---
title: "A conservative interaction between two particles has the potential energy function \\(U(r)\\) shown in the graph, where \\(r\\) is the separation distance between the particles and \\(U(r) \\to 0\\) as \\(r \\to \\infty\\). The curve has a local minimum of \\(-U_0\\) at \\(r = r_0\\), crosses the horizontal axis at \\(r_1  r_0\\). Which of the following statements correctly identifies the nature of the interparticle force and the energy required to liberate the system?"
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url: "https://nerd-notes.com/ubq/124087/"
date_modified: "2026-09-28T13:30:22+00:00"
---

# A conservative interaction between two particles has the potential energy function \(U(r)\) shown in the graph, where \(r\) is the separation distance between the particles and \(U(r) \to 0\) as \(r \to \infty\). The curve has a local minimum of \(-U_0\) at \(r = r_0\), crosses the horizontal axis at \(r_1  r_0\). Which of the following statements correctly identifies the nature of the interparticle force and the energy required to liberate the system?

A conservative interaction between two particles has the potential energy function \(U(r)\) shown in the graph, where \(r\) is the separation distance between the particles and \(U(r) \to 0\) as \(r \to \infty\). The curve has a local minimum of \(-U_0\) at \(r = r_0\), crosses the horizontal axis at \(r_1 < r_0\), and has an inflection point at \(r = r_i\) where \(r_i > r_0\). Which of the following statements correctly identifies the nature of the interparticle force and the energy required to liberate the system?

![A Cartesian coordinate graph with horizontal axis labeled r and vertical axis labeled U(r). The horizontal axis has tick marks labeled r_1, r_0, and r_i in order of increasing distance from the vertical axis. The vertical axis has a tick mark labeled -U_0 below the origin and 0 at the origin. A single smooth, continuous solid curve represents U(r). Starting at small r with very large positive values near the vertical axis, the curve drops steeply downward to the right, crosses the horizontal axis at r = r_1, and reaches its lowest point at coordinates (r_0, -U_0). To the right of r_0, the curve rises upward, passes through an inflection point at r = r_i where it changes from concave up to concave down, and approaches the horizontal axis asymptotically from below as r increases toward infinity. A horizontal dashed reference line extends from the minimum to the vertical axis at -U_0. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602222-ADOVe4.jpg)

- **A.** The force is repulsive for \(r < r_0\) because \(\dfrac{dU}{dr} < 0\), the attractive force has maximum magnitude at the inflection point \(r_i\), and the minimum kinetic energy required at \(r_0\) to liberate the particle to infinity is \(U_0\).
- **B.** The force is attractive for \(r < r_0\) because \(\dfrac{dU}{dr} < 0\), the attractive force has maximum magnitude at the inflection point \(r_i\), and the minimum kinetic energy required at \(r_0\) to liberate the particle to infinity is \(U_0\).
- **C.** The force is repulsive for \(r < r_0\) because \(\dfrac{dU}{dr} < 0\), the attractive force has maximum magnitude at the separation where \(U(r) = 0\), and the minimum kinetic energy required at \(r_0\) to liberate the particle to infinity is \(U_0\).
- **D.** The force is repulsive for \(r < r_0\) because \(\dfrac{dU}{dr} < 0\), the attractive force has maximum magnitude at the inflection point \(r_i\), and the minimum kinetic energy required at \(r_0\) to liberate the particle to infinity is \(\dfrac{1}{2}U_0\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124087/*