# Rotational Dynamics

## Alexei Gilchrist

The physics and mathematics of rotating systems.

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This thread supports half of a first year advanced physics course at Macquarie University. We examine in some detail the physics of rotating systems and objects.

*Status:* Many of lectures are typeset but no exercises yet. Supporting material as mindmaps available on link below

**Mindmaps:** /map/rotational-dynamics/. Many of the maps are 10Mb or more so please be patient. Scroll to zoom, click and drag to pan or otherwise use the controls on the side.

- A simple review of the notation and some of the expressions for kinematics and dynamics along a line, for classical physics.
- The centre of mass of a system of particles is derived as a fictitious point that behaves as if the mass of the entire system where concentrated there. Tricks for finding it are discussed.
- The description of 2D rotational motion is developed in analogue to linear kinematics.
- The description of 2D rotational motion is further developed in analogue to linear dynamics by introducing the rotational equivalents of force, momentum and mass.
- In a rotating coordinate system, a free particle moves in a way that appears to be affected by three forces: the centrifugal force, the Coriolis force and the Euler force. These forces arise only from the rotation of the coordinate system.
- A single object or particle moving under the influence of a central force.
- Representing rotations by matrices in 2D and 3D
- Study of the motion of a spinning solid object with not torques or forces applied.
- Given an object spinning around some axis with no external torques applied, how stable the rotation is depends around which principal axes the rotation is close to.
- An extremely simplified model of a cat (just four masses!) is presented, following Putterman and Raz [Am. J. Phys.
**76**, 1040 (2008)]. This simple model is enough to show how a deformable body is able to rotate itself despite conserving angular momentum. - Equations of motion for a gyroscope are derived by considering the torques.
- The equations of motion for a heavy symmetrical top are derived in terms of the Euler angles and using a Lagrangian approach. The solutions are examined numerically.