---
title: "A thin non-uniform rod of length \\(L\\) and total mass \\(M\\) lies on a frictionless horizontal surface and is pivoted at one end (\\(x = 0\\)) about a fixed vertical axle. The linear mass density of the rod is given by \\(\\lambda(x) = C x\\), where \\(C\\) is a constant. A small block of mass \\(m\\) slides across the surface with speed \\(v_0\\) perpendicular to the rod, strikes the free end (\\(x = L\\)), and sticks to it. Which of the following expressions represents the magnitude of the horizontal linear impulse exerted on the rod-block system by the pivot during the collision?"
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url: "https://nerd-notes.com/ubq/117677/"
date_modified: "2026-08-04T07:49:54+00:00"
---

# A thin non-uniform rod of length \(L\) and total mass \(M\) lies on a frictionless horizontal surface and is pivoted at one end (\(x = 0\)) about a fixed vertical axle. The linear mass density of the rod is given by \(\lambda(x) = C x\), where \(C\) is a constant. A small block of mass \(m\) slides across the surface with speed \(v_0\) perpendicular to the rod, strikes the free end (\(x = L\)), and sticks to it. Which of the following expressions represents the magnitude of the horizontal linear impulse exerted on the rod-block system by the pivot during the collision?

A thin non-uniform rod of length \(L\) and total mass \(M\) lies on a frictionless horizontal surface and is pivoted at one end (\(x = 0\)) about a fixed vertical axle. The linear mass density of the rod is given by \(\lambda(x) = C x\), where \(C\) is a constant. A small block of mass \(m\) slides across the surface with speed \(v_0\) perpendicular to the rod, strikes the free end (\(x = L\)), and sticks to it. Which of the following expressions represents the magnitude of the horizontal linear impulse exerted on the rod-block system by the pivot during the collision?

![A top-down view of a thin horizontal rod of length L pivoted at its left end by a fixed vertical pin labeled x = 0. The rod is shaded progressively darker toward its right free end labeled x = L to indicate increasing mass density. A small square block of mass m moves to the right toward the free end with a velocity vector arrow labeled v_0 perpendicular to the rod. Axis labels x = 0 and x = L appear near the ends. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829794-rVM9uF.jpg)

- **A.** \(\dfrac{M m v_0}{3(M + 2m)}\)
- **B.** \(\dfrac{M m v_0}{2(M + 3m)}\)
- **C.** \(\dfrac{4 M m v_0}{3(M + 2m)}\)
- **D.** \(\dfrac{2 M m v_0}{3(M + 2m)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117677/*