Introduction

01 Nov 2024

Motivation

Originally, I created this series as an excuse to evaluate some fun integrals and draw pretty diagrams. As I continued writing I really fell down a rabbit hole. Instead of climbing out, I kept digging until I came out the other side. Essentially, this series has become my collation, interpretation, and summarization of all the resources and information that I could find on moments of inertia as well as some of my own unique results.

In this series, I first give the relevant physics and mathematical background necessary to compute moments of inertia. Then I derive the inertia tensor for various interesting shapes. I separate them into curves ($\text{1D}$ objects), surfaces ($\text{2D}$ objects), and volumes ($\text{3D}$ objects). Most of the posts after the Background section are intended to be self-contained, so you can read them in any order. However, some of the more complicated shapes (such as triangles) will require results from previous posts. I will provide links when this is the case.

What is a Moment?

It’s a bit of a meme in physics that everyone thinks they understand moments until they are asked to explain them. Many resources say “a moment is a mathematical expression involving the product of a distance and physical quantity”, which at first is laughably vague, but the more you study physics the better of a description it becomes.

At the most basic level, we observe that objects with mass have an intrinsic resistance to change. Elegantly summarized as Newton’s First Law of Motion: An object at rest remains at rest, and an object in motion remains in motion. This is true for both translation and rotation. The moment of inertia is the mathematical quantity that measures an object’s resistance to change in rotational motion. An object’s mass is to translational motion as the object’s moment of inertia is to rotational motion. We can see this symmetry in the formulas for force and linear momentum compared to torque and angular momentum.

$$\begin{align} &F = m a &\qquad\scriptsize{\text{(force, mass, acceleration)}}& &\quad\qquad& \tau = I \alpha &&\quad\scriptsize{\text{(torque, rotational inertia, angular acceleration)}} \\[10pt] &p = m v &\quad\scriptsize{\text{(momentum, mass, velocity)}}& &\qquad\qquad& L = I \omega &&\quad\scriptsize{\text{(angular momentum, rotational inertia, angular vecloty)}} \end{align}$$

For more detail, the physics YouTuber Andrew Dotson has good videos on where the concept of a “moment” comes from and its interpretation (part 1 and part 2). I will also give my formulation of the moment of inertia in the next post.

Prerequisites

Vector Calculus

This series is going to require the computation of integrals. However, due to the setup, these integrals will be reasonably simple to compute. I am going to assume the reader is comfortable with standard integration techniques (e.g. the power rule).

The complexity in solving for the moment of inertia of an object will actually be parameterizing the object and setting up the corresponding integral, which requires vector calculus. Therefore, I highly recommend that the reader has studied vector calculus at one point (even if it was a while ago). The vector calculus is (usually) not difficult, so I will do my best to make the series accessible to those who have not.

If you have never studied vector calculus, the first chapter of David Griffiths’ Introduction to Electrodynamics has an excellent synopsis of this subject. Otherwise, any textbook or lecture videos will probably do.

Trigonometry

Setting up these integrals is going to require a good understanding of geometry and trigonometry. I am going to assume the reader is very comfortable with trigonometry and its identities (as to not belabor over every minute detail). If this is not the case, I have a series on trigonometry.

To prevent repetition in proofs, I will assert some trigonometric integrals. I leave it as an exercise to the reader to prove these results (or just google it).

$$\begin{align} &\int_{0}^{2\pi} \cos \theta \; d \theta \ \ = \int_{0}^{2\pi} \sin \theta \; d \theta = 0 &\qquad\qquad& \int_{0}^{\pi} \cos \theta \; d \theta = 0 && \int_{0}^{\pi} \sin \theta \; d \theta = 2 \\[10pt] &\int_{0}^{2\pi} \cos^2 \theta \; d \theta = \int_{0}^{2\pi} \sin^2 \theta \; d \theta = \pi &\qquad\qquad& \int_{0}^{\pi} \cos^2 \theta \; d \theta = \frac{\pi}{2} && \int_{0}^{\pi} \sin^2 \theta \; d \theta = \frac{\pi}{2} \\[10pt] &\int_{0}^{2\pi} \cos^3 \theta \; d \theta = \int_{0}^{2\pi} \sin^3 \theta \; d \theta = 0 &\qquad\qquad& \int_{0}^{\pi} \cos^3 \theta \; d \theta = 0 && \int_{0}^{\pi} \sin^3 \theta \; d \theta = \frac{4}{3} \end{align}$$

Notation and Conventions

I really try my best to have good notation. Especially when you have scalars, vectors, and matrices mixing it is easy to get lost. Having good notation and conventions really speeds up understanding.

Scalars

Any regular-font character is a scalar, e.g. $M$, $r$, or $\phi$. Latin characters ($a$, $b$, $c$, etc) will typically refer to lengths while Greek characters ($\alpha$, $\beta$, $\gamma$, etc) will typically refer to angles.

Vectors

Any bold-face character is a vector, e.g. $\b{r}$ or $\b{\omega}$. Any bold-face character with a hat is a unit vector, e.g. $\u{x}$.

One definition of a vector is a mathematical object with a magnitude and direction. We actually have good notation for both of these concepts. Given any vector $\b{v}$, it’s magnitude is written $\abs{\b{v}}$, but is often denoted just by the unbolded character $v$ (since it is a scalar). Its direction is denoted by its unit vector $\u{v}$, which points in the same direction as $\b{v}$ but has magnitude $1$. We can relate all of these objects as follows.

$$\u{v} = \frac{\b{v}}{\abs{\b{v}}} = \frac{\b{v}}{v} \qquad\qquad \b{v} = \abs{\b{v}} \; \u{v} = v \; \u{v}$$

Matrices

Any regular-font capitalized character between square brackets is a matrix, e.g. $\m{A}$.

I will be using the notation $I_{\b{\omega}}$ and $\m{I}$ to refer to the moment of inertia of an object, and it is important to understand the difference. $I_{\b{\omega}}$ is a scalar quantity refering to the moment of inertia with respect to some fixed axis of rotation $\b{\omega}$. $\m{I}$ refers to the inertia tensor, which is a matrix quantity that describes the moment of inertia with respect to any axis. In particular,

$$I_{\b{\omega}} = \u{\omega}^T \; \m{I} \; \u{\omega}$$

We will discuss this more in a future post.

Another thing to mention. Typically in linear algebra, $\m{I}$ is used to denote the identity matrix. As to not conflate variables, I will use $\m{\mathbb{I}}$. In particular, since we are working with $\text{3D}$ coordiantes, it will be the $3 {\times} 3$ identity matrix.

$$\m{\mathbb{I}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$