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title: "A particle of mass \\(m\\) is constrained to move along the \\(x\\)-axis under the influence of a conservative potential energy function given by \\(U(x) = c|x|\\), where \\(c\\) is a positive constant. The particle is released from rest at position \\(x = x_0\\) (where \\(x_0 > 0\\)) and undergoes periodic oscillations between \\(x = -x_0\\) and \\(x = x_0\\). Which of the following integral expressions correctly represents the period \\(T\\) of the particle’s oscillation?"
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url: "https://nerd-notes.com/ubq/120967/"
date_modified: "2026-08-23T04:44:52+00:00"
---

# A particle of mass \(m\) is constrained to move along the \(x\)-axis under the influence of a conservative potential energy function given by \(U(x) = c|x|\), where \(c\) is a positive constant. The particle is released from rest at position \(x = x_0\) (where \(x_0 > 0\)) and undergoes periodic oscillations between \(x = -x_0\) and \(x = x_0\). Which of the following integral expressions correctly represents the period \(T\) of the particle’s oscillation?

A particle of mass \(m\) is constrained to move along the \(x\)-axis under the influence of a conservative potential energy function given by \(U(x) = c|x|\), where \(c\) is a positive constant. The particle is released from rest at position \(x = x_0\) (where \(x_0 > 0\)) and undergoes periodic oscillations between \(x = -x_0\) and \(x = x_0\). Which of the following integral expressions correctly represents the period \(T\) of the particle's oscillation?

- **A.** \(T = \sqrt{\dfrac{2m}{c}} \int_0^{x_0} \dfrac{dx}{\sqrt{x_0 - x}}\)
- **B.** \(T = \sqrt{\dfrac{8m}{c}} \int_0^{x_0} \dfrac{dx}{\sqrt{x_0 - x}}\)
- **C.** \(T = \sqrt{\dfrac{8m}{c}} \int_0^{x_0} \dfrac{dx}{\sqrt{x}}\)
- **D.** \(T = \sqrt{\dfrac{m}{2c}} \int_0^{x_0} \dfrac{dx}{\sqrt{x_0 - x}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120967/*