---
title: "A non-uniform solid cone has total mass \\(M\\), height \\(H\\), and base radius \\(R\\). The volumetric mass density of the cone varies linearly with the distance \\(z\\) from the apex according to \\(\\rho(z) = Cz\\), where \\(C\\) is a positive constant and \\(0 \\le z \\le H\\). Which of the following expressions represents the rotational inertia of the cone about its central axis of symmetry?"
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url: "https://nerd-notes.com/ubq/117629/"
date_modified: "2026-08-04T07:49:22+00:00"
---

# A non-uniform solid cone has total mass \(M\), height \(H\), and base radius \(R\). The volumetric mass density of the cone varies linearly with the distance \(z\) from the apex according to \(\rho(z) = Cz\), where \(C\) is a positive constant and \(0 \le z \le H\). Which of the following expressions represents the rotational inertia of the cone about its central axis of symmetry?

A non-uniform solid cone has total mass \(M\), height \(H\), and base radius \(R\). The volumetric mass density of the cone varies linearly with the distance \(z\) from the apex according to \(\rho(z) = Cz\), where \(C\) is a positive constant and \(0 \le z \le H\). Which of the following expressions represents the rotational inertia of the cone about its central axis of symmetry?

![A schematic diagram of a solid cone with its apex pointing upward at the origin. The vertical central axis is labeled z, with z = 0 at the top apex and z = H at the flat circular base at the bottom. A thin horizontal disk slice of thickness dz is highlighted at a distance z below the apex. The radius of this slice is labeled r(z). The base of the cone at the bottom has a radius labeled R. A curved rotational arrow encircles the central vertical z-axis at the top to indicate rotation about the central axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785829762-ZL9wVN.jpg)

- **A.** \(\dfrac{3}{10} M R^2\)
- **B.** \(\dfrac{1}{3} M R^2\)
- **C.** \(\dfrac{1}{2} M R^2\)
- **D.** \(\dfrac{2}{3} M R^2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117629/*