Question

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 coefficient
$A_{1},\, A_{2}=$ Area of rods
$Y_{1},\, Y_{2}=$ Young modulus)

• A. $\frac{A_{1}}{A_{2}}=\frac{\alpha_{1} Y_{1}}{\alpha_{2} Y_{2}}$
• B. $\frac{A_{1}}{A_{2}}=\frac{L_{1} \alpha_{1} Y_{1}}{L_{2} \alpha_{2} Y_{2}}$
• C. $\frac{A_{1}}{A_{2}}=\frac{L_{2} \alpha_{2} Y_{2}}{L_{1} \alpha_{1} Y_{1}}$
• D. $\frac{A_{1}}{A_{2}}=\frac{\alpha_{2} Y_{2}}{\alpha_{1} Y_{1}}$

Answer 1

Answer: (D)

$Y=\frac{\text { Stress }}{\text { Strain }}=\frac{T / A}{\Delta l / l}$
$T=\frac{Y \cdot \Delta l}{l} A=Y \cdot A \alpha \Delta T$
In both the rods tension will be same so
$T_{1}=T_{2}$, Hence $Y_{1} A_{1} \alpha_{1} \Delta T$
$=Y_{2} A_{2} \alpha_{2} \Delta T$
$\frac{A_{1}}{A_{2}}=\frac{Y_{2} \alpha_{2}}{Y_{1} \alpha_{1}}$