# Angular frequency for Potential Energy Function

• rogeralms
In summary, the conversation discusses finding the angular frequency for a mass oscillating on the x-axis with a given potential energy function. The solution involves taking the derivative of the potential energy with respect to x and using the approximation that the oscillation occurs around x=0, leading to a simplified formula for the angular frequency.
rogeralms

## Homework Statement

A mass moves along the x-axis with potential energy
U(x)= - U0 a^2 / (a^2 + x^2). What is the angular frequency assuming the oscillation is small enough to be harmonic?

## Homework Equations

w^2 = k/m with w as the angular frequency

F= -kx = -(gradient) U

## The Attempt at a Solution

Since this is one-dimensional we take the derivative of U with respect to x.

I get -(gradient) U = -2 U0 a^2 x / (a^2 + x^2)^2

Therefore k= 2 U0 a^2 / (a^2 + x^2)^2

The correct answer does not have an x term in it.

w (omega) = k/m = (2 U0 / m a^2) ^ (1/2)

Is there a binomial expansion that would essentially eliminate the x term in the denominator?

Thanks for any help.

The oscillation happens around x=0, so you can approximate the gradient there and neglect the x^2. That is exactly the approximation required by the problem statement.

## 1. What is angular frequency for potential energy function?

Angular frequency for potential energy function is a measure of how fast a particle is oscillating around its equilibrium position in a potential energy field. It is represented by the symbol ω and is equal to the square root of the spring constant divided by the mass of the particle.

## 2. How is angular frequency related to potential energy?

Angular frequency and potential energy are directly proportional to each other. This means that as the angular frequency increases, the potential energy also increases. Similarly, as the angular frequency decreases, the potential energy decreases as well.

## 3. What is the formula for calculating angular frequency for potential energy function?

The formula for calculating angular frequency for potential energy function is ω = √(k/m), where ω is the angular frequency, k is the spring constant, and m is the mass of the particle.

## 4. How does changing the mass affect the angular frequency for potential energy function?

Changing the mass of the particle affects the angular frequency for potential energy function in an inverse manner. This means that as the mass increases, the angular frequency decreases and vice versa.

## 5. How can angular frequency for potential energy function be used in real-world applications?

Angular frequency for potential energy function is used in various real-world applications, such as in designing and analyzing oscillating systems like springs, pendulums, and mass-spring systems. It is also used in fields like engineering, physics, and mechanics to understand the behavior of systems in potential energy fields.

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