---
title: "An object of mass \\(m\\) on a horizontal frictionless surface is attached to a non-linear spring that exerts a restoring force of magnitude \\(F = bx^3\\) on the object, where \\(b\\) is a positive constant and \\(x\\) is the displacement from equilibrium. The object is initially released from rest at amplitude \\(A_1\\) and oscillates with period \\(T_1\\). The experiment is repeated with an initial release from rest at amplitude \\(A_2 = 2A_1\\), resulting in a period \\(T_2\\). Which of the following correctly compares \\(T_2\\) to \\(T_1\\) and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/122765/"
date_modified: "2026-09-28T11:02:52+00:00"
---

# An object of mass \(m\) on a horizontal frictionless surface is attached to a non-linear spring that exerts a restoring force of magnitude \(F = bx^3\) on the object, where \(b\) is a positive constant and \(x\) is the displacement from equilibrium. The object is initially released from rest at amplitude \(A_1\) and oscillates with period \(T_1\). The experiment is repeated with an initial release from rest at amplitude \(A_2 = 2A_1\), resulting in a period \(T_2\). Which of the following correctly compares \(T_2\) to \(T_1\) and provides the correct physical justification?

An object of mass \(m\) on a horizontal frictionless surface is attached to a non-linear spring that exerts a restoring force of magnitude \(F = bx^3\) on the object, where \(b\) is a positive constant and \(x\) is the displacement from equilibrium. The object is initially released from rest at amplitude \(A_1\) and oscillates with period \(T_1\). The experiment is repeated with an initial release from rest at amplitude \(A_2 = 2A_1\), resulting in a period \(T_2\). Which of the following correctly compares \(T_2\) to \(T_1\) and provides the correct physical justification?

![A schematic of a horizontal spring-mass oscillator on a horizontal surface. On the left, a vertical wall is attached to a horizontal coil spring. The right end of the spring connects to a rectangular block of mass \(m\). A horizontal dashed centerline extends beneath the block. A vertical tick mark on the ground directly below the block's center is labeled \(x = 0\). Above the spring, the label \(F = -b x^3\) is shown. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790593371-hc1Pd6.jpg)

- **A.** \(T_2 < T_1\), because the average restoring force increases by a larger factor than the displacement, causing the average speed of the object to increase by a larger factor than the distance traveled per cycle.
- **B.** \(T_2 = T_1\), because the period of an oscillating system depends only on the mass \(m\) and the force constant \(b\), remaining independent of amplitude for any symmetric restoring force.
- **C.** \(T_2 > T_1\), because the distance traveled by the object during each full cycle is doubled while the mass of the oscillating object remains unchanged.
- **D.** \(T_2 > T_1\), because the cubic force increases the stored potential energy, which requires a longer time interval to convert into kinetic energy during each oscillation.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/122765/*