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Longitudinal Strain

Two elastic r...

Question

Two elastic 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 is - $(α_{1}, α_{2}$ = linear expansion coefficient, $A_{1}, A_{2}$ = Area of rods, $Y_{1}, Y_{2}$ = Young's modulus, $L_{1}, L_{2}$ = lengths )

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

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

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

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

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Solution

The correct option is D $\frac{A_{1}}{A_{2}} = \frac{α_{2} Y_{2}}{α_{1} Y_{1}}$
Since, tension in the two rods will be same, hence,
$A_{1} Y_{1} α_{1} Δ θ = A_{2} Y_{2} α_{2} Δ θ$
$\Rightarrow A_{1} Y_{1} α_{1} = A_{2} Y_{2} α_{2}$
$\frac{A_{1}}{A_{2}} = \frac{α_{2} Y_{2}}{α_{1} Y_{1}}$

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α_{1} = 2 α_{2}, Y_{1} = Y_{2}

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