(Solved): solve the first problem ( under radial acceleration ) not the second. New solution please The elas ...

The elastic bar spins with angular velocity $$?$$ about an axis, as shown in the figure below. The radial acceleration at a generic point $$x$$ along the bar is $$a(x)={?}^{2}x$$. Under this radial acceleration, the bar stretches along $$x$$ with displacement function $$u(x)$$. The displacement $$u(x)$$ is governed by the following equations: $$\{\begin{array}{ll}\frac{d}{dx}(?(x))+?a(x)=0& \text{ PDE }\\ ?(x)=E\frac{du}{dx}& \text{ Hooke’s law }\end{array}$$ where $$?(x)$$ is the axial stress in the rod, $$?$$ 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 $$BC$$ sor this physical problem. 2. The displacement function $$u(x)$$. 3. The stress function $$?(x)$$. An elastic bar of length $$L$$ spins with angular velocity $$?$$ about an axis, as shown in the figure below. The radial acceleration at a generic point $$x$$ along the bar is $$a(x)={?}^{2}x$$. Due to this radial acceleration, the bar stretches along $$x$$ with displacement function $$u(x)$$. The displacement $$u(x)$$ is governed by the following equations: $$\{\begin{array}{ll}\frac{d}{dx}(?(x))+?a(x)=0& \text{ PDE }\\ ?(x)=E\frac{du}{dx}& \text{ Hooke’s law }\end{array}$$ where $$?(x)$$ is the axial stress in the rod, $$?$$ 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 free at $$x=L$$. Determine: 1. Appropriate $$BC$$ s for this physical problem. 2. The displacement function $$u(x)$$. 3. The stress function $$?(x)$$. An elastic bar of length $$L$$ spins with angular velocity $$?$$ about an axis, as shown in the figure below. The radial acceleration at a generic point $$x$$ along the bar is $$a(x)={?}^{2}x$$. Due to this radial acceleration, the bar stretches along $$x$$ with displacement function $$u(x)$$. The displacement $$u(x)$$ is governed by the following equations: $$\{\begin{array}{ll}\frac{d}{dx}(?(x))+?a(x)=0& \text{ PDE }\\ ?(x)=E\frac{du}{dx}& \text{ Hooke’s law }\end{array}$$ where $$?(x)$$ is the axial stress in the rod, $$?$$ 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 free at $$x=L$$. Determine: 1. Appropriate $$BC$$ s for this physical problem. 2. The displacement function $$u(x)$$. 3. The stress function $$?(x)$$. The elastic bar spins with angular velocity $$?$$ about an axis, as shown in the figure below. The radial acceleration at a generic point $$x$$ along the bar is $$a(x)={?}^{2}x$$. Under this radial acceleration, the bar stretches along $$x$$ with displacement function $$u(x)$$. The displacement $$u(x)$$ is governed by the following equations: $$\{\begin{array}{ll}\frac{d}{dx}(?(x))+?a(x)=0& \text{ PDE }\\ ?(x)=E\frac{du}{dx}& \text{ Hooke’s law }\end{array}$$ where $$?(x)$$ is the axial stress in the rod, $$?$$ 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 $$BC$$ sor this physical problem. 2. The displacement function $$u(x)$$. 3. The stress function $$?(x)$$.