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A rod of mass M, cross-sectional area A, length l0, linear expansion coefficient alpha, and Young's modulus Y is connected at one end to a fixed wall and at the other end to a spring of constant k (other end of spring is fixed). Initially the spring is at natural length. When the temperature rises by delta_T, find the compressive stress developed in the rod.

1. [k * Y * alpha * delta_T * l0] / [k*l0 + Y*A]
2. Y * alpha * delta_T
3. [k*l0 + Y*A] / [k*l0] * Y*alpha*delta_T
4. [k*l0 + Y] / [k*l0] * Y*alpha*delta_T

Correct answer: [k * Y * alpha * delta_T * l0] / [k*l0 + Y*A]

Solution

Let x be the spring compression. Rod's thermal expansion alpha*l0*delta_T is split: spring compression x and rod elastic compression. Force balance gives k*x = Y*A*(alpha*l0*delta_T - x)/l0, from which stress = k*x/A = k*Y*alpha*delta_T*l0 / (k*l0 + Y*A).

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