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Q.

Two elastic rods are joined between fixed supports as shown in fig. The condition for no change in the lengths of individual rods with the increase of temperature $(α_{1}, α_{2} =$ linear expansion coefficients, $A_{1}, A_{2} =$ areas of cross sections of rods, $Y_{1}, Y_{2} =$ Young’s module) is

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a

$\frac{A_{1}}{A_{2}} = \frac{α_{2} Y_{2}}{α_{1} Y_{1}}$

b

$\frac{A_{1}}{A_{2}} = \frac{L_{1} α_{1} Y_{1}}{L_{2} α_{2} Y_{2}}$

c

$\frac{A_{1}}{A_{2}} = \frac{α_{1} Y_{1}}{α_{2} Y_{2}}$

d

$\frac{A_{1}}{A_{2}} = \frac{L_{2} α_{2} Y_{2}}{L_{1} α_{1} Y_{1}}$

answer is D.

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Detailed Solution

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Since tension in the two rods will be same

$A_{1} Y_{1} α_{1} {θ=A}_{2} Y_{2} α_{2} {θ,A}_{1} Y_{1} α_{1} {=A}_{2} Y_{2} α_{2}$

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Two elastic rods are joined between fixed supports as shown in fig. The condition for no change in the lengths of individual rods with the increase of temperature ( α1,α2=linear expansion coefficients, A1,A2= areas of cross sections of rods, Y1,Y2= Young’s module) is