---
title: "The potential energy \\(U\\) of a particle interacting in a conservative field is given by \\(U(x) = \\alpha x^{-n}\\), where \\(x > 0\\) is the position of the particle, and \\(\\alpha\\) and \\(n\\) are unknown positive constants. A student measures \\(U\\) at several positions \\(x\\) to experimentally determine both \\(n\\) and \\(\\alpha\\). Which of the following graphs should the student plot to yield a linear relationship, and how are the parameters \\(n\\) and \\(\\alpha\\) determined from the line?"
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url: "https://nerd-notes.com/ubq/123977/"
date_modified: "2026-09-28T13:29:49+00:00"
---

# The potential energy \(U\) of a particle interacting in a conservative field is given by \(U(x) = \alpha x^{-n}\), where \(x > 0\) is the position of the particle, and \(\alpha\) and \(n\) are unknown positive constants. A student measures \(U\) at several positions \(x\) to experimentally determine both \(n\) and \(\alpha\). Which of the following graphs should the student plot to yield a linear relationship, and how are the parameters \(n\) and \(\alpha\) determined from the line?

The potential energy \(U\) of a particle interacting in a conservative field is given by \(U(x) = \alpha x^{-n}\), where \(x > 0\) is the position of the particle, and \(\alpha\) and \(n\) are unknown positive constants. A student measures \(U\) at several positions \(x\) to experimentally determine both \(n\) and \(\alpha\). Which of the following graphs should the student plot to yield a linear relationship, and how are the parameters \(n\) and \(\alpha\) determined from the line?

- **A.** Plot \(\ln U\) on the vertical axis versus \(\ln x\) on the horizontal axis; the slope is \(-n\) and the vertical intercept is \(\alpha\).
- **B.** Plot \(\ln U\) on the vertical axis versus \(\ln x\) on the horizontal axis; the slope is \(-n\) and the vertical intercept is \(\ln\alpha\).
- **C.** Plot \(\ln U\) on the vertical axis versus \(\ln x\) on the horizontal axis; the slope is \(n\) and the vertical intercept is \(\ln\alpha\).
- **D.** Plot \(\ln U\) on the vertical axis versus \(x\) on the horizontal axis; the slope is \(-n\) and the vertical intercept is \(\ln\alpha\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123977/*