## PHY 3221 "Mechanics I" Class Diary for Fall 2018

 Date Notes Wednesday 8/22 Introductions, course policies. Syllabus. UF grading policy for assigning grade points. Friday 8/24 Reading material: sections 1.1-1.3. Difference between classical mechanics and quantum mechanics, relativity. Space and time. Cartesian coordinates. Vectors and vector operations: addition, subtraction, dot and cross products. Orthogonal vectors <=> dot product vanishes. (anti-)collinear vectors <=> vector product vanishes. Exercises: Problems 1.1, 1.4 (answer: 55.15 deg), 1.9. Monday 8/27 Reading material: sections 1.4-1.6. Differentiation of vectors. Position, velocity, acceleration (problem 1.19). Equation of the orbit (trajectory) (problem 1.11). Reference frames (problem 1.26). Newton's laws. Review of differential equations (problems 1.24, 1.25). Wednesday 8/29 Newton's second law in Cartesian coordinates. Motion under the influence of a constant force. Example 1.1: Block sliding down an incline. Third law and conservation of momentum. Friday 8/31 HW 1 is due in class (except for Problems 1.46 and 1.49 - they can be turned in together with the second HW on 09/10.) Reading material: section 1.7. Two-dimensional polar coordinates (problems 1.42, 1.43). Newton's second law in polar coordinates. Example 1.2: An oscillating skateboard. Problems 1.41, 1.45. Projectile motion (problem 1.35). END OF CHAPTER 1. Additional recommended problems for self-study: 1.3; 1.6; 1.7; 1.12; 1.13; 1.15; 1.18; 1.20; 1.36, 1.37, 1.38, 1.39, 1.47; 1.48. Monday 9/3 Labor Day - no class. Wednesday 9/5 QUIZ DAY Reading material: section 2.2. Examples of motion under different types of forces: a) constant; b) time-dependent; c) position-dependent; Problem 2.12, as an exercise do Problem 2.13 (harmonic oscillator). d) velocity-dependent. Problem 2.7, as an exercise do Problem 2.8. Friday 9/7 Reading material: read Section 2.1 on your own including the corresponding HW problems (2.2, 2.3 and 2.4). Reading material: Section 2.2. Velocity-dependent forces. Terminal velocity. Horizontal motion with linear drag. Vertical motion with linear drag. Monday 9/10 HW 2 is due in class Section 2.3: Equation for the trajectory in a linear medium, equation for the horizontal range, solving for the range using perturbation theory in 1/tau. Wednesday 9/12 Reading material: section 2.4. Quadratic air resistance: the general case (coupled equations); solutions for purely horizontal motion and for purely vertical motion. Friday 9/14 Reading material: section 2.5 and the end of section 2.7 (page 71). We will be skipping section 2.6 and instead solving the equations by using the method outlined in problem 2.54. Motion of a point charge in a constant magnetic field in the z direction. Equations of motion. Solving for the motion along the z direction. Equations of motion in the transverse plane. Decoupling the x and y equations. General solution of a second order linear differential equation with constant coefficients. Solution for the velocity and for the position. Trajectory. Cyclotron frequency. Additional recommended problems: 2.55. END OF CHAPTER 2. Additional recommended problems for self-study in preparation for the first exam: 2.5; 2.6; 2.9; 2.15; 2.23; 2.24; 2.26; 2.27; 2.29; 2.30; 2.31; 2.32; 2.36; 2.37; 2.38; 2.39; 2.40. Monday 9/17 HW 3 is due in class Reading material: sections 3.1-3.3. Momentum conservation (section 1.5 and problem 1.28). Example 3.1: inelastic collision of two bodies. Section 3.2: Rocket motion. Thrust. Wednesday 9/19 QUIZ DAY Section 3.3. Computing the center of mass of point mass configurations and extended bodies. Examples: Problems 3.15, 3.16 and 3.17. The CM of a solid cone (example 3.2). Friday 9/21 Reading material: sections 3.4-3.5. Review: cross product of two vectors. Area of a triangle, Problem 3.24 (see also problem 1.18). Kinematical variables for rotational motion. Angular velocity, angular acceleration, angular momentum, torque, moment of inertia. Conservation of angular momentum. Kepler's second law. Problem 3.27. Section 3.5: Angular momentum for systems of particles. END OF CHAPTER 3. Additional recommended problems from Chapter 3 for self-study in preparation for the exam: 3.1, 3.2, 3.5, 3.7, 3.9, 3.10, 3.12, 3.22, 3.24, 3.25, 3.26, 3.31, 3.33. Monday 9/24 HW 4 is due in class Review session for the first exam. Review of Quiz 2. Example 3.3. Lump of Putty colliding with a turntable. Parallel axis theorem. Example 3.4. A sliding and spinning dumbbell. Problem 3.35. A disk rolling down an inclined plane. Problem 3.32. Moment of inertia of a uniform sphere. Problem 2.55 (the basic principle of a velocity selector). Wednesday 9/26 First Exam: Chapters 1-3. Friday 9/28 Brief review of the exam. Begin Chapter 4. Reading material: sections 4.1-4.4. Kinetic energy and work. Work-KE theorem. Example 4.1. Three line integrals. Monday 10/1 Conservative forces: definition. Conservative forces and potential energy. Conservation of mechanical energy. Section 4.3. Force as the gradient of potential energy. Section 4.4. The curl condition for conservative forces. Wednesday 10/3 Reading material: sections 4.5-4.8. Main discussion topic: Conservative forces and potential energy. Constant forces. Problem 4.17. Central forces (Coulomb force: Example 4.5 and problem 4.22; Gravity force: Problem 4.21). Finding the potential energy from a given force field. Checking if the force is conservative, computing the potential energy. Cartesian versus spherical coordinates, Problem 4.43. Calculating the force from a potential energy (Problems 4.12, 4.13). Direction of the gradient, direction of the force. Problem 4.18. Friday 10/5 Sections 4.6 and 4.7: One-dimensional systems. Graphs of the potential energy. Equilibrium points. Stable and unstable equilibrium. Turning points. Example 4.3 (block sliding down a plane). Example 4.6 (dropping a stone from a tower). Monday 10/8 HW 5 is due in class Reading material: sections 4.9-4.10. Sec. 4.5: Time-dependent potential energy. Problem 4.26. Review of central forces (section 4.8). Problem 4.44. Energy of interaction between two particles. Problems 4.49, 4.50. Total energy of a two-particle system. Elastic collisions, Example 4.8 (see also problem 3.5). Problems 4.46, 4.47. The energy of a multiparticle system. Rigid bodies. Example 4.9: A cylinder rolling down an incline. END OF CHAPTER 4. Additional recommended problems from Chapter 4 for self-study in preparation for the exam: 4.5, 4.10, 4.11, 4.19, 4.20, 4.23, 4.31, 4.35, 4.42. Wednesday 10/10 QUIZ DAY Begin Chapter 5. Section 5.1. Hooke's law. Expansion of the potential energy around a stable equilibrium position. Example 5.1: Cube balanced on a cylinder. Friday 10/12 Reading material: section 5.2. Mathematics of oscillations. Solving linear second order differential equations with constant coefficients. General solution to the homogeneous equation. Characteristic equation. Particular solutions to the inhomogeneous equation. Simple harmonic oscillator. Two forms of the solution. Monday 10/15 HW 6 is due in class Reading material: section 5.2. The third form of the harmonic oscillator solution. The case of negative k (hyperbolic motion; for a review of hyperbolic functions, see Problems 2.33 and 2.34.) Determining the integration constants from the initial conditions (Problem 5.7). Finding the frequency of small oscillations (Problem 5.2, see also Problem 5.10). Example 5.2. A bottle in a bucket. Energy considerations. Wednesday 10/17 QUIZ DAY Reading material: sections 5.3-5.4. Section 5.3. Two-dimensional oscillators. Section 5.4. Damped oscillators. Undamped, underdamped, overdamped and critically damped oscillator. Dependence of the decay parameter on beta (see also Problem 5.20). Problems: 5.22, 5.25, 5.27. Friday 10/19 Reading material: sections 5.5-5.6. Section 5.5. Driven damped oscillations. General solution to the homogeneous equation and a particular solution to the inhomogeneous equation (see also Problem 5.34). Section 5.6. Resonance. Resonant frequency (see also problem 5.40). Width of the resonance, Q-factor. The phase at the resonance. Monday 10/22 HW 7 is due in class Review session for the second exam. Additional problems from Chapter 5: 5.1, 5.2, 5.3, 5.6, 5.8, 5.9, 5.10, 5.11, 5.16, 5.21, 5.22, 5.23, 5.25, 5.26, 5.28, 5.42, 5.43. Wednesday 10/24 Second Exam. Chapters 4 and 5 (excluding sections 5.7, 5.8 and 5.9). Friday 10/26 Go over some of the problems from the second exam. Solving problems with dimensional analysis. You can refer to these notes, in particular see examples 3.1, 3.2 and 3.3. Monday 10/29 Reading material: section 6.1. Formulation of mechanics problems as boundary value problems. Comparing different possible trajectories and choosing the optimal one. "Cost" function for a given trajectory. Analogy with vehicle routing. Wednesday 10/31 Reading material: section 6.2. Snell's law (Problem 6.4). begin the derivation of the Euler-Lagrange equation. Friday 11/2 Homecoming - no class. Monday 11/5 Reading material: sections 6.3-6.4 and section 7.1. Finish the derivation of the Euler-Lagrange equations. Applications of the Euler-Lagrange equations. A particle moving in one dimension under a conservative force (compare to example 7.1). Shortest path between two points in 2 dimensions (Example 6.1) and in 3 dimensions (Problem 6.27). The brachistochrone (Example 6.2). Isochronous curves. The brachistochrone as a cycloid (Problem 6.14). Fun videos: The brachistochrone version 1. The brachistochrone version 2. Wednesday 11/7 !!!EXTRA CREDIT EXAM!!! Friday 11/9 HW 8 is due in class A first integral when the function does not depend on y (the dependent variable), Problem 6.10 and momentum conservation in mechanics. A first integral when the function does not depend on x (the independent variable), Problem 6.20 and energy conservation in mechanics. Problem 6.6: infinitesimal line element in different coordinate systems. Problem 6.9. Monday 11/12 Veteran's day - no class. Wednesday 11/14 QUIZ DAY Reading material: section 7.1-7.6. Sec. 7.1: Lagrange's equations for unconstrained motion. Different formulations of classical mechanics: Newtonian, Hamilton's principle, Lagrange's equations. Sec. 7.3: Constrained systems in general. Generalized coordinates. Generalized forces and generalized momenta (see Sec. 7.6). Counting the relevant number of degrees of freedom. Friday 11/16 HW 9 can be handed in today in class Reading material: sections 7.5-7.7. The four steps to be followed in solving any problem by the Lagrangian method (page 258). Example 7.2. One particle in two dimensions using polar coordinates. Examples: Sec. 7.2: Plane pendulum. Monday 11/19 HW 9 is due in class More examples practicing the Lagrangian method. Example 7.5: A block sliding on a sliding wedge. Cyclic (ignorable) coordinates. Example 7.3: Atwood's machine. Wednesday 11/21 Pre-Thanksgiving holiday - no class. Friday 11/23 Thanksgiving holiday - no class. Monday 11/26 More examples practicing the Lagrangian method. The example of Fig. 7.4 and Problem 7.30: pendulum in an accelerating railroad car. Unnatural (rheonomous, forced) coordinates. Natural (scleronomous) coordinates. Example 7.6: Bead on a spinning wire hoop (see also Example 7.7 and problems 7.26 and 7.28). Wednesday 11/28 Reading material: sections 7.8 and 7.10 (skip 7.9). 7.8: More about conservation laws. Noether's theorem and cyclic coordinates. Conservation of total momentum. Conservation of energy. Hamiltonian. 7.10: Lagrange multipliers and constraint forces. Example 7.8: Atwood's machine using a Langange multiplier. Problems 7.51 and 7.52. Friday 11/30 HW 10 is due in class Review session for the third exam: Example 7.4: particle on a cylinder. Problem 7.34: car connected to a massive spring. Problem 7.36: plane pendulum on a spring. Problems 7.16, 7.18, 7.37, 7.48, 7.51, 7.52. Suggested additional problems: Chapter 6: Problems 6.6, 6.7, 6.8, 6.11, 6.12, 6.18, 6.19. Chapter 7: Problems 7.1, 7.3, 7.9, 7.11, 7.15, 7.17, 7.21, 7.28. Monday 12/3 Third Exam. Chapters 6 and 7 + dimensional analysis. Wednesday 12/5 Brief review of the exam. Overview of the material to be covered in the Spring semester.