PHY5210 W08

Midterm 2 Review

The exam will cover chapter 7, and chapter 10, sections 10.1 to 10.4.

Mathematical Techniques

• Calculus of variations: the Euler-Lagrange equation.
• Full and partial derivatives.
• 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.

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 and 6. 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
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. Constraints and the method of Lagrange multipliers.

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.