Module Vi: Introduction To 3d Rigid Body Motion

This section introduces the complexities of 3D rigid body motion, highlighting angular velocity as a vector, the moment of inertia as a tensor, and the unique dynamics involved.

Overview

This section introduces the complexities of 3D rigid body motion, highlighting the distinctions between 2D and 3D rotation, including angular velocity vectors and moment of inertia tensors.

Angular Velocity Vector And Its Rate Of Change

This section covers the concept of angular velocity as a vector in 3D rigid body motion and introduces angular acceleration as its rate of change.

Angular Velocity Vector Ω⃗

In 3D rigid body motion, angular velocity becomes a vector that describes rotation around an arbitrary axis, leading to more complex dynamics.

Rate Of Change: Angular Acceleration Α⃗

This section introduces the concept of angular acceleration in three-dimensional motion, illustrating the fundamental differences from two-dimensional motion.

Moment Of Inertia Tensor I

The moment of inertia tensor extends the concept of inertia to three dimensions, capturing the complexities of angular motion in 3D space.

Tensor Form

In 3D rigid body motion, angular velocity is represented as a vector, and moment of inertia takes the form of a tensor, leading to more complex dynamics than in 2D.

Off-Diagonal Terms

This section explores off-diagonal terms in the moment of inertia tensor, highlighting their significance in 3D rigid body motion.

Key Insight

In 3D rigid body motion, angular velocity, moment of inertia, and angular momentum are treated as vector and tensor properties, leading to complex motions not seen in 2D.

Example: Rod Executing Conical Motion

Setup

This section discusses the differences between 2D and 3D rigid body motion, focusing on angular velocity, moment of inertia, and the dynamics associated with 3D motion.

Observation

This section covers the complexities of 3D rigid body motion, emphasizing the distinction of angular velocity as a vector and the moment of inertia as a tensor.

Why 2d Formulation Fails

The section explains why 2D rigid body motion concepts are insufficient to describe the complexities of 3D motion, specifically regarding angular velocity and moment of inertia.

Conclusion & Summary

In 3D rigid body motion, angular velocity and moment of inertia are vector and tensor quantities respectively, representing the complexities of arbitrary rotation axes.