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Question

Two steel rods and an aluminium rod of equal length l0 and equal cross-section are joined rigidly at their ends, as shown in the figure below. All the rods are in a state of zero tension at 0°C. Find the length of the system when the temperature is raised to θ. Coefficient of linear expansion of aluminium and steel are αa and αs, respectively. Young's modulus of aluminium is Ya and of steel is Ys.
Steel
Aluminium
Steel
Figure 23-E1

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Solution

Given:
Initial length of the system (two identical steel rods + an aluminium rod) at 0 °C = l0
Coefficient of linear expansion of steel and aluminium are αs and αAl​, respectively.
Temperature is raised by θ.
So, the change in temperature, θ = θ - 0 °C = θ
Young's modulus of steel and aluminium are γs and γAl, respectively.
If l be the final length of the system at temperature θ,
strain on the system =l-l0l0 ​ ...(1)
Young's modulus = Stress/ Strain
Therefore, the total strain on the system =Total stress on the systemTotal Young's modulus of the system

Now, total stress = stress due to the two steel rods + stress due to the aluminium rod
Stress=FA=αYθ
Total stress = γsαsθ + γsαsθ + γAlαAlθ
= 2γsαsθ + γAlαAlθ ...(2)
Young's modulus of the system,
Y = γs + γs + γAl = 2γs + γAl ...(3)
Using (1), (2) and (3), we get:
Strain on the system =2γsαsθ+γAlαAl θ2γs+γAll-l0l0=2γsαsθ+γAlαAlθ2γs+γAll=l0 1+2γsαsθ+γAlαAlθ2γs+γAl
Therefore, the final length of the system will be l0 1+2γsαsθ+γAlαAlθ2γs+γAl, where l0 is its initial length.

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