- #1

Sefrez

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- 0

## Homework Statement

Find the frequency of small oscillations around the minimum of the potential

U(x)=1-e^(-x^2)

## Homework Equations

Force is the negative of the gradient of the potential...

## The Attempt at a Solution

Given the problem statement bit, "around the minimum," I take this as a hint to find the taylor expansion of U(x) to approximate the potential at the minimum.

In doing the taylor expansion at 0, I get that:

U(x) ≈ x^2

The force on a particle in this potential is given by:

F = -dU/dx = -2x.

And so we have that:

F + 2x = 0 => mx'' + 2x = 0. Solving this differential equation, we have something of the form:

x = A*cos(√(2/m)t - [itex]\phi[/itex])

So, we have, the angular frequency to be: ω = √(2/m).

Finally, the frequency is then: [itex]\nu[/itex] = ω/(2∏) = √(2/m)/(2∏) = (2m∏^2)^(-1/2)

Does this seem correct? I was a little confused that the frequency is dependent on the mass, but then I see that the potential given is independent of mass. But that's a bit odd. Thanks.