---
title: "A particle of mass \\(m\\) moves along the \\(x\\)-axis in a potential energy field given by \\(U(x) = -\\dfrac{U_0}{\\cosh^2(x/a)}\\), where \\(U_0\\) and \\(a\\) are positive constants. The particle is displaced a small distance from its equilibrium position at \\(x = 0\\) and released from rest. What is the angular frequency \\(\\omega\\) of the resulting small oscillations?"
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url: "https://nerd-notes.com/ubq/117673/"
date_modified: "2026-08-04T07:49:49+00:00"
---

# A particle of mass \(m\) moves along the \(x\)-axis in a potential energy field given by \(U(x) = -\dfrac{U_0}{\cosh^2(x/a)}\), where \(U_0\) and \(a\) are positive constants. The particle is displaced a small distance from its equilibrium position at \(x = 0\) and released from rest. What is the angular frequency \(\omega\) of the resulting small oscillations?

A particle of mass \(m\) moves along the \(x\)-axis in a potential energy field given by \(U(x) = -\dfrac{U_0}{\cosh^2(x/a)}\), where \(U_0\) and \(a\) are positive constants. The particle is displaced a small distance from its equilibrium position at \(x = 0\) and released from rest. What is the angular frequency \(\omega\) of the resulting small oscillations?

![A plot of potential energy U(x) versus position x on Cartesian axes. The horizontal axis is labeled x with an origin at 0. The vertical axis represents U(x) with a tick mark labeled -U_0. A symmetric curve opens upward from a global minimum at (0, -U_0) and flattens horizontally toward 0 as x goes to positive and negative infinity. A dashed curve overlays the minimum around x = 0, representing a parabolic approximation. No other labels, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829789-n3sRql.jpg)

- **A.** \(\omega = \dfrac{1}{2a} \sqrt{\dfrac{U_0}{m}}\)
- **B.** \(\omega = \dfrac{1}{a} \sqrt{\dfrac{U_0}{2m}}\)
- **C.** \(\omega = \dfrac{1}{a} \sqrt{\dfrac{U_0}{m}}\)
- **D.** \(\omega = \dfrac{1}{a} \sqrt{\dfrac{2U_0}{m}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117673/*