---
title: "A particle of mass \\(m\\) is constrained to move along the x-axis. The particle experiences a conservative force such that its potential energy \\(U(x)\\) is given by the function  \\[ U(x) = Ax^4 – Bx^2 \\]  where \\(A\\) and \\(B\\) are positive constants, and \\(U(0) = 0\\)."
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url: "https://nerd-notes.com/ubq/117744/"
date_modified: "2026-08-04T07:55:45+00:00"
---

# A particle of mass \(m\) is constrained to move along the x-axis. The particle experiences a conservative force such that its potential energy \(U(x)\) is given by the function

\[ U(x) = Ax^4 – Bx^2 \]

where \(A\) and \(B\) are positive constants, and \(U(0) = 0\).

A particle of mass \(m\) is constrained to move along the x-axis. The particle experiences a conservative force such that its potential energy \(U(x)\) is given by the function

\[ U(x) = Ax^4 - Bx^2 \]

where \(A\) and \(B\) are positive constants, and \(U(0) = 0\).

**Part a)** **Indicate** whether the equilibrium position at \(x = 0\) is stable or unstable. - [ ] Stable - [ ] Unstable **Justify** your answer qualitatively by considering the values or shape of the potential energy function near \(x = 0\). *(2 points)*

**Part b)** **Derive** an expression for the force \(F(x)\) exerted on the particle as a function of position \(x\). Express your answer in terms of \(A\), \(B\), \(x\), and fundamental constants, as appropriate. *(2 points)*

**Part c)** Using your expression from part (b), **support your claim** from part (a) regarding the stability of the equilibrium at \(x = 0\). *(2 points)*

**Part d)** **Derive** an expression for the positive position \(x_{eq}\) where the particle is in a stable equilibrium. Express your answer in terms of \(A\) and \(B\). *(2 points)*

**Part e)** The particle is placed at position \(x_1\), where \(0 < x_1 < x_{eq}\), and released from rest. A student claims that the particle will subsequently pass through the origin (\(x = 0\)). **Indicate** whether the student's claim is correct or incorrect. - [ ] Correct - [ ] Incorrect **Justify** your answer using principles of energy conservation. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117744/*