A particle under double-exponential potential

Question

A particle of mass m is constrained to move along the x-axis. It's potential energy is given by:

U(x) = U0 [ e−2β(x−x0) − 2e−β(x−x0) ]

where U₀, β, and x₀, are known parameters.

a) Assume that the particle is subject to a weak frictional force. What would be the position of the particle after a very long time?

b) Now ignore the existence of the friction. Find an expression for the natural frequency (in Radians/second) of the particle around its equilibrium position

Answer 1

First ... the potential function you quote is actually a single exponential ... by definition, force is dU/dx and if we assume this frictional force is linear, the distance x can be found by setting f=dU/dx and solving algebraicly. If f=0 then we have a separable differential equation du/dx=mx'' that can be solved by direct integration