Motion of a mass m confined to the x-axis (Hamiltonian)

[jmm5872]

Consider a mass m confined to the x-axis and subject to a force F_x=kx where k>0.

Write down and sketch the potential energy U(x) and describe the possible motions of the mass. (Distinguish between the cases that E>0 and E<0.


It is the part in parenthesis that confuses me. I can't picture what a negative value of energy would be.

I know the potential is U(x) = (1/2)kx², and that Total energy is kinetic plus potential (E = T + U). I also assume that the potential is always positive. If this is true, then the only was for the energy to be negative is to have the kinetic be negative and larger than the potential.

Does this refer to the case when the mass is moving in the -x direction giving T = -(1/2)mv²?

It seems to me that the motion should be the same whether the energy is negative or positive since this is a classical mass and confined to the parabolic potential well. It seems like it should oscillate back and forth for any energy.

For some reason I don't think I am picturing this correctly.

 

[vela]

You've written down the potential for the force F = -kx, but your force doesn't have the negative sign. The potential should be U(x) = -1/2 kx². Now E<0 should make sense to you.

 

[jmm5872]

The problem specifically states that k > 0, and F = kx, which gives U = 1/2 kx^2. I don't really understand what you are trying to say, or where the F = -kx that you wrote comes from.

 

[vela]

I'm saying your belief that F=kx implies U = 1/2 kx² is wrong. Look up how to find the potential from a force or vice versa.

 

[jmm5872]

Ah, okay, I understand now. I forgot that the potential is opposite the sign of force! Thank you!

F = -dU/dx