---
title: "A particle of mass \\(m\\) is constrained to move along the \\(x\\)-axis in a region where its potential energy is given by \\(U(x) = \\alpha x^4\\), where \\(\\alpha\\) is a positive constant. The particle is released from rest at \\(x = A\\) and oscillates periodically about the equilibrium point \\(x = 0\\). If the oscillation amplitude \\(A\\) is increased, which of the following statements correctly describes the effect on the period of oscillation and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/120973/"
date_modified: "2026-08-23T04:44:53+00:00"
---

# A particle of mass \(m\) is constrained to move along the \(x\)-axis in a region where its potential energy is given by \(U(x) = \alpha x^4\), where \(\alpha\) is a positive constant. The particle is released from rest at \(x = A\) and oscillates periodically about the equilibrium point \(x = 0\). If the oscillation amplitude \(A\) is increased, which of the following statements correctly describes the effect on the period of oscillation and provides the correct physical justification?

A particle of mass \(m\) is constrained to move along the \(x\)-axis in a region where its potential energy is given by \(U(x) = \alpha x^4\), where \(\alpha\) is a positive constant. The particle is released from rest at \(x = A\) and oscillates periodically about the equilibrium point \(x = 0\). If the oscillation amplitude \(A\) is increased, which of the following statements correctly describes the effect on the period of oscillation and provides the correct physical justification?

![A graph of potential energy \(U\) versus position \(x\) on bare axes. The horizontal axis is labeled \(x\) at the right end, and the vertical axis is labeled \(U(x)\) at the top end. The intersection is labeled \(0\). A single solid curve shaped like a symmetric, flat-bottomed U-shaped bowl opens upward, centered at \(x = 0\) where \(U(0) = 0\). Two symmetric vertical dashed lines drop from the curve to the horizontal axis: one at \(-A\) on the negative horizontal axis and one at \(+A\) on the positive horizontal axis. A horizontal dashed line connects the points \((-A, U(A))\) and \((+A, U(A))\), intersecting the vertical axis at a tick labeled \(E\). No other labels, gridlines, lines, or markings appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460293-kysRTQ.jpg)

- **A.** The period remains unchanged because for any symmetric bound potential well, the period of oscillation is independent of amplitude.
- **B.** The period decreases because the restoring force magnitude scales as \(x^3\), causing the average speed of the particle to increase by a greater factor than the path length.
- **C.** The period increases because a larger amplitude requires the particle to travel a greater distance each cycle while the restoring force increases linearly with displacement.
- **D.** The period increases because the mechanical energy at the turning points requires more work to be done to traverse the well, which increases the travel time.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120973/*