How does frequency depend on the potential energy?

[JoAuSc]

Let's say you have a particle confined to one dimension in a potential field U(x). At t=0, the particles initial kinetic energy is K_o. U(x) is such that the particle is trapped between two points x=a and x=b; this means U(a) = U(b) = K_o, and the slopes of U(x) at a and b are such that the particle is kept between a and b. How would you find the frequency of this system?

 

[Gokul43201]

From -dU/dx, you know the force on the particle, F(x). With it's mass, you can find the acceleration a(x). From a(x) and the total distance, x = |b-a|, you can find the time taken, since you know v(a) = v(b) = 0. Twice this time is the inverse of the frequency.

 

[robphy]

By doing a Taylor expansion for U(x), you may also be able to approximate your potential energy function by "a simple harmonic oscillator with spring constant k".