The elastic bar from Problem 1 spins with angular velocity w about an axis, as shown in the figure below. The radial acceleration at a generic point x along the bar is a(x) = w²x. Under this radial acceleration, the bar stretches along x with displacement function u(x). The displacement u(x) is governed by the following equations: √ d dx (σ(x)) + pa(x) = 0 PDE (o(x) = Edu Hooke's law (2) where σ(x) is the axial stress in the rod, p is the mass density, and E is the (constant) Young's modulus. The bar is pinned on the rotation axis at x = 0 and it is also pinned at x = L. ω L 4
The elastic bar from Problem 1 spins with angular velocity ω about an axis, as shown in the figure below. The radial acceleration at a generic point x along the bar is a(x) = ω2x. Under this radial acceleration, the bar stretches along x with displacement function u(x). The displacement d u(x) is governed by the following equations: dx (σ(x)) + ρa(x) = 0 PDE σ(x) = E du dx Hooke’s law (2) where σ(x) is the axial stress in the rod, ρ is the mass density, and E is the (constant) Young’s modulus. The bar is pinned on the rotation axis at x = 0 and it is also pinned at x = L. Determine: 1. Appropriate BCs for this physical problem. 2. The displacement function u(x). 3. The stress function σ(x). SIDE QUESTION: I saw a tutor solve it before but I didn't understand why the tutor did not divide E under the second term (c1x) before finding u(x). The tutor only divided E under first term. please explain and thank you
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