Newton-Euler Equations for Rigid Bodies | Center of Mass & Inertia | ESM 5314 L10

Newton-Euler Equations for a Rigid Body | Center of Mass & Inertia Tensor | ESM 5314 Lecture 10 Treat rigid bodies as continuous mass distributions and derive the mass moments — total mass (zeroth), center of mass (first), and moment of inertia tensor (second). Apply Newton-Euler's laws (Euler's 1st and 2nd Laws) to write the full equations of motion for a 3D rigid body, then express Euler's 2nd Law in the body-fixed frame as three coupled first-order ODEs for the angular velocity components. ▶️ Chapters: 0:00 Intro: Continuous mass distributions and rigid bodies 1:25 Total mass as the zeroth moment of the mass distribution 3:52 Center of mass formula for a continuous body 7:18 Worked example: Triangular plate of uniform density 11:21 Setting up the double integral for the center of mass 19:40 Evaluating the integral: x-coordinate of center of mass 23:18 Composite shapes: Decomposing complex bodies into simpler parts 31:40 Composite body example: Estimating center of mass by parts 36:05 Moment of inertia for composite shapes (parallel axis theorem) 41:03 Newton-Euler approach to rigid body dynamics 43:03 Euler's 1st Law: Translational dynamics (superparticle theorem) 44:53 Euler's 2nd Law: Rotational dynamics equation 51:17 Parallels between translational and rotational equations of motion 56:40 Mass moments summarized: Zeroth (mass), first (c.o.m.), second (inertia) 1:01:08 Euler's 2nd Law in body-fixed frame: 3 first-order ODEs for ω 📘 What you'll learn: Write total mass and center of mass as integrals over a continuous body Compute the center of mass of a triangular plate using double integrals Use composite body decomposition to estimate mass moments for complex shapes Apply the parallel axis theorem for the moment of inertia Derive Euler's 1st and 2nd Laws for a rigid body Express the rotational dynamics in the body-fixed frame as 3 coupled ODEs 🎓 Course: Intermediate/Analytical Dynamics ESM 5314: Lagrangian & Rigid Body Dynamics (Virginia Tech, graduate-level) 🔗 Full course playlist (29 lectures):    • Lagrangian Mechanics & Rigid Body Dynamics   📄 Lecture notes (PDF): https://drive.google.com/drive/folder... 📖 References: Goldstein, Poole & Safko, Classical Mechanics (3rd ed., Ch. 5) N. Jeremy Kasdin & Derek A. Paley, Engineering Dynamics: A Comprehensive Introduction (textbook used in course) 👨‍🏫 Instructor: Dr. Shane Ross, Virginia Tech (Caltech PhD) Research: https://ross.aoe.vt.edu Follow on X: https://x.com/RossDynamicsLab Subscribe: https://www.youtube.com/c/RossDynamic... ▶️ Previous: Moment of Inertia Tensor & Principal Axis Frame (Lecture 9)    • Moment of Inertia Tensor & Principal Axis ...   ▶️ Next: Rotational Dynamics About an Arbitrary Reference Point | Planar Rigid Body Motion | Car Jump Example (Lecture 11)    • Rotational Dynamics About Arbitrary Refere...   🔗 Related courses: Lagrangian & Rigid Body Dynamics:    • Lagrangian Mechanics & Rigid Body Dynamics   Hamiltonian Dynamics:    • Hamiltonian Mechanics: Full Graduate Cours...   Local Bifurcation Theory:    • Local Bifurcation Theory: Center Manifolds...   Three-Body Problem:    • Three-Body Problem: Trajectory Design & Lo...   Nonlinear Dynamics & Chaos:    • Nonlinear Dynamics & Chaos — Full Course F...   Recorded: Fall 2020 #NewtonEuler #RigidBody #CenterOfMass #MomentOfInertia #InertiaTensor #EulersEquations #MassMoments #CompositeBody #AnalyticalDynamics #RotationalDynamics