Solving One Dimensional Wave Motion with Mass m

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Homework Help Overview

The problem involves a particle of mass m moving in one dimension under a specific potential energy function, U(x) = -Uoe^-(x/l)². The initial conditions specify that the mass starts at position 0.01l with zero velocity. The goal is to determine the position of the mass over time.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the applicability of Lagrangian and Hamiltonian methods, with some expressing uncertainty about these approaches. There is a suggestion to sketch the potential and consider approximations based on familiar potentials. One participant presents a potential solution involving harmonic motion, while another questions the validity of a spring constant that varies with position.

Discussion Status

The discussion is active, with participants exploring different interpretations of the problem and sharing insights. Some guidance has been offered regarding the potential and its approximation, and there is an acknowledgment of the harmonic oscillator concept. However, there is no explicit consensus on the approach or solution.

Contextual Notes

Participants note that the course does not expect knowledge of advanced methods like Lagrangian or Hamiltonian mechanics, which may influence the discussion and approaches taken.

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Homework Statement



A particle of mass m moves in one dimension subject to the potential energy given by:

U(x) = -Uoe-(x/l)2

where Uo is a positive constant with units of energy. Supposing that at time t=0 the mass has zero velocity and is located at position 0.01l, what is the position of the mass for all subsequent times?

Homework Equations



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The Attempt at a Solution



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Looks like an instanton to me. Anyways, don't you just do the normal lagrangian/hamiltonian methods here? Its just a classical problem, right?
 
This is a second year engineering physics course on Vibrations and Waves, that does not expect any knowledge of Lagrangian method (I barely know) and Hamiltonian method (don't know).
 
Sorry that was basically overkill, I had different idea in mind when i looked at it quickly before.

Sketch the potential, where does the mass start out? Is it reasonable to approximate the potential by one that you know well?
 
Thanks Jrosen! You have really helped me on this problem.

I got something like:

x = 0.01l cos (wt)

where w^2 = k/m

where k = U'' = 4(x/l)2 Uo e-(x/l)2
 
It wasn't so bad was it? That looks right to me, harmonic oscillator all the way. In the future try to realize if you are patient and think it through a little you will figure it out. Don't be in a hurry to get the answer, enjoy the process of figuring it out for yourself!
 
I just noticed you wrote your spring constant as a function of x, but that wouldn't be right, would it?
 

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