# Is this the correct approach? (finding frequency of oscillation)

• Sefrez
In summary, the problem asks to find the frequency of small oscillations around the minimum of the given potential, U(x)=1-e^(-x^2). In order to do so, the taylor expansion of U(x) is taken to approximate the potential at the minimum. The resulting differential equation is solved to find the angular frequency, ω, which is then used to calculate the frequency, ν. The frequency is found to be dependent on the mass, but this is due to the potential being independent of mass.
Sefrez

## 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 - $\phi$)

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

Finally, the frequency is then: $\nu$ = 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.

This all seems entirely correct to me.

Your teach can come up with any kind of potential he wants!

## 1. What is the correct approach for finding frequency of oscillation?

The correct approach for finding frequency of oscillation depends on the specific system or phenomenon being studied. Some common methods include measuring the period of oscillation and using it to calculate frequency, using Fourier analysis to determine the dominant frequency component, or using resonance to find the natural frequency of the system.

## 2. How do you measure the period of oscillation?

The period of oscillation can be measured by recording the time it takes for one complete cycle of oscillation to occur. This can be done by using a stopwatch or timer to track the time between two consecutive peaks or troughs in the oscillation.

## 3. What is the difference between frequency and period of oscillation?

Frequency and period of oscillation are two different ways of measuring the same phenomenon. Frequency is a measure of how many cycles of oscillation occur in a given amount of time, usually expressed in Hertz (Hz). Period, on the other hand, is the time it takes for one complete cycle of oscillation to occur and is usually expressed in seconds (s). They are inversely related, meaning that as one increases, the other decreases.

## 4. Can resonance be used to find the natural frequency of a system?

Yes, resonance can be used to find the natural frequency of a system. Resonance occurs when a system is subjected to a periodic force at its natural frequency, causing it to vibrate with a large amplitude. By adjusting the frequency of the applied force and observing the amplitude of the system's response, we can determine the natural frequency of the system.

## 5. How does Fourier analysis help in finding frequency of oscillation?

Fourier analysis is a mathematical method used to break down a complex signal into its individual frequency components. By applying Fourier analysis to a signal representing an oscillatory system, we can identify the dominant frequency component and determine the frequency of oscillation. This method is particularly useful when dealing with signals that have multiple frequency components.

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