1. Classical Mechanics Lecture 9 Today's Examples: a) Conservative Forces b) Potential and Mechanical energy Today's Concepts : a) Energy and Friction b) Potential energy & force Mechanics Lecture 9, Slide 1 Midterm 2 will be held on March 13. Covers units 4-9 Unit 8 Homework Due Sunday March 8 11:30 PM. No extension! Unit 9 Homework Due Thursday March 12 11:30 PM. No extension!

2. Practice Exams Mechanics Lecture 8, Slide 2 Phys 1500 Exams - Spring 2013: http://www.physics.utah.edu/~springer/phys1500/exams/MidtermExam2.pdf http://www.physics.utah.edu/~springer/phys1500/exams/MidtermExam2.pdf - Solutions: http://www.physics.utah.edu/~springer/phys1500/exams/MidtermExam2Soln.pdf http://www.physics.utah.edu/~springer/phys1500/exams/MidtermExam2Soln.pdf Phys 2210 Exams - Practice : http://www.physics.utah.edu/~woolf/2210_Jui/rev2.pdfhttp://www.physics.utah.edu/~woolf/2210_Jui/rev2.pdf - Spring 2015: http://www.physics.utah.edu/~woolf/2210_Jui/ex2.pdfhttp://www.physics.utah.edu/~woolf/2210_Jui/ex2.pdf

3. Energy Conservation Problems in general Mechanics Lecture 8, Slide 3 For systems with only conservative forces acting E mechanical is a constant

4. Energy Conservation Problems in general Mechanics Lecture 8, Slide 4  conservation of mechanical energy can be used to “easily” solve problems. (for conservative forces) ALWAYS!  Identify important configurations i.e where potential is minimized  U=0.  Define coordinates: where is U=0?  Identify important configurations, i.e starting point where mass is motionless  K=0  Problem usually states the configurations of interest!

5. Pendulum Problem Mechanics Lecture 8, Slide 5 Using Work Formalism Using Conservation of Mechanical energy Conserve Energy from initial to final position

6. Pendulum Problem Mechanics Lecture 8, Slide 6 Don’t forget centripetal acceleration …required to maintain circular path. At bottom of path: Tension is …”what it has to be!”

7. Pendulum Problem Mechanics Lecture 8, Slide 7

8. Pendulum Problem Mechanics Lecture 8, Slide 8 Kinetic energy of mass prior to string hitting peg is conserved. Set h=0 to be at bottom equilibrium position

9. Pendulum Problem Mechanics Lecture 8, Slide 9 Radius for centripetal acceleration has been shortened to L/5 !

10. Loop the Loop Mechanics Lecture 8, Slide 10 To stay on loop, the normal force, N, must be greater than zero.

11. Mechanics Lecture 8, Slide 11  Mass must start higher than top of loop

12. Mechanics Lecture 8, Slide 12

13. Mechanics Lecture 8, Slide 13

14. Mechanics Lecture 8, Slide 14

15. Mechanics Lecture 8, Slide 15

16. Mechanics Lecture 8, Slide 16

17. Gravity and Springs…Oh my! Mechanics Lecture 8, Slide 17 Solve Quadratic Equation for x! Conservation of Energy Coordinate System Initial configuration Final configuration

18. Pendulum 2 Mechanics Lecture 8, Slide 18 Given speed and tension at certain point in pendulum trajectory can use conservation of mechanical energy to solve for L and m….

19. Pendulum 2 Mechanics Lecture 8, Slide 19

20. Pendulum 2 Mechanics Lecture 8, Slide 20

21. Pendulum 2 Mechanics Lecture 8, Slide 21

22. Main Points Mechanics Lecture 8, Slide 22

23. Main Points Mechanics Lecture 8, Slide 23

24. Mechanics Lecture 9, Slide 24 Macroscopic Work done by Friction Using Newton’s 2 nd Law

25. Mechanics Lecture 9, Slide 25 Macroscopic Work done by Friction Using Work-Kinetic Energy Theorem

26. Mechanics Lecture 9, Slide 26 Macroscopic Work done by Friction Using Work-Kinetic Energy Theorem

27. Mechanics Lecture 9, Slide 27 Macroscopic Work done by Friction Using Work-Kinetic Energy Theorem

28. Mechanics Lecture 9, Slide 28

29. Mechanics Lecture 9, Slide 29

30. f f Mechanics Lecture 9, Slide 30

31. “Heat” is just the kinetic energy of the atoms! Mechanics Lecture 9, Slide 31

32. Macroscopic Work: This is not a new idea – it’s the same “work” you are used to. Applied to big (i.e. macroscopic) objects rather than point particles (picky detail) We call it “macroscopic” to distinguish it from “microscopic”. Characterize Microscopic effect with coefficient of friction Mechanics Lecture 9, Slide 32 Characterize the microscopic effect with coefficient of friction 

33. Incline with Friction:Newton’s law and kinematics Mechanics Lecture 8, Slide 33 Using Newton’s 2 nd Law

34. Incline with Friction: Work-Kinetic Energy Mechanics Lecture 8, Slide 34 Using Work-Kinetic Energy Theorem Same result as using Newton’s Law

35. H N1N1 mg N2N2   mg must be negative m Work by Friction :W friction <0 Mechanics Lecture 9, Slide 35 What is the macroscopic work done on the block by friction during this process? A) mgH B) –mgH C)  k mgD D) 0

36. Checkpoint Mechanics Lecture 8, Slide 36 What is the total macroscopic work done on the block by all forces during this process? A) mgH B) –mgH C)  k mgD D) 0 Mechanics Lecture 9, Slide 36 D m H

37. Mechanics Lecture 8, Slide 37

38. Force from Potential Energy:1D Mechanics Lecture 8, Slide 38

39. Force from Potential Energy Mechanics Lecture 8, Slide 39

40. Force from Potential Energy in 3-d Mechanics Lecture 8, Slide 40 Gradient operator

41. Potential Energy vs. Force Mechanics Lecture 9, Slide 41

42. Potential Energy vs. Force Mechanics Lecture 9, Slide 42

43. Potential Energy vs. Force Mechanics Lecture 9, Slide 43

44. Potential Energy vs. Force Mechanics Lecture 9, Slide 44

45. Demo Potential Energy vs. Force Mechanics Lecture 9, Slide 45

46. Clicker Question A. B. C. D. Mechanics Lecture 8, Slide 46 Suppose the potential energy of some object U as a function of x looks like the plot shown below. Where is the force on the object zero? A) (a) B) (b) C) (c)D) (d) U(x)U(x) x (a)(b) (c)(d)

47. Clicker Question A. B. C. D. Mechanics Lecture 8, Slide 47 Suppose the potential energy of some object U as a function of x looks like the plot shown below. Where is the force on the object in the +x direction? A) To the left of (b) B) To the right of (b) C) Nowhere U(x)U(x) x (a)(b) (c)(d)

48. Clicker Question A. B. C. D. Mechanics Lecture 8, Slide 48 Suppose the potential energy of some object U as a function of x looks like the plot shown below. Where is the force on the object biggest in the –x direction? A) (a) B) (b) C) (c)D) (d) U(x)U(x) x (a)(b) (c)(d)

49. Equilibrium Mechanics Lecture 8, Slide 49

50. Equilibrium points Mechanics Lecture 8, Slide 50

51. Equilibrium points Mechanics Lecture 8, Slide 51

52. Equilibrium points Mechanics Lecture 8, Slide 52

53. Block on Incline Mechanics Lecture 8, Slide 53

54. Block on Incline Mechanics Lecture 8, Slide 54

55. Block on Incline Mechanics Lecture 8, Slide 55

56. Block on Incline Mechanics Lecture 8, Slide 56

57. Energy Conservation Problems in general Mechanics Lecture 8, Slide 57 For systems with only conservative forces acting E mechanical is a constant

58. Gravitational Potential Energy Mechanics Lecture 8, Slide 58

59. Gravitational Potential Problems Mechanics Lecture 8, Slide 59  conservation of mechanical energy can be used to “easily” solve problems.  Define coordinates: where is U=0? as  Add potential energy from each source.

60. Trip to the moon Mechanics Lecture 8, Slide 60

61. Trip to the moon Mechanics Lecture 8, Slide 61 Can ignore effect of moon for this problem at level of precision for SmartPhysics

62. Trip to the moon Mechanics Lecture 8, Slide 62 …or you can practice solving the quadratic equation with many terms!!!

63. Trip to the moon Mechanics Lecture 8, Slide 63 Can NOT ignore effect of moon for this problem since the rocket is AT the moon in the end !!!!

64. Trip to the moon Mechanics Lecture 8, Slide 64

65. Trip to the moon Mechanics Lecture 8, Slide 65

66. Trip to the moon Mechanics Lecture 8, Slide 66

67. Block on Incline 2 Mechanics Lecture 8, Slide 67

68. Block on Incline 2 Mechanics Lecture 8, Slide 68

69. Block on Incline 2 Mechanics Lecture 8, Slide 69

70. Block on Incline 2 Mechanics Lecture 8, Slide 70