1. -Conservation of angular momentum -Relation between conservation laws & symmetries Lect 4

2. Rotation

3. d1d1 d2d2 The ants moved different distances: d 1 is less than d 2

4. Rotation Both ants moved the Same angle:  1 =  2 (=  )    Angle is a simpler quantity than distance for describing rotational motion

5. Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  velocity v change in d elapsed time = angular vel.  change in  elapsed time =

6. Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration a change in v elapsed time = angular accel.  change in  elapsed time = velocity vangular vel. 

7. Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration aangular accel.  velocity vangular vel.  Moment of inertia = mass x (moment-arm) 2 mass m resistance to change in the state of (linear) motion Moment of Inertia I (= mr 2 ) resistance to change in the state of angular motion M x moment arm

8. Moment of inertial M M x rr I  Mr 2 r = dist from axis of rotation I=small I=large (same M) easy to turn harder to turn

9. Moment of inertia

10. Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration aangular accel.  velocity vangular vel.  Force F (=ma) torque  (=I  ) torque = force x moment-arm Same force; bigger torque Same force; even bigger torque mass mmoment of inertia I

11. Teeter-Totter F F but Boy’s moment-arm is larger.. His weight produces a larger torque Forces are the same..

12. Torque = force x moment-arm “Line of action” “ Moment Arm” = d  = F x d F

13. Opening a door F difficult F easy d small d large

14. Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration aangular accel.  velocity vangular vel.  Force F (=ma) torque  (=I  ) mass mmoment of inertia I momentum p (=mv) angular mom. L  (=I  ) Angular momentum is conserved: L=const I  = I  L= p x moment-arm = I  x p

15. Conservation of angular momentum II II II

16. High Diver II II II

17. Conservation of angular momentum II II

19. Angular momentum is a vector Right -hand rule

20. Torque is also a vector wrist by pivot point Fingers in F direction F Thumb in  direction another right -hand rule F pivot point  is out of the screen example:

21. Conservation of angular momentum L has no vertical component No torques possible Around vertical axis  vertical component of L= const Girl spins: net vertical component of L still = 0

22. Turning bicycle L L These compensate

23. Spinning wheel F  wheel precesses away from viewer

24. Angular vs “linear” quantities Linear quantity symb. Angular quantity symb. distance d angle  acceleration aangular accel.  velocity vangular vel.  Force F (=ma) torque  (=I  ) mass mmoment of inertia I momentum p (=mv) kinetic energy ½ mv 2 angular mom. L  (=I  ) rotational k.e. ½ I   I  V KE tot = ½ mV 2 + ½ I  2

25. Hoop disk sphere race

26. I I I hoop disk sphere

27. Hoop disk sphere race I I I KE = ½ mv 2 + ½ I  2 hoop disk sphere

28. Hoop disk sphere race Every sphere beats every disk & every disk beats every hoop

29. Kepler’s 3 laws of planetary motion Orbits are elipses with Sun at a focus Equal areas in equal time Period 2  r 3 Johannes Kepler 1571-1630

30. Basis of Kepler’s laws Laws 1 & 3 are consequences of the nature of the gravitational force The 2 nd law is a consequence of conservation of angular momentum A 1 =r 1 v 1 T r1r1 v1v1 A 2 =r 2 v 2 T r2r2 v2v2 L 1 =Mr 1 v 1 L 2 =Mr 2 v 2 L 1 =L 2  v 1 r 1 =v 2 r 2

31. Symmetry and Conservation laws Lect 4a

32. Hiroshige 1797-1858 36 views of Fuji View 4View 14

33. Hokusai 1760-1849 24 views of Fuji View 18View 20

34. Temple of heaven (Beijing)

35. Snowflakes 60 0

36. Kaleidoscope rotate by 45 0 Start with a random pattern Use mirrors to repeat it over & over The attraction is all in the symmetry Include a reflection

37. Rotational symmetry No matter which way I turn a perfect sphere It looks identical  

38. Space translation symmetry Mid-west corn field

39. Time- translation symmetry in music repeat repeat again & again

40. Prior to Kepler, Galileo, etc God is perfect, therefore nature must be perfectly symmetric: Planetary orbits must be perfect circles Celestial objects must be perfect spheres

41. Kepler: planetary orbits are ellipses; not perfect circles

42. Galileo: There are mountains on the Moon; it is not a perfect sphere!

43. Critique of Newton’s Laws What is an inertial reference frame?: a frame where the law of inertia works. Circular Logic!! Law of Inertia (1 st Law): only works in inertial reference frames.

44. Newton’s 2 nd Law F = m a But what is F? whatever gives you the correct value for m a Is this a law of nature? or a definition of force? ?????

45. But Newton’s laws led us to discover Conservation Laws! Conservation of Momentum Conservation of Energy Conservation of Angular Momentum These are fundamental (At least we think so.)

46. Newton’s laws implicitly assume that they are valid for all times in the past, present & future Processes that we see occurring in these distant Galaxies actually happened billions of years ago Newton’s laws have time-translation symmetry

47. The Bible agrees that nature is time-translation symmetric T he thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun Ecclesiates 1.9

48. Newton believed that his laws apply equally well everywhere in the Universe Newton realized that the same laws that cause apples to fall from trees here on Earth, apply to planets billions of miles away from Earth. Newton’s laws have space-translation symmetry

49. rotational symmetry F F a a F = m aF = m a Same rule for all directions (no “preferred” directions in space.) Newton’s laws have rotation symmetry

50. Symmetry recovered Symmetry resides in the laws of nature, not necessarily in the solutions to these laws.

51. Emmy Noether 1882 - 1935 Conserved quantities: stay the same throughout a process Symmetry: something that stays the same throughout a process Conservation laws are consequences of symmetries

52. Symmetries  Conservation laws Symmetry Conservation law Rotation  Angular momentum Space translation  Momentum Time translation  Energy

53. Noether’s discovery: Conservation laws are a consequence of the simple and elegant properties of space and time! Content of Newton’s laws is in their symmetry properties