Intermediate Mechanics Physics 321

1. Intermediate MechanicsPhysics 321 Richard Sonnenfeld New Mexico Tech :00

2. Two particles with central forces :35

3. y y m m h x x O

4. Class #12 of 30 • Finish up problem 4 of exam • Status of course • Lagrange’s equations • Worked examples • Atwood’s machine :72

5. Physics Concepts • Classical Mechanics • Study of how things move • Newton’s laws • Conservation laws • Solutions in different reference frames (including rotating and accelerated reference frames) • Lagrangian formulation (and Hamiltonian form.) • Central force problems – orbital mechanics • Rigid body-motion • Oscillations • Chaos :04

6. Mathematical Methods • Vector Calculus • Differential equations of vector quantities • Partial differential equations • More tricks w/ cross product and dot product • Stokes Theorem • “Div, grad, curl and all that” • Matrices • Coordinate change / rotations • Diagonalization / eigenvalues / principal axes • Lagrangian formulation • Calculus of variations • “Functionals” • Lagrange multipliers for constraints • General Mathematical competence :06

7. Joseph LaGrangeGiuseppe Lodovico Lagrangia The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure.Preface to Mécanique Analytique. Joseph Lagrange [1736-1813]  (Variational Calculus, Lagrangian Mechanics, Theory of Diff. Eq’s.) Greatness recognized by Euler and D’Alembert 1788 – Wrote “Analytical Mechanics”. You’re taking his course. Rescued from the guillotine by Lavoisier – who was himself killed. Lagrange Said:“It took the mob only a moment to remove his head; a century will not suffice to reproduce it.” If I had not inherited a fortune I should probably not have cast my lot with mathematics. I do not know. [summarizing his life's work] :45 :08

8. Lagrange’s Equation Works best for conservative systems Eliminates the need to write down forces of constraint ASSUMES THAT q-I are consistent with forces of constraint!! Automates the generation of differential equations (physics for mathematicians) Is much more impressive to parents, employers, and members of the opposite sex :12

9. Lagrange’s Kitchen Mechanics “Cookbook” for Lagrangian Formalism • Write down T and U in anyconvenient coordinate system. It is better to pick “natural coords”, but isn’t necessary. 2) Write down constraint equations Reduce 3N or 5N degrees of freedom to smaller number. 3) Define the generalized coordinates One for each degree of freedom 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables :17

10. Degrees of Freedom for Multiparticle Systems • 5-N for multiple rigid bodies • 3-N for multiple particles :20

11. Atwood’s MachineReverend George Atwood – Trinity College, Cambridge / 1784 Two masses are hung from a frictionless, massless pulley and released. • Describe their acceleration and motion. • Imagine the pulley is a disk of radius R and moment of inertia I. Solve again. :45 :25

12. Atwood’s MachineLagrangian recipe m1 m2 :40

13. Atwood’s MachineLagrangian recipe m1 m2 :45

14. Atwood’s MachineSimulation :45 :50

15. The simplest Lagrangian problem 1) Write down T and U in anyconvenient coordinate system. 2) Write down constraint equations 3) Define the generalized coordinates 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables v0 m A ball is thrown at v0 from a tower of height s. Calculate the ball’s subsequent motion g :65

16. Class #12 Windup • Office hours today 4-6 • Wed 4-5:30 :72

17. Atwood’s Machine with massive pulleyLagrangian recipe R I m1 m2 :70

18. Class #13 of 30 • Lagrange’s equations • Worked examples • Pendulum with sliding support • You solve it • T7-17 Atwood’s machine with massive pulley • T7-4, 7-16 Masses on ramps :02

19. Pendulum with sliding support-I x1 m1 L z2 m2 x2 A pendulum with mass m2 and length L is suspended from a block of mass m1 resting on a frictionless plane. What is the frequency of small oscillations of the pendulum? What is the center of mass motion? :10

20. Pendulum with sliding support-II :15

21. Pendulum with sliding support-III :20

22. Pendulum with sliding support-IV :25

23. Pendulum with sliding support-V x1 m1 L z2 m2 x2 A pendulum with mass m2 and length L is suspended from a block of mass m1 resting on a frictionless plane. What is the frequency of small oscillations of the pendulum? :40

24. Pendulum with sliding support-VI x1 m1 L z2 m2 x2 A pendulum with mass m2 and length L is suspended from a block of mass m1 resting on a frictionless plane. What is the center of mass motion? :45

25. Atwood’s MachineLagrangian recipe R I m1 m2 m1 m2 :60

26. T7-17 Atwood’s Machine with massive pulleyLagrangian recipe R I m1 m2 1) Write down T and U in anyconvenient coordinate system. 2) Write down constraint equations 3) Define the generalized coordinates 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables :70

27. T7-4, T7-16 Masses on ramps y y m m h x x O A block of mass “m” starts from rest and slides down a ramp of height “h” and angle “theta”. Calculate acceleration “a” at top of ramp, time “t” to get to bottom of ramp and velocity “v” at bottom of ramp. Use the Lagrangian formalism. Do the same for a rolling disk (mass “m”, radius “r”) :65

28. Class #13 Windup • New variables can be introduced so long as add additional constraints • Generalized coordinates do not need to be of same type (e.g. angle / position). • Office hours today 3-5 • Wed 4-5:30 :72

29. Test #2 of 4 • Thurs. 10/17/02 – in class • Bring an index card 3”x5”. Use both sides. Write anything you want that will help. • You may bring your last index card as well. • Lagrangian method (most of exam) • Line integrals / curls • Generalized forces / Lagrange multipliers / constraint forces • Tuesday 10/15 – Real-time review / problem session :08

30. Class #14 of 30 • Extensions of the Lagrangian Method • Generalized force and momenta • Plausibility of the Lagrange equations • Lagrange Multipliers • Calculating forces of constraint • Worked example • You solve it (Taylor 7-50). :72

31. Atwood’s MachineLagrangian recipe m1 m2 :40

32. T7-16 Rolling mass on ramp y m q x :65

33. Generalized Force and Momentum Traditional Generalized Force Momentum Newton’s Law :72

34. Lagrange Multipliers Lagrangian method as practiced so far eliminates the need to write down forces of constraint BECAUSE it assumes that are consistent with forces of constraint!! Forces of constraint of form may be found by method of Lagrange multipliers :12

35. Lagrange Multipliers Lagrangian method as practiced so far eliminates the need to write down forces of constraint BECAUSE it assumes that are consistent with forces of constraint!! Forces of constraint of form may be found by method of Lagrange multipliers :12

36. Lagrange’s Dining Room Mechanics “Cookbook” for Lagrange Multipliers • Write down T and U in anyconvenient coordinate system. 2) Write down constraints of form 3) Define the generalized coordinates 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s and solve for lambda 8) The terms give the constraint forces :17

37. Atwood’s MachineLagrangian recipe m1 m2 :40

38. Atwood’s MachineLagrange Multiplier Recipe m1 m2 :40

39. Atwood’s MachineLagrange Multiplier Recipe m1 m2 :40

40. Atwood’s MachineLagrange Multiplier Recipe m1 m2 :40

41. Taylor 7-50 A mass m1 rests on a frictionless horizontal table. Attached to it is a string which runs horizontally to the edge of the table, where it passes over a frictionless small pulley and down to where it supports a mass m2. Use as coordinates x and y, the distances of m1 and m2 from the pulley. These satisfy the constraint equations f(x,y)=x+y=const. Write down the two modified Lagrange equations and solve them for x’’, y’’ and the Lagrange multiplier Lambda. Find the tension forces on the two masses. :12

42. Class #14 Windup • Exam next Thursday :72

43. Class #15 of 30 • Exam Review • Taylor 7.50, 7.8, 7.20, 7.29, ODE’s, 4.4, 4.7, 4.15, 4.19, 4.20 :72

44. Test #2 of 4 • Thurs. 10/17/02 – in class • Bring an index card 3”x5”. Use both sides. Write anything you want that will help. • You may bring your last index card as well. • Anything on last 3 homeworks • Lagrangian method (most of exam) • Line integrals / curls • Generalized forces / Lagrange multipliers / constraint forces • Tuesday 10/15 – Real-time review / problem session :08

45. Taylor 7-50 m2 m1 A mass m1 rests on a frictionless horizontal table. Attached to it is a string which runs horizontally to the edge of the table, where it passes over a frictionless small pulley and down to where it supports a mass m2. Use as coordinates x and y, the distances of m1 and m2 from the pulley. These satisfy the constraint equations f(x,y)=x+y=const. Write down the two modified Lagrange equations and solve them for x’’, y’’ and the Lagrange multiplier Lambda. Find the tension forces on the two masses. :12

46. Atwood’s MachineLagrange Multiplier Recipe -- CORRECTED m1 m2 :40

47. Class #14 Windup - CORRECTED • Exam next Thursday :72

48. ODE Summary Math Physics :55

49. Class #15 Windup • HW due Thursday • Tues 3-5, Wed 4-5:30 • Bring up to two index cards • Midterm grades will be posted on web • Only one (hard) HW problem • Happy 49ers! :72

50. Class #17 of 30 • Central Force Motion • Gravitational law • Properties of Inverse-square forces • Center of Mass motion • Angular momentum conservation • Lagrangian for Central forces • Reduced Mass and CM reference frame • Central Forces DVD • BRIEF Exam Review :72