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title: "A thin, non-uniform rod of length \\(L\\) lies on a frictionless horizontal table and is pivoted about a fixed, frictionless vertical axis passing through one end at \\(x = 0\\). The linear mass density of the rod is given by \\(\\lambda(x) = \\lambda_0 \\left(\\dfrac{x}{L}\\right)\\), where \\(\\lambda_0\\) is a positive constant and \\(x\\) is the distance from the pivot. The rod rotates in the horizontal plane with a constant angular speed \\(\\omega\\). What is the total kinetic energy of the rotating rod in terms of \\(\\lambda_0\\), \\(L\\), and \\(\\omega\\)?"
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url: "https://nerd-notes.com/ubq/120899/"
date_modified: "2026-08-23T04:44:11+00:00"
---

# A thin, non-uniform rod of length \(L\) lies on a frictionless horizontal table and is pivoted about a fixed, frictionless vertical axis passing through one end at \(x = 0\). The linear mass density of the rod is given by \(\lambda(x) = \lambda_0 \left(\dfrac{x}{L}\right)\), where \(\lambda_0\) is a positive constant and \(x\) is the distance from the pivot. The rod rotates in the horizontal plane with a constant angular speed \(\omega\). What is the total kinetic energy of the rotating rod in terms of \(\lambda_0\), \(L\), and \(\omega\)?

A thin, non-uniform rod of length \(L\) lies on a frictionless horizontal table and is pivoted about a fixed, frictionless vertical axis passing through one end at \(x = 0\). The linear mass density of the rod is given by \(\lambda(x) = \lambda_0 \left(\dfrac{x}{L}\right)\), where \(\lambda_0\) is a positive constant and \(x\) is the distance from the pivot. The rod rotates in the horizontal plane with a constant angular speed \(\omega\). What is the total kinetic energy of the rotating rod in terms of \(\lambda_0\), \(L\), and \(\omega\)?

![A top-down diagram showing a horizontal rod of length \(L\) extending from a pivot at \(x = 0\) to its free end at \(x = L\). At the left end at \(x = 0\), a small solid circle represents the fixed vertical pivot. A curved counterclockwise arrow near the pivot is labeled \(\omega\). The rod is shaded with a smooth horizontal grayscale gradient that starts very light gray at \(x = 0\) and becomes progressively darker gray toward the right end at \(x = L\). A horizontal dimension line beneath the rod spans its entire length and is labeled \(L\). Centered directly above the rod is the text \(\lambda(x) = \lambda_0\left(\dfrac{x}{L}\right)\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787460251-0t1iqF.jpg)

- **A.** \(\dfrac{1}{12} \lambda_0 L^3 \omega^2\)
- **B.** \(\dfrac{1}{8} \lambda_0 L^3 \omega^2\)
- **C.** \(\dfrac{1}{6} \lambda_0 L^3 \omega^2\)
- **D.** \(\dfrac{1}{4} \lambda_0 L^3 \omega^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120899/*