3.25 The two-dimensional displacement field in a body is given byu₁ (X) = X₁ [X²X₂ +0₁ (20² + 30²X₂ X³)],2 (20² + 3}{0²X2 — — X³ + 3}{G₁X³²X₂),where c₁ and ₂ are constants. Find the linear and nonlinear Green-Lagrange strains.U₂(X) =-X₂ (20² +

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Answered: 3.25 The two-dimensional displacement…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = = x²ex. (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of vand x. Show that dE/dt = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion mv = F) (iii) Assume the particle starts in x0 = 0 with positive initial velocity vo > 0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ô, with 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 ༧(2) vo m (iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > v. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the…
Consider the following function: z = α x+ β y^2, where alpha and beta are constant parameters. Write the total differential dz and evaluate the partial derivatives
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3.25 The two-dimensional displacement field in a body is given by u₁ (X) = X₁ [X²X₂ +0₁ (20² + 30²X₂ X³)], 2 (20² + 3}{0²X2 — — X³ + 3}{G₁X³²X₂), where c₁ and ₂ are constants. Find the linear and nonlinear Green-Lagrange strains. U₂(X) =-X₂ (20² +