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QPages 12-14: Q30: HC Verma: Concepts of Physics - Volume 2 Heat and Temperature Pages 12-14: Q30 Solution: Two steel rods and an aluminium rod of equal length and equal…

⚠️ Common Exam Traps & Conceptual Pitfalls

Students often average the linear expansion coefficients by the number of rods (or by cross‑section) and write $\Delta L/l_0=\frac{2\alpha_s+\alpha_a}{3}\theta$, ignoring that the rods are forced to have the same final strain. The correct length comes from imposing zero net force and equal strain, which gives a stiffness‑weighted effective coefficient $\alpha_{eff}=\frac{2Y_s\alpha_s+Y_a\alpha_a}{2Y_s+Y_a}$. Remember: when different materials are bonded, compatibility (same strain) and equilibrium (sum of internal forces zero) replace a simple arithmetic mean of $\alpha$.

Two steel rods and an aluminium rod of equal length l0l_0 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\theta. Coefficient of linear expansion of aluminium and steel are αa\alpha_a and αs\alpha_s, respectively. Young's modulus of aluminium is YaY_a and of steel is YsY_s.

Step-by-Step Verified Solution

**Answer:** Length =l0[1+2αsYs+αaYa2Ys+Yaθ]= l_0[1 + \frac{2\alpha_s Y_s + \alpha_a Y_a}{2Y_s + Y_a}\theta] **Step-by-Step Explanation:** For the composite rod (two steel, one aluminium in parallel), the total force is zero. The effective expansion coefficient is the weighted average by stiffness YAYA. Total stiffness = 2YsA+YaA=A(2Ys+Ya)2Y_s A + Y_a A = A(2Y_s + Y_a). Net thermal force = A(2Ysαs+Yaαa)θA(2Y_s\alpha_s + Y_a\alpha_a)\theta. Effective strain = 2Ysαs+Yaαa2Ys+Yaθ\frac{2Y_s\alpha_s + Y_a\alpha_a}{2Y_s + Y_a}\theta. Length = l0[1+2Ysαs+Yaαa2Ys+Yaθ]l_0[1 + \frac{2Y_s\alpha_s + Y_a\alpha_a}{2Y_s + Y_a}\theta].

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