Physics / advanced
Rotation in three dimensions
Connect inertia tensors, body frames, torque-free motion, gyroscopes, and rotational stability through concrete rigid-body problems.
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Module 01 advanced
Rotation in three dimensions
Connect inertia tensors, body frames, torque-free motion, gyroscopes, and rotational stability through concrete rigid-body problems.
- 01
Inertia tensors and principal axes
Calculate rotational response from mass distribution, identify principal directions, and distinguish angular momentum from angular velocity.
24 min - 02
Angular velocity and rotation matrices
Connect a body-to-space rotation matrix to the instantaneous angular velocity and differentiate vectors in rotating frames.
24 min - 03
Euler angles and coordinate singularities
Use a declared Z-X-Z orientation chart and identify when angle coordinates fail despite smooth physical motion.
24 min - 04
Euler equations for rigid bodies
Derive rotational equations in principal body axes and use torque, energy, and angular momentum as independent checks.
24 min - 05
Torque-free precession of a symmetric body
Solve the symmetric free rotor and distinguish body-component circulation from precession of the symmetry axis around fixed angular momentum.
26 min - 06
Why the intermediate axis is unstable
Linearize torque-free Euler equations and distinguish oscillatory stability from exponential growth without invoking a torque.
24 min - 07
Reduce the heavy symmetric top to one angle
Use cyclic momenta and an effective potential to separate spin, precession, and nutation for a top with a fixed frictionless pivot.
24 min - 08
Separate slow precession from nutation
Derive the fast-spin precession approximation, inspect its regime of validity, and distinguish a steady cone from a nodding axis.
24 min - 09
Account for energy in rolling contact
Derive acceleration and friction for ideal rolling, then distinguish zero contact work from the redistribution of translational and rotational energy.
24 min - 10
Solve an off-center collision with angular impulse
Combine linear impulse, angular impulse, and contact restitution while checking which angular momentum can be conserved during impact.
24 min