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(D) When the temperature increases by $\Delta T$,the rods tend to expand. Since they are fixed between rigid supports,they experience a compressive fo

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$\frac{A_1}{A_2} = \frac{\alpha_2 Y_2}{\alpha_1 Y_1}$

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(D) When the temperature increases by $\Delta T$,the rods tend to expand. Since they are fixed between rigid supports,they experience a compressive force $F$. For the total length to remain unchanged,the sum of the thermal expansion and the compression due to the force must be zero for each rod,or more simply,the force $F$ generated in both rods must be equal.The thermal expansion is $\Delta L_{thermal} = L \alpha \Delta T$.The compression due to force $F$ is $\Delta L_{comp} = \frac{F L}{A Y}$.For the total length to remain constant,the net change in length must be zero: $\Delta L_{thermal} - \Delta L_{comp} = 0$,which implies $F = A Y \alpha \Delta T$.Since the rods are in series,the force $F$ is the same in both rods:$F_1 = F_2$$A_1 Y_1 \alpha_1 \Delta T = A_2 Y_2 \alpha_2 \Delta T$$\frac{A_1}{A_2} = \frac{\alpha_2 Y_2}{\alpha_1 Y_1}$

Two rods are joined between fixed supports as shown in the figure. The condition for no change in the total length of the system with an increase in temperature is given by:
($\alpha_1, \alpha_2$ = linear expansion coefficients,$A_1, A_2$ = cross-sectional areas of rods,$Y_1, Y_2$ = Young's moduli)

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Two rods are joined between fixed supports as shown in the figure. The condition for no change in the total length of the system with an increase in temperature is given by:($\alpha_1, \alpha_2$ = linear expansion coefficients,$A_1, A_2$ = cross-sectional areas of rods,$Y_1, Y_2$ = Young's moduli)