A uniform elastic rod of natural length L, density ρ and Young's modulus Y rotates with constant angular speed ω about one end in a horizontal plane. Assume the elastic extension is sufficiently small that centrifugal forces may be calculated using the natural coordinates. What is the total elongation of the rod?
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Question

A uniform elastic rod of natural length L, density ρ and Young's modulus Y rotates with constant angular speed ω about one end in a horizontal plane. Assume the elastic extension is sufficiently small that centrifugal forces may be calculated using the natural coordinates. What is the total elongation of the rod?

Options

G1 Physics · Accepted Answer

Consider a section at distance x from the axis. The tension at this section must provide centripetal force to the part of the rod lying from x to L. Therefore, T(x) = ∫ from x to L of ρAω²r dr. Hence, T(x) = [ρAω²/2](L²−x²). The elemental extension is d(ΔL) = [T(x) dx] / [AY]. Therefore, ΔL = [ρω²/(2Y)] ∫₀ᴸ(L²−x²)dx. The integral is L³−L³/3 = 2L³/3. Thus, ΔL = [ρω²/(2Y)] [2L³/3] = [ρω²L³] / [3Y]. Correct option: B.