)^{4}} \\ & =\frac{4}{81}: \frac{1}{16}: \frac{36}{625}=0.05: 0.0625: 0.0576 \end{aligned} $$
i.e. $Q _B$ is maximum.
