PHY5210 W15

Midterm 2 Review

The exam will cover chapter 7, 10, and 11.

Mathematical Techniques

  • angular velocity vector
  • addition of angular velocities
  • Linear algebra
    1. Matrix representation of vectors and tensors
    2. Matrix multiplication
    3. Determinant of a matrix
    4. Eigenvalue problem: finding eigenvalues and eigenvectors
  • Calculus of variations: the Euler-Lagrange equation.
  • Full and partial derivatives.
  • Determining v² from time derivative of (x(q), y(q), z(q)).
  • Square of path length, ds², in cartesian, cylindrical, and spherical coordinates.
  • Shortcut relation: v² = ds²/dt².
  • Vector triple product, BAC-CAB rule.
  • Kronecker delta and summation notation.
  • Taking discrete sums to integrals.
  • Volume, surface, and line integrals (for mass, CM, and inertia tensor).

    • Physics Topics

    1. Relation between ω and v for pure rolling.
    2. 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.
      5. Generalized momentum and generalized force.
    3. Definitions of center of mass, and inertia tensor (moments and products of inertia).
    4. Center of mass and moment of inertia tensor of composite objects.
    5. Parallel axis theorem for the inertia tensor.
    6. Conservation of linear and angular momentum for a rigid body.
    7. General motion of a rigid body, 2 cases:
      1. motion of the CM, and rotation about the CM
      2. rotation about a fixed point
    8. Fixed axis rotation: Γ = dL/dt and L = Iω
    9. 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
    10. Coupled oscillators
    11. Normal frequencies and normal modes
    12. Weak coupling limit
    13. Handling cases with many coupled oscillators
    14. Normal coordinates

    Problem Sets 5 to 7

    Review any questions on these problems.

    Below is the list of review topics for the first exam covering chapter 9, 6, and 7.1. This might be useful when studying for this exam.

    Mathematical Techniques

    1. Note that it is expected that you can handle vector algebra (addition, scalar multiplication, dot and cross products, componenets, unit vectors, and cartesian, cylindirical and spherical coordinates) and differentiation and integration involving vectors.
    2. angular velocity vector
    3. addition of angular velocities
    4. Rotating Coordinate Frames
      1. derivative of a vector: (du/dt)non-rotating = (du/dt)rotating + Ω × u
    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 × (r × Ω)
      2. Coriolis force 2mv × Ω
      3. Transverse force m r × dtΩ
    3. Relation between ω and v for pure rolling
    4. Solving minimization / maximization problems with the Euler-Lagrange equation.
    5. 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

    © 2015 Robert Harr