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 α_a and α_s, respectively. Young's modulus of aluminium is Y_a and of steel is Y_s.

Steel

Aluminium

Steel

Figure 23-E1

Answer 1

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 α_s and α_Al​, respectively.
Temperature is raised by θ.
_​So, the change in temperature, $∆\theta$ = θ $-$ 0 °C = θ
Young's modulus of steel and aluminium are γ_s and γ_Al, respectively.
If l be the final length of the system at temperature θ,
strain on the system $=\frac{\left(l-l_0\right)}{l_0}$ ​ ...(1)
Young's modulus = Stress/ Strain
Therefore, the total strain on the system $=\frac{\text{T}\mathrm{otal}\mathrm{stress}\text{on the}\mathrm{system}}{\text{T}\mathrm{otal}\mathrm{Young}\text{'}\mathrm{s}\mathrm{modulus}\mathrm{of}\text{the}\mathrm{system}}$

Now, total stress = stress due to the two steel rods + stress due to the aluminium rod
$\text{S}\mathrm{tress}=\frac{F}{A}=\alpha Y∆\theta$
Total stress = γ_sα_sθ + γ_sα_sθ + γ_Alα_Alθ
= 2γ_sα_sθ + γ_Alα_Alθ ...(2)
Young's modulus of the system,
Y = γ_s + γ_s + γ_Al = 2γ_s + γ_Al ...(3)
Using (1), (2) and (3), we get:
$$\begin{gathered} \mathrm{Strain}\text{on the}\mathrm{system}=\frac{2{\gamma }_s{\alpha }_s\theta +{\gamma }_{Al}{\alpha }_{Al}\theta }{2{\gamma }_s+{\gamma }_{Al}} \\ \Rightarrow \frac{l-l_0}{l_0}=\frac{2{\gamma }_s{\alpha }_s\theta +{\gamma }_{Al}{\alpha }_{Al}\theta }{2{\gamma }_s+{\gamma }_{Al}} \\ \Rightarrow l=l_0\left[1+\frac{2{\gamma }_s{\alpha }_s\theta +{\gamma }_{Al}{\alpha }_{Al}\theta }{2{\gamma }_s+{\gamma }_{Al}}\right] \end{gathered}$$
Therefore, the final length of the system will be $l_0\left[1+\frac{2{\gamma }_s{\alpha }_s\theta +{\gamma }_{Al}{\alpha }_{Al}\theta }{2{\gamma }_s+{\gamma }_{Al}}\right]$, where l₀ is its initial length.