Classical Mechanics Lecture 19

1. Classical Mechanics Lecture 19 Today’s Concept: Angular Momentum

2. Schedule • One unit per lecture! • I will rely on you watching and understanding pre-lecture videos!!!! • Lectures will only contain summary, homework problems, clicker questions, Example exam problems…. Midterm 3 Wed Dec 11

3. Translational/Rotational Motion Comparison

4. Main Points

6. Angular Momentum of point particle Orbits? Angular Momentum Conserved

7. Conservation of Angular Momentum Similar to Conservation of Linear Momentum

8. http://hyperphysics.phy-astr.gsu.edu/hbase/conser.htm http://en.wikipedia.org/wiki/Emmy_Noether http://en.wikipedia.org/wiki/Noether%27s_theorem http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation Spatial Translation Symmetry Conservation of Linear Momentum Conservation of Angular Momentum Spatial Rotational Symmetry Conservation of Energy Temporal Translation Symmetry

9. Angular Momentum of Point Particle moving in a circle

10. Angular Momentum of System of Particles

11. Angular Momentum

12. Comparison to Linear Momentum

13. Kinetic energy

14. Right Hand Rule -Again

15. Angular Momentum of Solid Objects

16. Angular Momentum of Solid Objects

17. Total Angular Momentum of object that is rotating and revolving

18. Total Angular Momentum of object that is rotating and revolving

19. Precession of Equinoxes Because of Torque On Earth’s Equatorial Bulge In about 13,000 years Vega will be the “North Star” Your Zodiac sign is probably incorrect! http://hyperphysics.phy-astr.gsu.edu/hbase/solar/earthprecess.html

20. Angular Momentum We have shown that for a system of particles Momentum is conserved if What is the rotational version of this? The rotational analogue of forceFis torque Define the rotational analogue of momentum p to beAngular Momentum: For a symmetric solid object

21. Torque & Angular Momentum where and In the absence of external torques Total angular momentum is conserved

22. Example – Disk dropped on Disk =

23. What about Kinetic Energy? same x2

24. CheckPoint M M M 2M In both cases shown below a solid disk of mass M, radius R, and initial angular velocity wois dropped onto an initially stationary second disk having the same radius. In Case 2 the mass of the bottom disk is twice as big as in Case 1. If there are no external torques acting on either system, in which case is the final kinetic energy of the system biggest? A) Case 1 B) Case 2 C) Same Same initial L Case 1 Case 2

25. CheckPoint M M M 2M In which case is the final kinetic energy of the system biggest? A) Case 1 B) Case 2 C) Same Case 1 Case 2 A) L is equal but bigger mass in case 2

26. Point Particle moving in a Straight Line axis D

27. axis D Direction given by right hand rule(out of the page in this case)

28. Point Particle moving in a Straight Line

29. Playground Example After w Before M Top View R v m Disk Kid

30. CheckPoint The magnitude of the angular momentum of a freely rotating disk around its center is L. You toss a heavy block onto the disk along the direction shown. Friction acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. What is the magnitude of the final angular momentum of the disk-block system: A) >L B) =L C) <L Top View Instead of a block, imagine it’s a kid hopping onto a merry go round

31. CheckPoint What is the magnitude of the final angular momentum of the disk-block system: A) >L B) =L C) <L Top View B) Initial total momentum is L (L + 0*m*v) Because there is no external force, this momentum is conserved.

32. CheckPoint What is the magnitude of the final angular momentum of the disk-block system: A) >L B) =L C) <L Top View B) Initial total momentum is L (L + 0*m*v) Because there is no external force, this momentum is conserved.

33. CheckPoint The magnitude of the angular momentum of a freely rotating disk around its center is L. You toss a heavy block onto the disk along the direction shown. Friction acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. What is the magnitude of the final angular momentum of the disk-block system: A) >L B) =L C) <L Top View Instead of a block, imagine it’s a kid hopping onto a merry go round

34. CheckPoint What is the magnitude of the final angular momentum of the disk-block system: A) >L B) =L C) <L Top View C) The block adds momentum going in the opposite direction, so the total momentum is smaller than the momentum of the disk.

35. Clicker Question A student holding a heavy ball sits on the outer edge a merry go round which is initially rotating counterclockwise. Which way should she throw the ball so that she stops the rotation? • A) To her left • B) To her right • C) Radially outward C A B w top view: initial final

36. Clicker Question w2 w1 A student is riding on the outside edge of a merry-go-round rotating about a frictionless pivot. She holds a heavy ball at rest in her hand. If she releases the ball, the angular velocity of the merry-go-round will: A) Increase B) Decrease C) Stay the same

37. Pivot/Putty problem Angular Velocity Kinetic Energy Not conserved! Angular Momentum Conserved!

38. Pivot/Putty problem Solve!

40. m2 m1 w0

41. m2 m1 wo

42. Solve for wf m2 m1 w0 wf

44. M wf

45. Just like for linear momentum M wf for angular momentum

46. Person on Disk

47. Person on Disk