---
title: "A non-uniform slender rod of length \\(L\\) lies on a horizontal, frictionless table. The rod is attached to a fixed vertical pivot at one end, located at \\(x = 0\\). The linear mass density \\(\\lambda\\) of the rod varies with distance \\(x\\) from the pivot according to the equation \\(\\lambda(x) = c x^2\\), where \\(c\\) is a positive constant."
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url: "https://nerd-notes.com/ubq/117748/"
date_modified: "2026-08-04T07:55:55+00:00"
---

# A non-uniform slender rod of length \(L\) lies on a horizontal, frictionless table. The rod is attached to a fixed vertical pivot at one end, located at \(x = 0\). The linear mass density \(\lambda\) of the rod varies with distance \(x\) from the pivot according to the equation \(\lambda(x) = c x^2\), where \(c\) is a positive constant.

A non-uniform slender rod of length \(L\) lies on a horizontal, frictionless table. The rod is attached to a fixed vertical pivot at one end, located at \(x = 0\). The linear mass density \(\lambda\) of the rod varies with distance \(x\) from the pivot according to the equation \(\lambda(x) = c x^2\), where \(c\) is a positive constant.

![A top-down view of a thin horizontal rod. The left end is attached to a pivot, indicated by a small dark circle. A coordinate axis is shown below the rod, with \(x = 0\) aligned with the pivot and \(x = L\) aligned with the right end. At the right end, a straight arrow labeled \(\vec{F}_0\) points upwards, perpendicular to the rod. The text \(\lambda(x) = c x^2\) is written above the rod to indicate its variable mass density. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830155-TVPWmF.jpg)

**Part a)** Using integral calculus, **derive** an expression for the rotational inertia \(I\) of the rod about the pivot. Express your answer in terms of \(c\), \(L\), and fundamental constants. *(2 points)*

**Part b)** **Derive** an expression for the total mass \(M\) of the rod in terms of \(c\), \(L\), and fundamental constants. *(2 points)*

**Part c)** **Show that** the rotational inertia \(I\) of the rod about the pivot can be expressed as \(I = \dfrac{3}{5}ML^2\). *(1 points)*

**Part d)** A constant force of magnitude \(F_0\) is applied to the free end of the rod at \(x = L\). The force is always directed perpendicular to the rod, as shown in Figure 1. **Derive** an expression for the angular acceleration \(\alpha\) of the rod. Express your answer in terms of \(M\), \(L\), \(F_0\), and fundamental constants. *(2 points)*

**Part e)** The rod starts from rest at \(t = 0\). **Derive** an expression for the angular displacement \(\Delta\theta\) of the rod as a function of time \(t\). Express your answer in terms of \(M\), \(L\), \(F_0\), \(t\), and fundamental constants. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117748/*