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Two rods of equal cross-sections, one of copper and the other of steel are joined to form a composite rod of length $2.0 m$ at $20 ° C$ , the length of the copper rod is $0.5 m$ . When the temperature is raised to $120 ° C$ , the length of composite rod increases to $2.002 m$ . If the composite rod is fixed between two rigid walls and thus not allowed to expand, it is foundthat the length of the component rods also do not change with increase in temperature. calculate the Young's modulus of steel. Given Young's modulus of copper = $1.3 \times 10^{11} N / m^{2}$ , the coefficient of linear expansion of copper $α_{c} = 1.6 \times 10^{- 5} / {}^{\circ}c .$

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$2.6 \times 10^{11} Pa$

$1.6 \times 10^{10} Pa$

$1.3 \times 10^{10} Pa$

$0.9 \times 10^{10} Pa$

answer is A.

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Detailed Solution

The change in length of copper rod due to change in temprature from $20 ° C$ to $120 ° C$ .

$Δ l_{1} = l_{c} α_{c} Δ T; 0.5 α_{c} (120 - 20) = 50 α_{c}$

For steel, $Δ l_{2} = l_{s} α_{s} Δ T = 1.5 α_{s} (120 - 20) = 150 α_{s}$

Total change in length, $Δ l = Δ l_{1} + Δ l_{2} = 50 α_{c} + 150 α_{s}$

It is given that $Δ l = 0.002 m$

$50 α_{c} + 150 α_{s} = 0.002 or, α_{s} = \frac{0.002 - 50 α_{c}}{150} = \frac{0.002 - 50 \times 1.6 \times 10^{- 5}}{150} = 0.8 \times 10^{- 5} / {}^{\circ}c F = Y A α Δ T; Y_{s} α_{s} = Y_{c} α_{c} Y_{s} = 2.6 \times 10^{11} N / m^{2}$

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