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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) = ω 2 x. 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)) + 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).

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Chapter 19, Problem 7MC

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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) = ω 2 x. 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)) + 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).

The heated rod from Problem 3 is subject to a volumetric heatingh(x) = h0xLin units of [Wm−3], as shown in the figure below. Under theheat supply the temperature of the rod changes along x with thetemperature function T(x). The temperature T(x) is governed by thefollowing equations:(−ddx (q(x)) + h(x) = 0 PDEq(x) = −kdTdx Fourier’s law of heat conduction(4)where q(x) is the heat flux through the rod and k is the (constant)thermal conductivity. Both ends of the bar are in contact with a heatreservoir at zero temperature. Determine:1. Appropriate BCs for this physical problem.2. The temperature function T(x).3. The heat flux function q(x).

A heated rod of length L is subject to a volumetric heating h(x) = h0xLinunits of [Wm−3], as shown in the figure below. Under the heat supply thetemperature of the rod changes along x with the temperature functionT(x). The temperature T(x) is governed by the following equations:(−ddx (q(x)) + h(x) = 0 PDEq(x) = −kdTdx Fourier’s law of heat conduction(3)where q(x) is the heat flux through the rod and k is the (constant)thermal conductivity. The left end of the bar is in contact with a heatreservoir at zero temperature, while the right end of the bar is thermallyinsulated. Determine:1. Appropriate BCs for this physical problem.2. The temperature function T(x).3. The heat flux function q(x).