1. More VPython… Wheel Falling clay Barbell

2. Three fundamental principles (1)Momentum principle (2)Energy principle (3)?.......

3. Angular momentum principle Instantaneous version: where is the torque ( 力矩 ) around the point A. Greek letter “tau”

4. Torque 力矩 It is hard to close the door by pushing here… …and much easier here, further from the hinges.

5. Easier Harder

6. Easier Harder!

8. Your turn… A yo-yo is in the xy plane. You pull up on the string with a force of magnitude 0.6 N. What is the direction of the torque you exert on the (yo-yo + string) system, around the CM?

9. Your turn… A yo-yo is in the xy plane. You pull up on the string with a force of magnitude 0.6 N. What is the magnitude of the torque you exert?

10. The component of the force perpendicular to r is F sin φ. Then Calculating torque: method 1 Direction of torque from right-hand rule.

11. Calculating torque: method 2 “lever arm”

12. Calculating torque: method 3 Magnitude:

13. Example

16. Example…

17. Your turn…

22. What if our system is the whole universe? System No “surroundings”, so the angular momentum principle says:

23. The angular momentum of the universe is conserved. System

24. Angular momentum is transferred between system and surroundings. Never created, never destroyed. For any choice of system:

25. If the system has no interactions with the surroundings, there are no external torques on the system, so: System Surroundings

26. If the system has no interactions with the surroundings, there are no external torques on the system, so: System Surroundings

27. Conserved quantity Momentum Energy Angular momentum Symmetry of physical laws Translation Time Emmy Noether’s theorem (1915) Rotation

28. Conservation of angular momentum 角动量守恒

29. Any external torques acting on the skater are small, and we will ignore them. Angular momentum principle (around skater’s COM):

30. The skater spins faster when she pulls her arms in. Where does the extra energy come from?

31. Example: A bullet fired into a door A police officer fires a bullet into the center of a door. The bullet gets stuck inside the door. What is the angular speed of the door just after it is hit by the bullet? v = 400 m/s m = 0.01 kg M = 15 kg d = 1.0 m A