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A particle of mass m moves along the x-axis. Its potential energy at any point x is V(x) = V₁x²e¯x², Vo = constant. i) Find the force on the particle. ii) Find all the points on the x-axis where the particle can be in equilibrium. Determine whether each equilibrium point is stable or unstable. Sketch the potential energy and label the equilibrium points. ¡¡¡) Determine the maximum total energy E。 that the particle can have and still execute bounded motion. If E < E is the motion necessarily bounded? Explain.

A particle of mass m moves along the x-axis. Its potential energy at any point x is V(x) = V₁x²e¯x², Vo = constant. i) Find the force on the particle. ii) Find all the points on the x-axis where the particle can be in equilibrium. Determine whether each equilibrium point is stable or unstable. Sketch the potential energy and label the equilibrium points. ¡¡¡) Determine the maximum total energy E。 that the particle can have and still execute bounded motion. If E < E is the motion necessarily bounded? Explain.

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A particle of mass m moves along the x-axis. Its potential energy at any point x is V(x) = Vox²e-² Vo = constant. i) Find the force on the particle. ii) Find all the points on the x-axis where the particle can be in equilibrium. Determine whether each equilibrium point is stable or unstable. Sketch the potential energy and label the equilibrium points. iii) Determine the maximum total energy Eo that the particle can have and still execute bounded motion. If E < Eo is the motion necessarily bounded? Explain. iv) Sketch the qualitatively different possible phase portraits (vary E).
Transcribed Image Text:A particle of mass m moves along the x-axis. Its potential energy at any point x is V(x) = Vox²e-² Vo = constant. i) Find the force on the particle. ii) Find all the points on the x-axis where the particle can be in equilibrium. Determine whether each equilibrium point is stable or unstable. Sketch the potential energy and label the equilibrium points. iii) Determine the maximum total energy Eo that the particle can have and still execute bounded motion. If E < Eo is the motion necessarily bounded? Explain. iv) Sketch the qualitatively different possible phase portraits (vary E).
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Please answer the remaining subpart (part (iv)).

A particle of mass m moves along the x-axis. Its potential energy at any point x is V(x) = Vox²e-² Vo = constant. i) Find the force on the particle. ii) Find all the points on the x-axis where the particle can be in equilibrium. Determine whether each equilibrium point is stable or unstable. Sketch the potential energy and label the equilibrium points. iii) Determine the maximum total energy Eo that the particle can have and still execute bounded motion. If E < Eo is the motion necessarily bounded? Explain. iv) Sketch the qualitatively different possible phase portraits (vary E).
Transcribed Image Text:A particle of mass m moves along the x-axis. Its potential energy at any point x is V(x) = Vox²e-² Vo = constant. i) Find the force on the particle. ii) Find all the points on the x-axis where the particle can be in equilibrium. Determine whether each equilibrium point is stable or unstable. Sketch the potential energy and label the equilibrium points. iii) Determine the maximum total energy Eo that the particle can have and still execute bounded motion. If E < Eo is the motion necessarily bounded? Explain. iv) Sketch the qualitatively different possible phase portraits (vary E).
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