Solving One Dimensional Wave Motion with Mass m

[zheng89120]

Homework Statement

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

U(x) = -U_oe^-(x/l)2

where U_o 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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[jrosen13]

Looks like an instanton to me. Anyways, don't you just do the normal lagrangian/hamiltonian methods here? Its just a classical problem, right?

 

[zheng89120]

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).

 

[jrosen13]

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?

 

[zheng89120]

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 U_o e^-(x/l)2

 

[jrosen13]

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!

 

[jrosen13]

I just noticed you wrote your spring constant as a function of x, but that wouldn't be right, would it?