PHYS 172 Modern Mechanics Lecture 15 Multiparticle Systems

PHYS 172: Modern Mechanics Lecture 15 – Multiparticle Systems Summer 2012 Read 9. 1 – 9. 2

Quantizing two interacting atoms Classical harmonic oscillator: Quantum harmonic oscillator: U = (1/2)kss 2 w 0 = E 2 = 2 w 0 + E 0 E 1 = w 0 + E 0 = 12 w 0 ks m equidistant spacing Any value of A is allowed And any E is possible. = ground state h = 1. 05 ｴ 10 34 J s 2 p Energy levels: EN = N w 0 + 12 w 0

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

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

The Center of Mass rcm = + m 2 r 2 + m 3 r 3 + … mr 11 m 1 + m 2 + m 3 + …

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

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!

Motion of the Center of Mass 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. F net , ext =M tota la cm = d. Ptotal Center of Mass dt

Center of Mass Motion 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.

Question for Discussion

Question for Discussion

Clicker Question Through what distance did the force act on the Point Particl Equal masses A) 0. 03 m B) 0. 04 m C) 0. 07 m D) 0. 08 m E) 0. 10 m

Clicker Question Through what distance did the force act on the Real system? Equal masses A) 0. 03 m B) 0. 04 m C) 0. 07 m D) 0. 08 m E) 0. 10

Clicker Question Which is the energy equation for the Point Particle s Equal masses A) ΔKtrans = F*(0. 07 m) B) ΔKtrans = F*(0. 08 m) C) ΔKtrans + ΔKvib + ΔUspring = F*(0. 07 m) D) ΔKtrans + ΔKvib + ΔUspring = F*(0. 08 m)

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

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. The velocity of the center of mass vcm will be: A) Zero B) Directed to right C) Directed to left http: //www. punchstock. com/asset_images/95652058

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. The translational kinetic energy of the system will be: A) Zero B) > 0 C) < 0 http: //www. punchstock. com/asset_images/95652058

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

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

Rotation and vibration CM Rotation and vibration and translation

Gravitational potential energy of a multiparticle system Gravitational energy near the Earth’s surface M ycm

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