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

1. angular velocity vector
3. Rotating Coordinate Frames
1. derivative of a vector: (du/dt)non-rotating = (du/dt)rotating + Ω×u
2. Vector triple product, BAC-CAB rule
4. Linear algebra
1. Matrix representation of vectors and tensors
2. Matrix multiplication
3. Determinant of a matrix
4. Eigenvalue problem: finding eigenvalues and eigenvectors
5. Calculus of variations: the Euler-Lagrange equation.
6. square of path length, ds², in cartesian, cylindrical, and spherical coordinates.
7. shortcut relation: v² = ds²/dt²

Physics Topics

1. Linearly accelerating reference frame, fictitous force -mA0
2. Rotating reference frame, fictitous forces
1. Centrifugal force -mΩ×(Ω×r)
2. Coriolis force 2mv×Ω
3. Transverse force
3. Relation between ω and v for pure rolling
4. Definitions of center of mass, and moment of inertia tensor
5. Center of mass and moment of inertia tensor of composite objects
6. Parallel axis theorem for the inertia tensor
7. Conservation of linear and angular momentum for a rigid body
8. General motion of a rigid body, 2 cases:
1. motion of the CM, and rotation about the CM
2. rotation about a fixed point
9. Fixed axis rotation: Γ = dL/dt and L = Iω
10. General rotations
1. moment of inertia tensor
2. principal axis theorem
3. finding principal axes, eigenvalue problem
4. Euler's equations: I will give you the equations, but you must know how to use them.
5. zero torque problems
6. objects with axial symmetry
11. Solving minimization / maximization problems with the Euler-Lagrange equation.
12. Lagrangian Mechanics
1. Determining the number of generalized coordinates
2. Determining the Lagrangian, L = T - U, for those generalized coordinates
3. Finding the motion with Lagrange's equation ∂L/∂qi - (d/dt)(∂L/∂qidot) = 0
4. Recognizing a conserved quantity in the Lagrangian

Problem Sets 1 to 7

Review any questions on these problems.