2. A particle of mass m moves in a straight line under the action of a conservative force F(x)with potential energyU(x) = x²e¯x.(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 theequilibria on your plot.(ii) Calculate the total mechanical energy E of the system, in terms of v and x. Show thatdE/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 xo O with positive initial velocity vo > 0. Find the initialenergy Eo of the particle. Using (ii), show that the particle reaches x 2 only if vo v,with=8e-2v =mand in this case the particle's velocity in x =2 is8e-2v(2)=v²m

The text describes a one-dimensional motion of a particle under the influence of a conservative…

Answered: 2. A particle of mass m moves in a…

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…
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