---
title: "A small bead of mass \\(m\\) and positive charge \\(q\\) is constrained to move along a frictionless, nonconducting horizontal track aligned with the \\(x\\)-axis. An arrangement of fixed external charges produces an electric potential along the axis given by the function \\(V(x) = \\alpha x^2\\), where \\(\\alpha\\) is a positive constant and \\(V = 0\\) at \\(x = 0\\). The bead is released from rest at a small initial displacement \\(x = x_0\\). Which of the following expressions represents the period \\(T\\) of the resulting oscillation of the bead?"
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url: "https://nerd-notes.com/ubq/117032/"
date_modified: "2026-08-04T06:17:02+00:00"
---

# A small bead of mass \(m\) and positive charge \(q\) is constrained to move along a frictionless, nonconducting horizontal track aligned with the \(x\)-axis. An arrangement of fixed external charges produces an electric potential along the axis given by the function \(V(x) = \alpha x^2\), where \(\alpha\) is a positive constant and \(V = 0\) at \(x = 0\). The bead is released from rest at a small initial displacement \(x = x_0\). Which of the following expressions represents the period \(T\) of the resulting oscillation of the bead?

A small bead of mass \(m\) and positive charge \(q\) is constrained to move along a frictionless, nonconducting horizontal track aligned with the \(x\)-axis. An arrangement of fixed external charges produces an electric potential along the axis given by the function \(V(x) = \alpha x^2\), where \(\alpha\) is a positive constant and \(V = 0\) at \(x = 0\). The bead is released from rest at a small initial displacement \(x = x_0\). Which of the following expressions represents the period \(T\) of the resulting oscillation of the bead?

![A plot of electric potential V on the vertical axis versus position x on the horizontal axis. A upward-opening parabola centered at the origin (0,0) represents the electric potential curve V(x) = alpha * x^2. Directly below the origin, a horizontal line represents the x-axis track, on which a small spherical bead labeled with mass m and charge +q rests at a displaced position x_0. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785824222-hFYBmN.jpg)

- **A.** \(2\pi \sqrt{\dfrac{m}{q \alpha}}\)
- **B.** \(2\pi \sqrt{\dfrac{2m}{q \alpha}}\)
- **C.** \(2\pi \sqrt{\dfrac{m}{2q \alpha}}\)
- **D.** \(2\pi \sqrt{\dfrac{2q \alpha}{m}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117032/*