Newton-Euler Equations for a Rigid Body - Center of Mass & Inertia Tensor Calculation Worked Example

Dive into a comprehensive lecture on Newton-Euler equations for rigid bodies, focusing on center of mass and inertia tensor calculations. Explore the formulas for total mass and center of mass in continuous mass distributions. Work through a detailed example of calculating the center of mass for a flat triangular plate using integrals. Learn about decomposing complex rigid bodies into simpler shapes for mass moment calculations. Understand the Newton-Euler approach to rigid body dynamics, including Euler's 1st and 2nd Laws. Examine the parallels between translational and rotational motion equations. Study the mass moments of rigid bodies, from total mass to moment of inertia. Conclude with an analysis of Euler's 2nd Law in the body-fixed frame and its representation as first-order ODEs for angular velocity components.

Rigid bodies made of a continuous mass distribution are considered. We write the formulas for the total mass and center of mass. .
flat triangular plate of uniform density and use integrals do determine the center of mass. We discuss the idea of decomposing our a complicated rigid body into simpler rigid bodies for purposes of calculating the mass moments (such as the location of the center of mass and the moment of inertia tensor). .
Composite shapes: complicated rigid body approximated by simpler ones to estimate center of mass and moment of inertia.
The Newton-Euler approach to rigid body dynamics is introduced, including Euler's 1st Law for translational motion (a.k.a., the "superparticle theorem") and Euler's 2nd Law for rotational motion (a.k.a., the rotational dynamics equation, Euler's rotational equation). .
Parallels between the kinematic and dynamic equations of the translational and rotational motion of a rigid body..
The mass moments of a rigid body are summarized:.
Euler's 2nd Law, the rotational dynamics equation, in the body-fixed frame, and as a set of 3 first-order ODEs for the components of angular velocity..