Moment of Inertia Explained: Worked Examples, Parallel Axis Theorem | AOE 3144 Lecture 20

How to calculate the moment of inertia matrix (inertia tensor) and center of mass for rigid bodies — worked examples using integrals, the flat plate (planar body) approximation, the parallel axis theorem, and the composite body approach. These are the practical calculation skills needed before you can do any rigid body dynamics: every spacecraft, robot arm, or vehicle analysis starts with knowing the mass properties. We work through the standard techniques: 🎯 Direct integration: setting up and evaluating the integrals for mass, center of mass, and inertia matrix entries 🎯 The planar (flat plate) approximation: when a body is thin in one dimension, the integrals simplify dramatically 🎯 Worked example: circular flat plate / disk 🎯 Worked example: center of mass of a solid cone (full integral evaluation) 🎯 Worked example: moment of inertia of a planar body 🎯 The 3D parallel axis theorem: computing inertia about any point from the center-of-mass values 🎯 The composite body approach: complicated shapes as collections of simple ones — the foundation of how CAD software computes mass properties 🎯 Rigid body origami: folded planar shapes Plus a fun detour into squishy rabbit simulations (deformable body dynamics) and why chocolate Easter rabbits are not rigid bodies. This is Lecture 20 of an undergraduate spacecraft dynamics course (AOE 3144, Virginia Tech). ▶️ Chapters 0:00 Introduction and context 0:40 Formulas for mass and center of mass vector 4:42 Planar rigid body (thin plate approximation) 10:03 Circular flat plate / disk example 12:29 Table of standard bodies 14:53 Center of mass example: solid cone 26:25 Moment of inertia example: planar body 30:48 3D parallel axis theorem 34:44 Complicated shapes as collections of simpler ones (CAD) 37:50 Squishy rabbit simulations 39:11 Folded planar shape: rigid body origami 📘 What you'll learn How to set up and evaluate mass-property integrals for rigid bodies When and how to use the flat plate (planar) approximation How to compute the center of mass of 3D bodies (worked cone example) How to compute moment of inertia matrix entries by integration How the 3D parallel axis theorem works and when to use it How composite-body methods work (and how CAD software computes mass properties) 🎓 Course Space Vehicle Dynamics — AOE 3144, Virginia Tech Full playlist:    • Spacecraft Attitude Dynamics & Control | S...   📄 Lecture notes (PDF) https://drive.google.com/drive/folder... ▶️ Next (Lecture 21) — Euler's Equations of Rigid Body Dynamics Derived    • Euler's Equations of Rigid Body Dynamics D...   ▶️ Previous (Lecture 19) — Mass Moments of a Rigid Body: Analog to Statistical Moments    • Mass, Center of Mass & Inertia as Statisti...   📖 Reference textbooks Schaub & Junkins — Analytical Mechanics of Space Systems, 4th edition (2018), Chapter 4 https://arc.aiaa.org/doi/book/10.2514... Kasdin & Paley — Engineering Dynamics: A Comprehensive Introduction https://press.princeton.edu/books/har... ⸻ 👨‍🏫 Instructor Dr. Shane Ross Professor of Aerospace Engineering, Virginia Tech (Caltech PhD, former NASA/JPL and Boeing) Research: https://ross.aoe.vt.edu Follow: https://x.com/RossDynamicsLab Subscribe: https://www.youtube.com/user/RossDyna... ⸻ 🔗 Related courses Lagrangian & Rigid Body Dynamics (ESM 5314) —    • Lagrangian Mechanics & 3D Rigid Body Dynam...   Hamiltonian Dynamics —    • Hamiltonian Mechanics: Full Graduate Cours...   Nonlinear Dynamics & Chaos —    • Nonlinear Dynamics & Chaos — Full Course F...   Recorded: Spring 2021 #MomentOfInertia #CenterOfMass #InertiaTensor #InertiaMatrix #ParallelAxisTheorem #RigidBodyDynamics #CompositeBody #MassProperties #WorkedExamples #EngineeringDynamics #SchaubJunkins #AOE3144 #VirginiaTech