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  • Two rods of different materials, having coefficients of linear expansion <img alt="\mathrm{\alpha_1\, \, and \, \, \alpha_2}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Calp

Two rods of different materials, having coefficients of linear expansion \mathrm{\alpha_1\, \, and \, \, \alpha_2} and Young's modulii \mathrm{Y_1} and \mathrm{Y_2,} are fixed between two rigid walls. The rods are heated to the same temperature. There is no bending of rods. If \mathrm{\alpha_1: \alpha_2=2: 3}, the thermal stresses developed in the two rods will be equal provided \mathrm{Y_1: Y_2} is equal to

Option: 1

2: 3


Option: 2

1: 1


Option: 3

3: 2


Option: 4

4: 9


Answers (1)

best_answer

Young's modulus \mathrm{Y=\frac{\text { stress }}{\text { strain }}=\frac{\sigma}{\varepsilon}}, where \mathrm{\varepsilon=\frac{l}{L}.}
Now, if the temperature of a rod is increased by \mathrm{\theta}, the increase in its length due to thermal expansion is

                                      \mathrm{ l=\alpha L \theta }
\mathrm{\therefore \quad Strain \, \, \varepsilon=\frac{l}{L}=\alpha \theta}. Now stress is

                              \mathrm{ \sigma=Y \varepsilon=Y \alpha \theta }

For the two rods, the stress is

\mathrm{ \text { and } \quad \begin{aligned} \sigma_1 & =Y_1 \alpha_1 \theta \\ \sigma_2 & =Y_2 \alpha_2 \theta \end{aligned} }

But \mathrm{\sigma_1=\sigma_2 (given).} Hence\mathrm{ Y_1 \alpha_1 \theta=Y_2 \alpha_2 \theta\, \, or \, \, \frac{Y_1}{Y_2}=\frac{\alpha_2}{\alpha_1}=\frac{3}{2}}.

Hence the correct choice is (c).

Posted by

seema garhwal

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