---
title: "A thin, non-uniform rod of mass \\(M\\) and length \\(L\\) lies along the x-axis between \\(x = 0\\) and \\(x = L\\). The linear mass density of the rod is given by \\(\\lambda(x) = \\lambda_0 \\left(\\dfrac{x}{L}\\right)^2\\), where \\(\\lambda_0\\) is a constant. In terms of \\(M\\) and \\(L\\), what is the rotational inertia of the rod about an axis perpendicular to the rod and passing through the end \\(x = 0\\)?"
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url: "https://nerd-notes.com/ubq/117564/"
date_modified: "2026-08-04T07:49:06+00:00"
---

# A thin, non-uniform rod of mass \(M\) and length \(L\) lies along the x-axis between \(x = 0\) and \(x = L\). The linear mass density of the rod is given by \(\lambda(x) = \lambda_0 \left(\dfrac{x}{L}\right)^2\), where \(\lambda_0\) is a constant. In terms of \(M\) and \(L\), what is the rotational inertia of the rod about an axis perpendicular to the rod and passing through the end \(x = 0\)?

A thin, non-uniform rod of mass \(M\) and length \(L\) lies along the x-axis between \(x = 0\) and \(x = L\). The linear mass density of the rod is given by \(\lambda(x) = \lambda_0 \left(\dfrac{x}{L}\right)^2\), where \(\lambda_0\) is a constant. In terms of \(M\) and \(L\), what is the rotational inertia of the rod about an axis perpendicular to the rod and passing through the end \(x = 0\)?

![A thin horizontal rod lying along the x-axis with its left end at x = 0 and right end at x = L. A vertical dashed line representing the rotation axis passes through the left end at x = 0. The rod is shaded with a smooth gradient, starting light gray near x = 0 and becoming darker toward x = L to represent increasing mass density. A horizontal line below the rod indicates the axis labeled x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829746-9YJOVu.jpg)

- **A.** \(\dfrac{1}{5} M L^2\)
- **B.** \(\dfrac{3}{5} M L^2\)
- **C.** \(\dfrac{3}{4} M L^2\)
- **D.** \(\dfrac{5}{3} M L^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117564/*