---
title: "A thin, non-uniform rod of length \\(L\\) and total mass \\(M\\) lies along the \\(x\\)-axis between \\(x = 0\\) and \\(x = L\\). The linear mass density of the rod varies with position according to \\(\\lambda(x) = \\dfrac{2M}{L^2}x\\). A point particle of mass \\(m\\) is fixed on the \\(x\\)-axis at position \\(x = -d\\), where \\(d > 0\\). Which of the following expressions correctly represents the magnitude of the gravitational force exerted by the rod on the point particle?"
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url: "https://nerd-notes.com/ubq/123993/"
date_modified: "2026-09-28T13:30:03+00:00"
---

# A thin, non-uniform rod of length \(L\) and total mass \(M\) lies along the \(x\)-axis between \(x = 0\) and \(x = L\). The linear mass density of the rod varies with position according to \(\lambda(x) = \dfrac{2M}{L^2}x\). A point particle of mass \(m\) is fixed on the \(x\)-axis at position \(x = -d\), where \(d > 0\). Which of the following expressions correctly represents the magnitude of the gravitational force exerted by the rod on the point particle?

A thin, non-uniform rod of length \(L\) and total mass \(M\) lies along the \(x\)-axis between \(x = 0\) and \(x = L\). The linear mass density of the rod varies with position according to \(\lambda(x) = \dfrac{2M}{L^2}x\). A point particle of mass \(m\) is fixed on the \(x\)-axis at position \(x = -d\), where \(d > 0\). Which of the following expressions correctly represents the magnitude of the gravitational force exerted by the rod on the point particle?

![A horizontal axis labeled with an arrowhead pointing to the right represents the x-axis. A vertical tick mark on the axis is labeled x = -d, where a solid black circular dot representing a point particle of mass m is located. To the right, another vertical tick mark on the axis is labeled x = 0, marking the left end of a narrow horizontal rectangular bar representing a rod of length L. The bar extends along the axis to a vertical tick mark labeled x = L. The interior of the bar has a grayscale gradient shading that is light gray at the left end and becomes progressively darker toward the right end. A horizontal double-headed dimension arrow below the bar spans from x = 0 to x = L and is labeled L. A horizontal double-headed dimension arrow spans from x = -d to x = 0 and is labeled d. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790602203-AniBoQ.jpg)

- **A.** \(\dfrac{GMm}{L} \int_0^L \dfrac{1}{(x + d)^2}\,dx\)
- **B.** \(\dfrac{2GMm}{L^2} \int_0^L \dfrac{x}{(L + d - x)^2}\,dx\)
- **C.** \(\dfrac{2GMm}{L^2} \int_0^L \dfrac{x}{(x + d)^2}\,dx\)
- **D.** \(\dfrac{2GMm}{L^2} \int_d^{L+d} \dfrac{1}{x}\,dx\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/123993/*