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title: "A thin, rigid rod of length \\(L\\) lies along the positive \\(x\\)-axis with one end fixed at the origin \\(O\\). A distributed force acts along the rod such that the force acting on an infinitesimal segment \\(dx\\) located at position \\(x\\) is given by \\(d\\vec{F} = (\\beta x \\hat{j} + \\gamma x^2 \\hat{k})\\,dx\\), where \\(\\beta\\) and \\(\\gamma\\) are positive constants. Which of the following expressions represents the net torque vector \\(\\vec{\\tau}\\) exerted on the rod about the origin \\(O\\)?"
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url: "https://nerd-notes.com/ubq/120857/"
date_modified: "2026-08-23T04:43:41+00:00"
---

# A thin, rigid rod of length \(L\) lies along the positive \(x\)-axis with one end fixed at the origin \(O\). A distributed force acts along the rod such that the force acting on an infinitesimal segment \(dx\) located at position \(x\) is given by \(d\vec{F} = (\beta x \hat{j} + \gamma x^2 \hat{k})\,dx\), where \(\beta\) and \(\gamma\) are positive constants. Which of the following expressions represents the net torque vector \(\vec{\tau}\) exerted on the rod about the origin \(O\)?

A thin, rigid rod of length \(L\) lies along the positive \(x\)-axis with one end fixed at the origin \(O\). A distributed force acts along the rod such that the force acting on an infinitesimal segment \(dx\) located at position \(x\) is given by \(d\vec{F} = (\beta x \hat{j} + \gamma x^2 \hat{k})\,dx\), where \(\beta\) and \(\gamma\) are positive constants. Which of the following expressions represents the net torque vector \(\vec{\tau}\) exerted on the rod about the origin \(O\)?

![A three-dimensional Cartesian coordinate system with positive axes labeled x, y, and z. The x-axis extends horizontally to the right, the y-axis extends obliquely upward and to the right, and the z-axis extends vertically upward. A solid horizontal cylinder of length L representing a thin rod lies along the positive x-axis with its left end anchored at the origin labeled O and its right end at x = L. A small shaded segment of width dx is located at distance x from the origin. From this segment, two force component arrows originate: one pointing parallel to the positive y-axis labeled dF_y and one pointing vertically upward parallel to the positive z-axis labeled dF_z. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460221-mKyH63.jpg)

- **A.** \(-\dfrac{\gamma L^4}{4}\hat{j} + \dfrac{\beta L^3}{3}\hat{k}\)
- **B.** \(\dfrac{\gamma L^4}{4}\hat{j} + \dfrac{\beta L^3}{3}\hat{k}\)
- **C.** \(\dfrac{\gamma L^4}{4}\hat{j} - \dfrac{\beta L^3}{3}\hat{k}\)
- **D.** \(-\dfrac{\gamma L^4}{4}\hat{j} - \dfrac{\beta L^3}{3}\hat{k}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120857/*