• question_answer Two rods are joined between fixed supports as shown in the figure. Condition for no change in the lengths of individual rods with the increase of temperature will be (${{\alpha }_{1}},{{\alpha }_{2}}$= linear expansion co-efficient ${{\text{A}}_{\text{1}}}\text{,}{{\text{A}}_{\text{2}}}$ =Area of rods ${{\text{Y}}_{\text{1}}}\text{,}{{\text{Y}}_{\text{2}}}$= Young modulus) A) $\frac{{{\text{A}}_{\text{1}}}}{{{\text{A}}_{\text{2}}}}\text{=}\frac{{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}{{\text{Y}}_{\text{1}}}}{{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}{{\text{Y}}_{\text{2}}}}$                 B) $\frac{{{\text{A}}_{\text{1}}}}{{{\text{A}}_{\text{2}}}}\text{=}\frac{{{\text{L}}_{1}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}{{\text{Y}}_{\text{1}}}}{{{\text{L}}_{2}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}{{\text{Y}}_{\text{2}}}}$ C) $\frac{{{\text{A}}_{\text{1}}}}{{{\text{A}}_{\text{2}}}}\text{=}\frac{{{\text{L}}_{2}}{{\text{ }\!\!\alpha\!\!\text{ }}_{2}}{{\text{Y}}_{2}}}{{{\text{L}}_{1}}{{\text{ }\!\!\alpha\!\!\text{ }}_{1}}{{\text{Y}}_{1}}}$       D) $\frac{{{\text{A}}_{\text{1}}}}{{{\text{A}}_{\text{2}}}}\text{=}\frac{{{\text{ }\!\!\alpha\!\!\text{ }}_{2}}{{\text{Y}}_{2}}}{{{\text{ }\!\!\alpha\!\!\text{ }}_{1}}{{\text{Y}}_{1}}}$

 $Y=\frac{\text{Stress}}{\text{Strain}}=\frac{T/A}{\Delta \ell /\ell }$ $\text{T=}\frac{\text{Y}\text{. }\!\!\Delta\!\!\text{ l}}{\text{l}}\text{A=Y}\text{.A }\!\!\alpha\!\!\text{ }\!\!\Delta\!\!\text{ T}$ In both the rods tension will be same so ${{\text{T}}_{\text{1}}}\text{ = }{{\text{T}}_{\text{2}}}$ Hence ${{\text{Y}}_{\text{1}}}{{\text{A}}_{\text{1}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}\text{ }\!\!\Delta\!\!\text{ T = }{{\text{Y}}_{\text{2}}}{{\text{A}}_{\text{2}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}\text{ }\!\!\Delta\!\!\text{ T}$ $\frac{{{\text{A}}_{\text{1}}}}{{{\text{A}}_{\text{2}}}}\text{=}\frac{{{\text{Y}}_{\text{2}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{2}}}}{{{\text{Y}}_{\text{1}}}{{\text{ }\!\!\alpha\!\!\text{ }}_{\text{1}}}}$