# PHY6200 W07

## Midterm Review

The exam will cover chapter 9, chapter 10, sections 10.1 to 10.8, chapter 6, and chapter 7, sections 7.1 to 7.5.

### Mathematical Techniques

- angular velocity vector
- addition of angular velocities
- Rotating Coordinate Frames
- derivative of a vector: (d
**u**/dt)_{non-rotating} = (d**u**/dt)_{rotating} + **Ω**×**u**
- Vector triple product, BAC-CAB rule

- Linear algebra
- Matrix representation of vectors and tensors
- Matrix multiplication
- Determinant of a matrix
- Eigenvalue problem: finding eigenvalues and eigenvectors

- Calculus of variations: the Euler-Lagrange equation.
- square of path length, ds², in cartesian, cylindrical, and spherical coordinates.
- shortcut relation: v² = ds²/dt²

### Physics Topics

- Linearly accelerating reference frame, fictitous force -m
**A**_{0}
- Rotating reference frame, fictitous forces
- Centrifugal force -m
**Ω**×(**Ω**×**r**)
- Coriolis force 2m
**v**×**Ω**
- Transverse force

- Relation between ω and v for pure rolling
- Definitions of center of mass, and moment of inertia tensor
- Center of mass and moment of inertia tensor of composite objects
- Parallel axis theorem for the inertia tensor
- Conservation of linear and angular momentum for a rigid body
- General motion of a rigid body, 2 cases:
- motion of the CM, and rotation about the CM
- rotation about a fixed point

- Fixed axis rotation:
**Γ** = d**L**/dt and **L** = **I****ω**
- General rotations
- moment of inertia tensor
- principal axis theorem
- finding principal axes, eigenvalue problem
- Euler's equations: I will give you the equations, but you must know how to use them.
- zero torque problems
- objects with axial symmetry

- Solving minimization / maximization problems with the Euler-Lagrange equation.
- Lagrangian Mechanics
- Determining the number of generalized coordinates
- Determining the Lagrangian, L = T - U, for those generalized coordinates
- Finding the motion with Lagrange's equation ∂L/∂q
_{i} - (d/dt)(∂L/∂q_{i}dot) = 0
- Recognizing a conserved quantity in the Lagrangian

### Problem Sets 1 to 7

Review any questions on these problems.

© 2007 Robert Harr