A uniform vertical rod of natural length L, density ρ and Young's modulus Y is suspended from its upper end and stretches only under its own weight. Neglect any external load. What is its total elongation for small strain?
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Question

A uniform vertical rod of natural length L, density ρ and Young's modulus Y is suspended from its upper end and stretches only under its own weight. Neglect any external load. What is its total elongation for small strain?

Options

G1 Physics · Accepted Answer

Let x be the distance measured downward from the upper end. At section x, the tension equals the weight of the portion below the section. Therefore, T(x) = ρAg(L−x). An element dx undergoes extension d(ΔL) = [T(x) dx] / [AY]. Thus, d(ΔL) = [ρg(L−x) dx] / Y. Integrating from x = 0 to x = L, ΔL = [ρg/Y] ∫₀ᴸ(L−x)dx. Now, ∫₀ᴸ(L−x)dx = L²/2. Therefore, ΔL = [ρgL²] / [2Y]. Correct option: B.