1. PHYS 172: Modern Mechanics Lecture 15 – Multiparticle Systems Read 9.1 – 9.2 Summer 2012

2. Quantizing two interacting atoms Any value of A is allowed And any E is possible. Classical harmonic oscillator:Quantum harmonic oscillator: U = (1/2)k s s 2 100 EE  200 2 EE  Energy levels: 1 00 2 N EN  34 1.0510 Js 2 h   =  ｴ 0 s k m  1 00 2 E  equidistant spacing ground state

3. Time to Throw Things BALL We need to understand Center of Mass BATON

4. The Center of Mass where This is a weighted average of the positions -- each position appears in proportion to its mass

5. 112233 123 cm mrmrmr r mmm  =  … … The Center of Mass

6. Motion of the Center of Mass 1) Take one time derivative: Same as: (Good!)

7. Motion of the Center of Mass 1) Take a second time derivative: This says that the motion of the center of mass looks just like what would happen if all forces were applied to the total mass, as a point particle located at the center of mass position!

8.  F net, ext =M total a cm = dP total dt This says that the motion of the center of mass looks just like what would happen if all forces were applied to the total mass, as a point particle located at the center of mass position. Motion of the Center of Mass Center of Mass

9. Same Tension. Which puck will move faster? The centers of mass experience the same acceleration! HOWEVER: Hand #2 has to pull the string farther: W 2 > W 1. Where does this energy go? Rotational energy. The bottom spool is spinning. Center of Mass Motion

10. Question for Discussion

12. Through what distance did the force act on the Point Particle system? A) 0.03 m B) 0.04 m C) 0.07 m D) 0.08 m E) 0.10 m Clicker Question Equal masses

13. Through what distance did the force act on the Real system? A) 0.03 m B) 0.04 m C) 0.07 m D) 0.08 m E) 0.10 m Equal masses Clicker Question

14. Which is the energy equation for the Point Particle system? Equal masses Clicker Question A) Δ K trans = F*(0.07 m) B) Δ K trans = F*(0.08 m) C) Δ K trans + Δ K vib + Δ U spring = F*(0.07 m) D) Δ K trans + Δ K vib + Δ U spring = F*(0.08 m)

15. Kinetic energy of a multiparticle system Translational, motion of center of mass Vibration Rotation about center of mass Motion of parts relative to center of mass

16. Translational kinetic energy Translational kinetic energy: (motion of center of mass) (nonrelativistic case) Clicker: A system is initially at rest and consists of a man with a bottle sitting on ice (ignore friction). The man then throws the bottle away as shown. http://www.punchstock.com/asset_images/95652058 The velocity of the center of mass v cm will be: A)Zero B)Directed to right C)Directed to left

17. Translational kinetic energy Translational kinetic energy: (motion of center of mass) (nonrelativistic case) Clicker: A system is initially at rest and consists of a man with a bottle sitting on ice (ignore friction). The man then throws a bottle away as shown. http://www.punchstock.com/asset_images/95652058 The translational kinetic energy of the system will be: A)Zero B)> 0 C)< 0

18. Vibrational kinetic energy - Net momentum = 0 - Energy is constant (sum of elastic energy and kinetic energy)

19. Rotational kinetic energy - Net momentum = 0 - Energy is constant Motion around of center of mass

20. Rotation and vibration CM Rotation and vibration and translation

21. Gravitational potential energy of a multiparticle system Gravitational energy near the Earth’s surface M y cm

22. Example: Rotation and translation Assume all mass is in the rim Energy principle: =0 EXAMPLE: