A uniform disk of mass $m = 5 \\text{ kg}$ and radius $r = 0.2 \\text{ m}$ is spinning with an angular velocity $\\omega = 100 \\text{ rad/s}$ about its axis of symmetry. The axis is supported at one end and precesses horizontally due to gravity. The distance from the support to the center of the disk is $d = 0.4 \\text{ m}$. Determine the precession rate $\\Omega$.

A spacecraft is modeled as a uniform rectangular block with principal moments of inertia $I_x = 500 \text{ kg}\cdot\text{m}^2$, $I_y = 600 \text{ kg}\cdot\text{m}^2$, and $I_z = 200 \text{ kg}\cdot\text{m}^2$ about its mass center. Initially, it is rotating in space with an angular velocity $\mathbf{\omega} = (0.5 \hat{i} - 0.2 \hat{j} + 1.0 \hat{k}) \text{ rad/s}$. A small thruster malfunction briefly applies a constant moment $\mathbf{M} = (10 \hat{i} + 5 \hat{j} - 15 \hat{k}) \text{ N}\cdot\text{m}$ about the mass center. Determine the initial angular acceleration components of the spacecraft.

A spinning top is a classic demonstration of 3D rigid body kinetics. When stationary, a top placed on its tip immediately falls over due to gravity. However, when spinning rapidly, it remains upright and slowly "precesses" (wobbles) around the vertical axis. Explain this phenomenon using the concepts of angular momentum and gyroscopic torque.

In rotating machinery, such as car tires, turbines, or jet engine fans, proper balancing is critical. An unbalanced rotor causes severe vibrations, noise, and premature bearing failure. Differentiate between "static unbalance" and "dynamic unbalance" using 3D rigid body kinetics, and explain why a rotor that is statically balanced might still be dynamically unbalanced.